Combinatorial Constructions of Optimal Three ... - IEEE Xplore

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 1, JANUARY 2015

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Combinatorial Constructions of Optimal Three-Dimensional Optical Orthogonal Codes Lidong Wang and Yanxun Chang Abstract— In this paper, we study three-dimensional (u × v × w, k, λ) optical orthogonal codes (OOCs) with at most one optical pulse per wavelength/time plane (AM-OPP) restriction, which is denoted by AM-OPP 3-D (u × v × w, k, λ)-OOC. We build an equivalence relation between such an OOC and a certain combinatorial subject, called a w-cyclic group divisible packing of type (vw)u . By this link, the upper bound of the number of codewords is improved and some new combinatorial constructions are presented. As an application, the exact number of codewords of an optimal AM-OPP 3-D (u × v × w, 3, 1)-OOC is determined for any positive integers v, w, and u ≡ 2 (mod 6) with some possible exceptions. Index Terms— Three-dimensional optical orthogonal code, optimal, group divisible packing, w-cyclic.

I. I NTRODUCTION

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PTICAL code-division multiple access (OCDMA) systems attract much attention as they have several benefits such as asynchronous transmission, being flexible in network design, accommodation of burst traffic, etc. A main issue of the studies on OCDMA is to devise an optical code that can accommodate a large number of simultaneous users with a low error probability for a given code length. Optical orthogonal code (OOC) is one of such codes that have been designed for OCDMA. A one-dimensional optical orthogonal code (1-D OOC) is a set of binary sequences having good auto and crosscorrelations. The first systematical investigation on 1-D OOCs appeared in 1989 [11]. Since then there are many researches on 1-D OOCs (see [2], [4]–[10], [14], [15], [18], [19], [29]). 1-D OOCs spread optical pulses only in time domain. A drawback of 1-D OOCs is that the auto-correlation cannot be zero because there are multiple optical pulses within one period. The lower limit of auto-correlation in the 1-D codes is 1, and to achieve it, code length increases rapidly as the number of users increases. To overcome the limitation of 1-D OOCs, two-dimensional optical orthogonal codes (2-D OOCs) are proposed [28]. In 2-D OOCs, optical pulses are spread in Manuscript received May 6, 2014; revised October 17, 2014; accepted October 27, 2014. Date of publication November 20, 2014; date of current version December 22, 2014. L. Wang was supported by the NSFC under Grant 11401582. Y. Chang was supported by the NSFC under Grant 11271042 and Grant 11431003. L. Wang is with the Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China, and also with the Department of Basic Courses, Chinese People’s Armed Police Force Academy, Langfang 065000, China (e-mail: [email protected]). Y. Chang is with the Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China, and also with the Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing 100048, China (e-mail: [email protected]). Communicated by T. Helleseth, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2014.2368133

both time and wavelength domain. By employing another dimension (wavelength), 2-D OOC with single pulse per row is achieved and the performance of the 2-D OCDMA system is much improved compared to the 1-D OCDMA system. There is a considerable literature on 2-D OOC constructions. We refer the readers to the survey [23] for more details. To further improve the performance of 2-D OOCs, a third dimension (called the spatial dimension) for spreading is introduced by [17] using polarization techniques. With the spatial dimension, optical pulses can spread in space, wavelength and time. Optical codes which spreading in these three dimensions are called 3-D OOCs. We define a 3-D OOC formally as follows. Let u, v, w, k and λ be positive integers, where u, v and w are the number of spatial channels, wavelengths, and time slots, respectively. A three-dimensional (u × v × w, k, λ) optical orthogonal code (briefly 3-D (u×v×w, k, λ)-OOC), C, is a family of u × v × w (0, 1) arrays (called codewords) of Hamming weight k satisfying: for any two arrays A = [a(i, j, l)], B = [b(i, j, l)] ∈ C and any integer τ : v−1 w−1 u−1   

a(i, j, l)b(i, j, l + τ ) ≤ λ,

i=0 j =0 l=0

where either A = B or τ ≡ 0 (mod w), and the arithmetic l + τ is reduced modulo w. A wavelength/time plane is also called a spatial plane. There are several classes of 3-D OOCs of interests. The following two additional restrictions on the placement of pulses are often placed within the codewords of a 3-D OOC: • the one-pulse per plane (OPP) restriction: each codeword contains exactly one optical pulse per spatial plane. • the at-most one-pulse per plane (AM-OPP) restriction: each codeword contains at most one optical pulse per spatial plane. It is clear that 3-D OOCs with the OPP restriction must satisfy the AM-OPP restriction. In what follows, a 3-D OOC with the AM-OPP restriction is simply denoted by an AM-OPP 3-D OOC. The number of codewords of a 3-D OOC is called its size. For fixed values of u, v, w, k and λ, the largest possible size of an AM-OPP 3-D (u × v × w, k, λ)-OOC is denoted by (u × v × w, k, λ). Based on the Johnson bound [16] for constant weight codes, an upper bound of (u × v × w, k, λ) is given by Shum [25], as stated in the following lemma. Lemma 1.1 [25]: (u × v × w, k, λ) ≤ J (u × v × w, k, λ) holds for any positive integers u ≥ k > λ, where vw(u − λ) uv vw(u − 1) ···  · · · . J (u × v × w, k, λ) =   k k −1 k−λ

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An AM-OPP 3-D (u × v × w, k, λ)-OOC with (u × v × w, k, λ) codewords is said to be optimal. Furthermore, we say that an optimal AM-OPP 3-D (u × v × w, k, λ)-OOC is perfect if u(u − 1) · · · (u − λ) . (u × v × w, k, λ) = v λ+1 wλ k(k − 1) · · · (k − λ) We point out that an AM-OPP 3-D (u × 1 × w, k, λ)-OOC is called an AM-OPPW 2-D (u × w, k, λ)-OOC [22]. Wang and Yin [26] show that a perfect AM-OPPW 2-D OOC is equivalent to a semi-cyclic group divisible design. They obtained several infinite classes of perfect AM-OPPW 2-D (u × w, k, 1)-OOCs for k ∈ {2, 3, 4} by way of the corresponding designs. Several constructions for perfect AM-OPP 3-D (u × v × w, k, 1)-OOCs were provided in [25] by means of generalized Bhaskar Rao group divisible designs, and the existence spectrum for perfect AM-OPP 3-D (u × v × w, 3, 1)-OOCs was completely determined. We restate the results as follows. Theorem 1.2 [25]: A perfect AM-OPP 3-D (u×v×w, 3, 1)OOC exists if and only if (1) u = 3 : w is odd, or w is even and v is even; (2) u ≥ 4 : (u − 1)vw ≡ 0 (mod 2), u(u − 1)vw ≡ 0 (mod 3), and v ≡ 0 (mod 2) when u ≡ 2, 3 (mod 4) and w ≡ 2 (mod 4). Remark 1: The condition v ≡ 0 (mod 2) when u ≡ 2, 3 (mod 4) and w ≡ 2 (mod 4) is indeed equivalent to the condition (10) in Theorem 13 of [25]. In this paper, we focus on combinatorial constructions for general optimal AM-OPP 3-D (u × v × w, k, 1)-OOCs. As the main result of this paper, we determine the size of optimal AM-OPP 3-D (u ×v ×w, 3, 1)-OOCs for any positive integers v, w and u ≡ 2 (mod 6) with some possible exceptions. By the way, the known result on optimal AM-OPPW 2-D (u × w, 3, 1)-OOCs is also improved. The rest of this paper is organized as follows. In Section II, we build an equivalence relation between an AM-OPP 3-D (u × v × w, k, t − 1)-OOC and a w-cyclic group divisible t-packing of type (vw)u . By this link, we give an improvement on the upper bound of the size of optimal AM-OPP 3-D (u × v × w, 3, 1)-OOCs. In Section III, we introduce some auxiliary designs and constructive methods. In Section IV, we obtain some results of the auxiliary designs, which will be used in the sequel. In Section V, apply both recursive and direct constructions, we obtain the main result of this paper. Finally, we give a brief concluding remark in Section VI. II. C OMBINATORIAL D ESCRIPTION In this section, we define some terminologies and notations in design theory and state the link between AM-OPP 3-D (u × v × w, k, λ)-OOCs and the designs. Then we give a tighter upper bound of the size of AM-OPP 3-D (u × v× w, 3, 1)-OOCs. In what follows, we always assume that In = {0, 1, . . . , n − 1} and denote by Z n the additive group of integers modulo n. A. Combinatorial Tools Let K be a set of positive integers. A group divisible t-packing (t-GDP) is a triple (X, G, B) which satisfying the

following properties: (1) X is a set of v points; (2) G is a partition of X into subsets (called groups); (3) B is a collection of subsets of X (called blocks) such that each block has size from K and intersects any given group in at most one point, and each t-subset from t distinct groups is contained in at most one block of B. We use the notation GDP(t, K , v) to denote the t-GDP. If K = {k}, we will omit the braces to simply write k for K . If G contains u i groups of size gi , 1 ≤ i ≤ r , then we call g1u 1 g2u 2 · · · grur the group type (or type) of the GDP. If every admissible t-subset appears in exactly one block, the GDP is referred to as a group divisible t-design [21] and denoted by GDD(t, K , v). When t = 2, a GDP(t, K , v) is further briefly denoted by a K -GDP. A K -GDD of type 1v is also called a pairwise balance design and denoted by PBD(v, K ). A PBD(v, K ∪ {k ∗ }) is a PBD which has exactly one block of size k. We record some results concerning GDDs and PBDs, which will be used later. Lemma 2.1 [12]: Let g, t and w be positive integers. Then there exists a 3-GDD of type g t w1 if and only if the following conditions are all satisfied: (i ) t ≥ 3, or t = 2 and w = g; (ii ) w ≤ g(t −1); (iii ) g(t −1)+w ≡ 0 (mod 2); (i v) gt ≡ 0 (mod 2); and (v) g 2 t (t − 1) + 2gtw ≡ 0 (mod 6). Lemma 2.2 [1]: A PBD(u, {3, 4, 6}) exists for each u ≡ 0, 1 (mod 3). Lemma 2.3 [24]: There exists a 4-GDD of type 3n 61 for each n ≡ 1 (mod 4) and n ≥ 5. Next we introduce the concept of w-cyclic GDPs, which are very useful in the constructions of AM-OPP 3-D OOCs. Let w be a common divisor of gi , say gi = v i w, 1 ≤ i ≤ r . Suppose (X, G, B) is a GDP(t, K , v) of type g1u 1 g2u 2 · · · grur . If rthere is a permutation π on X which is the product of i=1 v i u i disjoint w-cycles, fixes every group, and leaves B invariant, then this design is said to be w-cyclic. Particularly, for a w-cyclic GDP(t, k, uvw) of type (vw)u , without loss of generality, we always identify X = Iu × Iv × Z w , G = {{i } × Iv × Z w : i ∈ Iu } and the permutation π : (i, j, x) → (i, j, x + 1) mod (−, −, w). Any permutation π partitions B into equivalence classes called the block orbits under π. A set of base blocks is an arbitrary set of representatives for these block orbits of B. A w-cyclic GDP(t, k, uvw) of type (vw)u is called optimal if it contains the largest possible number of base blocks. Example 2.4: There exists a 6-cyclic 3-GDP of type 63 with 5 base blocks. The required design is constructed on I3 × Z 6 with group set {{i } × Z 6 : i ∈ I3 }. We only list the base blocks as follows: {(0, 0), (1, 0), (2, 0)}, {(0, 0), (2, 1), (1, 2)}, {(0, 0), (1, 1), (2, 2)}, {(0, 0), (1, 3), (2, 5)}, {(0, 0), (2, 3), (1, 5)}.



We now establish the equivalence between an optimal AM-OPP 3-D (u × v × w, k, t − 1)-OOC and an optimal w-cyclic GDP(t, k, uvw) of type (vw)u .

WANG AND CHANG: COMBINATORIAL CONSTRUCTIONS OF OPTIMAL 3-D OOCs

Theorem 2.5: An optimal AM-OPP 3-D (u×v×w, k, t−1)OOC is equivalent to an optimal w-cyclic GDP(t, k, uvw) of type (vw)u . Proof: Suppose (X, G, B) is a w-cyclic GDP(t, k, uvw) of type (vw)u where X = Iu × Iv × Z w , G = {{i } × Iv × Z w : i ∈ Iu } and the permutation is (i, j, x) → (i, j , x + 1) mod (−, −, w). Let F be a set of base blocks of the design. For each base block B = {(i e , je , le ) : 1 ≤ e ≤ k} in F , we construct an u × v × w (0, 1)-matrix M B whose the (i, j, l) position is 1 if and only if (i, j, l) ∈ B. Since each block intersects any given group in at most one point and any two blocks intersect at most t − 1 points, the family {M B : B ∈ F } forms an AM-OPP 3-D (u × v × w, k, t − 1)-OOC. Conversely, given an AM-OPP 3-D (u × v × w, k, t − 1)-OOC, C, for each u × v × w-matrix M, we construct a k-subset B M of X where (i, j, l) ∈ B M if and only if the (i, j, l) position of M is 1. The AM-OPP restriction and the correlation property of the code guarantee that {B M : M ∈ C} forms a set of base blocks of a w-cyclic GDP(t, k, uvw) of type (vw)u . With the above observations, we can further know that the derived code from an optimal w-cyclic GDP(t, k, uvw) of  type (vw)u must be optimal, and vice versa. When the 3-D OOC is perfect, we have the following result from Theorem 2.5. Corollary 2.6: A perfect AM-OPP 3-D (u × v × w, k, t − 1)-OOC is equivalent to a w-cyclic GDD(t, k, uvw) of type (vw)u . Combining Theorem 1.2 and Corollary 2.6, the necessary and sufficient conditions for the existence of a w-cyclic 3-GDD of type (vw)u have been determined. Corollary 2.7: A w-cyclic 3-GDD of type (vw)u exists if and only if (1) u = 3 : w is odd, or w is even and v is even; (2) u ≥ 4 : (u − 1)vw ≡ 0 (mod 2), u(u − 1)vw ≡ 0 (mod 3), and v ≡ 0 (mod 2) when u ≡ 2, 3 (mod 4) and w ≡ 2 (mod 4). B. Upper Bounds For constructing an optimal AM-OPP 3-D (u × v × w, k, 1)-OOC, it suffices to construct the base blocks of an optimal w-cyclic k-GDP of type (vw)u . In the following, we shall improve the upper bound of (u × v × w, 3, 1). Lemma 2.8: (u × v × w, 3, 1) ≤ J (u × v × w, 3, 1) − 1 if one of the following conditions holds. (1) u = 3 : w is even and v is odd; (2) u ≥ 4 : u(u − 1)vw ≡ 0 (mod 3), u ≡ 2, 3 (mod 4), w ≡ 2 (mod 4) and v ≡ 1 (mod 2). Proof: Suppose there is an AM-OPP 3-D (u ×v ×w, 3, 1)OOC with J (u × v × w, 3, 1) codewords. Since the conditions of this lemma implies J (u×v×w, 3, 1) = u(u−1)v 2w/6, then this code must be perfect. However, Theorem 1.2 shows that no perfect AM-OPP 3-D (u ×v ×w, 3, 1)-OOCs exist for these two cases. Hence, (u × v × w, 3, 1) ≤ J (u × v × w, 3, 1) − 1 holds.  By Lemma 2.8, (3 × 1 × 6, 3, 1) ≤ 5. So, the 6-cyclic 3-GDP of type 63 given in Example 2.4 is optimal. Hence, (3 × 1 × 6, 3, 1) = 5.

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Lemma 2.9: Let u ≡ 2 (mod 3) and (u − 1)v 2 w ≡ 4 (mod 6). Then, (u × v × w, 3, 1) ≤ J (u × v × w, 3, 1) − 1. Proof: Suppose (X, G, B) is a w-cyclic 3-GDP of type (vw)u with J (u ×v ×w, 3, 1) = (u(u −1)v 2 w−2)/6 base blocks. Note that u(u − 1)v 2 w is congruent to 2 mod 6. It is readily checked that the pairs of X whose elements come from different groups can be partitioned into u(u − 1)v 2 w/2 pair orbits under the action of Z w . Since each base block contains three distinct pair orbits, the number of pair orbits which are not covered by any base block is u(u − 1)v 2 w/2 − 3(u(u − 1) v 2 w−2)/6 = 1. Let {a, b} be a representative of this pair orbit. Let ra denote the number of blocks containing a. Then, ra = ((u − 1)vw − 1)/2, which is not an integer. Thus (u × v × w, 3, 1) ≤ J (u × v × w, 3, 1) − 1.  Lemma 2.10: Let u ≡ 2 (mod 3), v ≡ 1, 2 (mod 3) and u(u −1)v 2 ≡ 0 (mod 4). Then, (u ×v ×2, 3, 1) ≤ J (u ×v× 2, 3, 1) − 1. Proof: Suppose (X, G, B) is a 2-cyclic 3-GDP of type (2v)u with J (u × v × 2, 3, 1) base blocks. As in Lemma 2.9, the pairs of X taken from different groups are partitioned into u(u − 1)v 2 pair orbits. It is readily checked that the number of pair orbits which are not covered by any base block is 2. Let {(u 1 , v 1 , 0), (u 2 , v 2 , w2 )} and {(u 1 , v 1 , 0), (u 3 , v 3 , w3 )} be the representatives of such two pair orbits. We claim that (u 2 , v 2 ) = (u 3 , v 3 ) and {w2 , w3 } = {0, 1}. Otherwise, r x = (2(u − 1)v − 1)/2, which is not an integer for each x ∈ {(u 2 , v 2 , w2 ), (u 3 , v 3 , w3 )}. On the other hand, the base blocks have two types: (i ) {(i 1 , j1 , 0), (i 2 , j2 , 0), (i 3 , j3 , 0)} and (ii ) {(i 1 , j1 , 0), Obviously, only each base (i 2 , j2 , 0), (i 3 , j3 , 1)}. block of type (ii ) contribute two pair orbits of type {(u 1 , v 1 , 0), (u 2 , v 2 , 1)}. Since the number u(u − 1)v 2 /2 of pair orbits of type {(u 1 , v 1 , 0), (u 2 , v 2 , 1)} is even, then the number of pair orbits of this type which are not covered by any base block must be even. It is contradict to that {w2 , w3 } = {0, 1}. Thus (u × v × 2, 3, 1) ≤ J (u × v× 2, 3, 1) − 1.  In the rest of this paper, we always assume that ⎧ J (u × v × w, 3, 1) − 1, ⎪ ⎪ ⎪ ⎪ if v ≡ 1 (mod 2), w ≡ 0 (mod 4), ⎪ ⎪ ⎪ ⎪ and u = 3; ⎪ ⎪ ⎪ ⎪ or v ≡ 1 (mod 2), w ≡ 2 (mod 4), ⎪ ⎪ ⎪ ⎪ ⎨ and u ≡ 3, 6, 7, 10 (mod 12); J ∗ (u ×v ×w, 3, 1) = or vw ≡ 6 (mod 12), w ≡ 2 (mod 4), ⎪ ⎪ and u ≡ 2, 11 (mod 12); ⎪ ⎪ ⎪ ⎪ or (u − 1)v 2 w ≡ 4 (mod 6), ⎪ ⎪ ⎪ ⎪ and u ≡ 2 (mod 3); ⎪ ⎪ ⎪ ⎪ or w = 2, u(u − 1)v 2 ≡ 8 (mod 12); ⎪ ⎪ ⎩ J (u × v × w, 3, 1), otherwise. Lemma 2.11: (u × v × w, 3, 1) ≤ J ∗ (u × v × w, 3, 1). Proof: Combining Lemmas 1.1 and 2.8-2.10, the conclusion then follows.  The size of an optimal 3-GDP of type v u , which also can be considered as a 1-cyclic 3-GDP of type v u , has been determined by Billington et al. [3], Yin [30]. Theorem 2.12 [3], [30]: (u × v × 1, 3, 1) = J ∗ (u × v × 1, 3, 1).

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III. AUXILIARY D ESIGNS AND F ILLING C ONSTRUCTIONS In this section, some recursive constructions for optimal 3-D OOCs will be given, called filling constructions. These constructions are the generalization of standard constructions in combinatorial design theory. Compared to 2-D OOCs, the structure of 3-D OOCs are more complex. Thus the following auxiliary designs to construct 3-D OOCs will be a little strange for the readers who first meet them. It is useful to notice that the concepts of holey group divisible designs and incomplete group divisible designs are two classes of special group divisible packings. A. Holey Group Divisible Designs Wei [27] introduced the concept of a holey group divisible design (HGDD) in 1993. A k-HGDD of type (n, g u ) is defined to be a quadruple (X, H, G, B) which satisfies the following properties: (1) X is an (ngu)-set (of points); (2) G is a partition of X into n subsets (called groups) of points gu each; (3) H is another partition of X into u subsets (called holes) of points gn each such that G ∩ H = g for any G ∈ G, H ∈ H; (4) B is a collection of k-subsets of X (called blocks) such that any pair of X from two distinct groups appears in a hole or exactly in one block but not both, and no other pairs of X occur in any block. Example 3.1: There exists a 3-HGDD of type (3, 13 ). Take X = I9 , G = {{0, 3, 6}, {1, 4, 7}, {2, 5, 8}} and H = {{0, 1, 2}, {3, 4, 5}, {6, 7, 8}}. Let B consists of the following 6 blocks {0, 4, 8}, {0, 5, 7}, {1, 3, 8}, {1, 5, 6}, {2, 4, 6}, {2, 3, 7}. It is readily checked that each pair from two distinct groups appears in a hole or exactly in one block.  Theorem 3.2 [27]: There exists a 3-HGDD of type (n, m t ) if and only if n, t ≥ 3, (t − 1)(n − 1)m ≡ 0 (mod 2), and t (t − 1)n(n − 1)m 2 ≡ 0 (mod 6). To illustrate why we introduce the concept of k-HGDDs. We first show how to use a k-HGDD to construct a k-GDP without the restriction of permutation groups. • Step 1: Start from a k-HGDD of type (n, m t ) (X, H, G, B). By the definition, each pair from two distinct groups appears in a hole or exactly in one block, and each pair contained in some group never occur in any block of B. • Step 2: Construct a k-GDP of type m n on the set H for each H ∈ H with group set {H ∩ G : G ∈ G}. Denote its block set by A H . • Step 3: Let A = ∪ H ∈H A H . It follows that each pair contained in one hole of H occurs in at most one block of A or exactly in one group, but not both. • Step 4: Let C = A ∪ B. Obviously, (X, G, C) is a k-GDP of type (mt)n . The basic idea of above construction is to fill in the holes of a k-HGDD with a k-GDP of type m n . Then this

construction can be termed as “Filling Construction”. Further more, if one hopes to obtain an optimal k-GDP of type (mt)n , it is necessary to input an optimal k-GDP of type m n . Note that the reverse is not true. For the purpose of constructing optimal m-cyclic k-GDP of type (mt)n , we need to modify above “Filling Construction” so that the initial HGDD admits some special permutations. Suppose (X, G, H, B) is a k-HGDD of type (n, (ht) pq ). If there is a permutation π on X which is the product of ntq disjoint hp-cycles, fixes every group, leaves B invariant, and partitions the pq elements of H into q equivalence classes, then this design is said to be (t, hp, q)-cyclic. Without loss of generality, we always identify X = In × It × Iq × Z hp , G = {{i } × It × Iq × Z hp : i ∈ In } and H = {In × It × {i } × {0+ j, p + j, . . ., (h −1) p + j } : (i, j ) ∈ Iq × Z p }. In this case, the permutation can be taken as (x, y, z, w) → (x, y, z, w+1) mod (−, −, −, hp). A (1, gu, 1)-cyclic k-HGDD of type (n, g u ) is also referred to as a semi-cyclic k-HGDD of type (n, g u ) and denoted by k-SCHGDD of type (n, g u ). Clearly, under the action of π, B can be partitioned into equivalence classes called the block orbits. A set of base blocks is a set of representatives for these block orbits of B. Example 3.3: There exists a (3, 3, 1)-cyclic 3-HGDD of type (6, 33 ). The desired design is constructed on I × Z 3 × Z 3 with group set {{i } × Z 3 × Z 3 : i ∈ I } and hole set {I × Z 3 × { j } : j ∈ Z 3 } where I = Z 5 ∪{∞}. Only initial base blocks are listed below. {(0, 0, 0), (1, 2, 1), (∞, 1, 2)}, {(0, 0, 0), (1, 0, 2), (3, 1, 1)}, {(0, 0, 0), (1, 1, 1), (∞, 2, 2)}, {(0, 0, 0), (1, 1, 2), (4, 1, 1)}, {(0, 0, 0), (2, 0, 2), (∞, 0, 1)}, {(0, 0, 0), (1, 0, 1), (3, 1, 2)}. All base blocks of the (3, 3, 1)-cyclic 3-HGDD of type (6, 33 ) are obtained by developing these base blocks by (+i, + j, −) mod (5, 3, −), where (i, j ) ∈ Z 5 × Z 3 and ∞ + 1 = ∞.  Although the notation “(t, hp, q)-cyclic” is complex, the following construction shows that this concept is powerful to construct hp-cyclic k-GDPs. Construction 3.4 (Filling Construction-I): Suppose that the following designs exist: (1) a (g, ht, 1)-cyclic k-HGDD of type (u, (gh)t ) with b base blocks; (2) an h-cyclic k-GDP of type (gh)u with f base blocks. Then, there exists an ht-cyclic k-GDP of type (ght)u with b + f base blocks. Proof: Start from a (g, ht, 1)-cyclic k-HGDD of type (u, (gh)t ) on Iu × Ig × Z ht with G = {{i } × Ig × Z ht : i ∈ Iu }, H = {Iu × Ig × {0 + j, t + j, . . . , (h − 1)t + j } : j ∈ Z t }. Denote the family of base blocks of this design by F . Let B be a set of base blocks of an h-cyclic k-GDP of type (gh)u on Iu × Ig × Z h with group set {{i } × Ig × Z h : i ∈ Iu }. For each B = {(ai , bi , ci ) : 1 ≤ i ≤ k} ∈ B, we take t B = {(ai , bi , tci ) : 1 ≤ i ≤ k}. Then F ∪ {t B : B ∈ B} forms a set of base blocks of an ht-cyclic k-GDP of type (ght)u on Iu × Ig × Z ht with group set {{i } × Ig × Z ht : i ∈ Iu }. 

WANG AND CHANG: COMBINATORIAL CONSTRUCTIONS OF OPTIMAL 3-D OOCs

Example 3.5: In this example, we construct an optimal 8-cyclic 3-GDP of type 83 . • Step 1: Construct a (1, 8, 1)-cyclic 3-HGDD of type (3, 24 ) on I3 × Z 8 with group set {{i } × Z 8 : i ∈ I3 } and hole set {I3 × { j, 4 + j } : 0 ≤ j ≤ 3}. We list the base blocks below. {(0, 0), (1, 1), (2, 2)}, {(0, 0), (1, 2), (2, 5)}, {(0, 0), (1, 3), (2, 1)}, {(0, 0), (1, 5), (2, 7)}, {(0, 0), (1, 6), (2, 3)}, {(0, 0), (1, 7), (2, 6)}. •

•

Step 2: Construct an optimal 2-cyclic 3-GDP of type 23 on I3 × Z 2 with group set {{i } × Z 2 : i ∈ I3 }. This design only contains one base blocks {(0, 0), (1, 1), (2, 1)}. Then apply Construction 3.4 with t = 4 to obtain a base block {(0, 0), (1, 4), (2, 4)}. Step 3: The 7 base blocks form a set of base blocks of a 8-cyclic 3-GDP of type 83 on I3 × Z 8 , where G = {{i } × Z 8 : i ∈ I3 }. By Lemma 2.11, (3 × 1 × 8, 3, 1) ≤ 7. Hence, it is optimal.

B. Incomplete Group Divisible Designs An incomplete group divisible design (IGDD) is a quadruple (X, Y, G, B) where X is a set (of points), Y is a subset (called a hole) of X, G is a partition of X into subsets (called groups), and B is a collection of subsets (called blocks) of X such that (1) each block intersects each group in at most one point; (2) no pair of distinct points of Y occurs in any block; (3) every pair of points from distinct groups with the exception of those in which both lie in Y occurs in exactly one block of B. Two types IGDD are needed here. A k-IGDD of type (g, h)u is an IGDD in which every block has size k and there are u groups of size g, each of which intersects the hole in h points. A k-IGDD of type g (n,t ) is another kind of IGDD in which every block has size k and there are n groups of size g, wherein t groups constitute the hole. Incomplete group divisible designs have been studied for many years. The existence of 3-IGDDs of type (g, h)u has been settled by Miao and Zhu [20]. Theorem 3.6 [20]: The necessary conditions for the existence of a 3-IGDD of type (g, h)u , u ≥ 3, g ≥ 2h, g(u − 1) ≡ 0 (mod 2), (g − h)(u − 1) ≡ 0 (mod 2), and u(u − 1)(g 2 − h 2 ) ≡ 0 (mod 6), are also sufficient. We introduce two kinds of k-IGDDs with special permutations in the following. (1) h-cyclic k-IGDD of type (gh, th)n : A k-IGDD of type (gh, th)n (X, Y, G, B) is said to be h-cyclic if there is a permutation on X which is the product of ng disjoint h-cycles, fixes every group, and leaves Y, B invariant. Without loss of generality, we can always identify X = In × Ig × Z h , G = {{i } × Ig × Z h : i ∈ In } and Y = In × It × Z h . In this case, the permutation can be taken as (i, j, x) → (i, j, x + 1) mod (−, −, h). B can be partitioned into some block orbits under the permutation. A set of base blocks is a set of representatives for these block orbits of B.

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Example 3.7: There exists a 2-cyclic 3-IGDD of type (4, 2)5 . This design is constructed on Z 5 × I2 × Z 2 with the group set {{i } × I2 × Z 2 : i ∈ Z 5 } and hole set Z 5 × {0} × Z 2 . Only initial base blocks are listed below. {(0, 0, 0), (1, 1, 0), (3, 1, 0)}, {(0, 0, 0), (2, 1, 0), (1, 1, 1)}, {(0, 0, 0), (4, 1, 0), (2, 1, 1)}, {(0, 0, 0), (3, 1, 1), (4, 1, 1)}. All base blocks of the 2-cyclic 3-IGDD of type (4, 2)5 are obtained by developing these base blocks by (+1, −, −) mod (5, −, −).  We exhibit our second “Filling Construction” via h-cyclic k-IGDDs of type (gh, th)u . Construction 3.8 (Filling Construction-II): Suppose that the following designs exist: (1) an h-cyclic k-IGDD of type (gh, th)u with f base blocks; (2) an h-cyclic k-GDP of type (th)u with b base blocks. Then, there exists an h-cyclic k-GDP of type (gh)u with b + f base blocks. Proof: Start from an h-cyclic k-IGDD of type (gh, th)u on Iu × Ig × Z h with group set G = {{i } × Ig × Z h : i ∈ Iu } and hole set Y = Iu × It × Z h . Denote the family of base blocks of this design by F . Construct an h-cyclic k-GDP of type (th)u on the hole Y with group set {{i } × It × Z h : i ∈ Iu }. Let B be a family of base blocks of this design. Then F ∪ B forms a set of base  blocks of the required h-cyclic k-GDP of type (ght)u . Example 3.9: In this example, we construct an optimal 2-cyclic 3-GDP of type 45 . • Step 1: Construct a 2-cyclic 3-IGDD of type (4, 2)5 on I5 × I2 × Z 2 with the group set {{i } × I2 × Z 2 : i ∈ I5 } and hole set I5 × {0} × Z 2 , which has 20 base blocks and exists by Example 3.7. • Step 2: Construct an optimal 2-cyclic 3-GDP of type 25 on I5 × {0} × Z 2 with group set {{i } × {0} × Z 2 : i ∈ I5 }. The 5 base blocks are listed below. {(0, 0, 0), (1, 0, 0), (2, 0, 0)}, {(0, 0, 0), (3, 0, 0), (4, 0, 0)}, {(1, 0, 0), (4, 0, 0), (2, 0, 1)}, {(0, 0, 0), (2, 0, 1), (4, 0, 1)}, {(0, 0, 0), (1, 0, 1), (3, 0, 1)}. •

Step 3: Apply Construction 3.8 to obtain a 2-cyclic 3-GDP of type 45 with 25 base blocks, which achieves the upper bound in Lemma 2.11 and is optimal. (2) h-cyclic k-IGDDs of type (gh)(n,t ): A k-IGDD of type (gh)(n,t ) (X, Y, G, B) is said to be h-cyclic if there is a permutation on X which is also a product of ng disjoint h-cycles, fixes every group, and leaves Y, B invariant. Without loss of generality, we can always identify X = In × Ig × Z h , G = {{i } × Ig × Z h : i ∈ In } and Y = It × Ig × Z h . In this case, the permutation can be taken as (i, j, x) → (i, j, x + 1) mod (−, −, h). Also, B can be partitioned into some block orbits under the permutation. A set of base blocks is a set of representatives for these block orbits of B.

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Example 3.10: There exists a 2-cyclic 3-IGDD of type 2(7,3). This design is constructed on I7 × Z 2 with the group set {{i } × Z 2 : i ∈ I7 } and hole set I3 × Z 2 . The base blocks are listed as follows: {(0, 0), (3, 0), (4, 0)}, {(0, 0), (5, 0), (6, 0)}, {(0, 0), (3, 1), (5, 1)}, {(0, 0), (4, 1), (6, 1)}, {(1, 0), (3, 0), (6, 0)}, {(1, 0), (4, 0), (3, 1)},

Construction 4.1: If there exists a k-SCHGDD of type (u, (gh)t ), then there exists a (g, h, t)-cyclic k-HGDD of type (u, (gh)t ). • Step 1: Start from a k-SCHGDD of type (u, (gh)t ) (X, G, H, B), where X = Iu × Z ght , G = {{i } × Z ght : i ∈ Iu } and H = {Iu ×{i, i +t, . . . , i +(gh −1)t} : i ∈ It }. Let F be a set of base blocks of this design. • Step 2: For each B ∈ F , let Bl = {(a, x +l) : (a, x) ∈ B}, 0 ≤ l ≤ t − 1. Denote

{(1, 0), (5, 0), (6, 1)}, {(1, 0), (4, 1), (5, 1)}, {(2, 0), (3, 0), (5, 1)}, {(2, 0), (4, 0), (6, 1)}, {(2, 0), (5, 0), (4, 1)}, {(2, 0), (6, 0), (3, 1)}.

A = ∪ B∈F {Bl : 0 ≤ l ≤ t − 1}. 

We exhibit our third “Filling Construction” via h-cyclic k-IGDDs of type (ht)(u,e) . Construction 3.11 (Filling Construction-III): Suppose that the following designs exist: (1) an h-cyclic k-IGDD of type (ht)(u,e) with b base blocks; (2) an h-cyclic k-GDP of type (ht)e with f base blocks. Then, there exists an h-cyclic k-GDP of type (ht)u with b + f base blocks. Proof: Start from an h-cyclic k-IGDD of type (ht)(u,e) on Iu × It × Z h with group set G = {{i } × It × Z h : i ∈ Iu } and hole set Y = Ie × It × Z h . Denote the family of base blocks of this design by F . Construct an h-cyclic k-GDP of type (ht)e on the hole Y with group set {{i } × It × Z h : i ∈ Ie }. Let B be a family of base blocks of this design. Then F ∪ B forms a set of base  blocks of the required h-cyclic k-GDP of type (ht)u . Example 3.12: In this example, we construct an optimal 2-cyclic 3-GDP of type 27 . • Step 1: Construct a 2-cyclic 3-IGDD of type 2(7,3) on I7 × Z 2 with the group set {{i } × Z 2 : i ∈ I7 } and hole set I3 × Z 2 , which has 12 base blocks and exists by Example 3.10. • Step 2: Construct an optimal 2-cyclic 3-GDP of type 23 on I3 × Z 2 with group set {{i } × Z 2 : i ∈ I3 }, which has 1 base block and exists by Example 3.5. • Step 3: Apply Construction 3.11 to obtain a 2-cyclic 3-GDP of type 27 with 13 base blocks, which achieves the upper bound in Lemma 2.11 and is optimal. IV. C ONSTRUCTIONS ON AUXILIARY D ESIGNS For applying Constructions 3.4, 3.8 and 3.11, we need some (t, hp, q)-cyclic k-HGDDs and the two kinds of h-cyclic k-IGDDs. In this section, we investigate their constructive methods, and apply these constructions to obtain some existence results. The main idea is the use of product constructions (weighting constructions and inflation constructions). These constructions are all basic constructions in combinatorial design theory. It is a routine matter of checking their correctness. Here we only focus on how these constructions work. A. (t, hp, q)-Cyclic k-HGDDs We first give some constructions on the existence of (t, hp, q)-cyclic k-HGDDs.

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Step 3: For each (a, x) ∈ X, define f ((a, x)) = (a, x 1 , x 2 ), where x = x 1 + x 2 t, 0 ≤ x 1 ≤ t − 1 and 0 ≤ x 2 ≤ gh − 1. Let C = ∪ A∈A f (A).

Then C forms a family of base blocks of a (1, gh, t)-cyclic k-HGDD of type (u, (gh)t ) on f (X), where the group set is { f (G) : G ∈ G} and the hole set is { f (H ) : H ∈ H}. • Step 4: Repeating this process, we can similarly obtain a (g, h, t)-cyclic k-HGDD of type (u, (gh)t ) from above (1, gh, t)-cyclic k-HGDD of type (u, (gh)t ). Construction 4.1 indicates that k-SCHGDDs are useful in the construction of (g, h, t)-cyclic k-HGDDs. In the following, we quote the known results on 3-SCHGDDs for later use. Theorem 4.2 [13]: There exists a 3-SCHGDD of type (n, m t ) if and only if n, t ≥ 3, (t − 1)(n − 1)m ≡ 0 (mod 2), and (t − 1)n(n − 1)m ≡ 0 (mod 6) except when (1) n ≡ 3, 7 (mod 12), m ≡ 1 (mod 2) and t ≡ 2 (mod 4); (2) n = 3, m ≡ 1 (mod 2) and t ≡ 0 (mod 2); (3) n = t = 3, m ≡ 0 (mod 2); (4) (n, m, t) ∈ {(5, 1, 4), (6, 1, 3)}; and possibly when (1) n = 8, m ≡ 2, 10 (mod 12) and t ≡ 7, 10 (mod 12); (2) t = 8, either m ≡ 1 (mod 2) and n ≡ 1, 3 (mod 6) and n ≥ 7, or m ≡ 3 (mod 6) and n ≡ 5 (mod 6); (3) n ≡ 1, 9 (mod 12), m ≡ 1 (mod 2) and t ≡ 2 (mod 4); (4) n ≡ 5 (mod 6) and n ≥ 11, either m ≡ 3 (mod 6) and t ≡ 2 (mod 4), or m ≡ 1, 5 (mod 6) and t ≡ 10 (mod 12). Inflation construction has been frequently used in the constructions of HGDDs. We generalize this method to construct (t, hp, q)-cyclic k-HGDDs. Construction 4.3 (Inflation-I): If there exist a (t, hp, q)cyclic k-HGDD of type (u, (ht) pq ) and a w-cyclic l-GDD of type (vw)k , then there exists a (vt, hpw, q)-cyclic l-HGDD of type (u, (vwht) pq ). • Step 1: Start from a (t, hp, q)-cyclic k-HGDD of type (u, (ht) pq ) (X, G, H, B) on X = Iu × It × Iq × Z hp , where G = {{i } × It × Iq × : i ∈ Iu }, and H = {Iu × It × Z hp {i } × {0 + j, p + j, . . . , (h − 1) p + j } : (i, j ) ∈ Iq × Z p }. Let F be a set of base blocks of this design. • Step 2: For each block B ∈ F , construct a w-cyclic l-GDD of type (vw)k on B × Iv × Z w with group set {{x} × Iv × Z w : x ∈ B}. Denote the family of base blocks of this design by A B .

WANG AND CHANG: COMBINATORIAL CONSTRUCTIONS OF OPTIMAL 3-D OOCs

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Step 3: Let A = ∪ B∈F A B . For each A ∈ A and each (a, b, c, d, e, f ) ∈ A, define a mapping τ : (a, b, c, d, e, f ) → (a, b + te, c, d + hp f ).

Define τ (A) = {τ (x) : x ∈ A}. Let C = ∪ A∈A τ (A). Then C forms a set of base blocks of the required (vt, hpw, q)-cyclic l-HGDD of type (u, (vwht) pq ) over Iu × Ivt × Iq × Z hpw , where G  = {{i } × Ivt × Iq × Z hpw : i ∈ Iu }, and H = {Iu × Ivt × {i } × {0 + j, p + j, . . . , (hw − 1) p + j } : (i, j ) ∈ Iq × Z p }. Now we are ready to apply these constructions to obtain some infinite classes of (t, hp, q)-cyclic 3-HGDDs, which are essential to our work. Lemma 4.4: Let t ≥ 3. Then there exists a (g, h, t)-cyclic 3-HGDD of type (u, (gh)t ) if u, g, h satisfy one of following conditions. (1) (u, g, h) = (3, 2, 4); (2) (u, g, h) = (5, 3, 4); (3) (u, g) = (5, 3) and h ≡ 2 (mod 6); (4) (g, h) = (2, 2) and u ≡ 0, 1 (mod 3). Proof: When (u, g, t) = (3, 2, 3) and h ∈ {2, 4}, take a 3-HGDD of type (3, 13 ) from Theorem 3.2, which is also a (1, 1, 3)-cyclic 3-HGDD of type (3, 13 ). Inflate it by an h-cyclic 3-GDD of type (2h)3 in Corollary 2.7. Then we obtain a (2, h, 3)-cyclic 3-HGDD of type (3, (2h)3 ) by Construction 4.3. For other values, by Theorem 4.2, there exists a 3-SCHGDD of type (u, (gh)t ). Hence, by Construction 4.1, there exists a (g, h, t)-cyclic 3-HGDD of type (u, (gh)t ).  Lemma 4.5: There exists a (g, 3, 1)-cyclic 3-HGDD of type (6, g 3 ) for any positive integer g ≥ 2. Proof: When g = 3, the required design can be found in Example 3.3. When g ≡ 0 (mod 3) and g ≥ 6, start from a (3, 3, 1)-cyclic 3-HGDD of type (6, 33 ). Inflate it by a 1-cyclic 3-GDD of type (g/3)3 from Corollary 2.7. By Construction 4.3, we obtain the required design. When (g, 3) = 1, Z 3g is isomorphic to Z g × Z 3 under the mapping τ : x → (x (mod g), x (mod 3)). By Theorem 4.2, there exists a 3-SCHGDD of type (6, g 3 ). Let F be the set of base blocks of this design. Hence we obtain a 3-HGDD of type (6, g 3 ) on I6 × Z g × Z 3 with group set {{i } × Z g × Z 3 : i ∈ I6 } and hole set {I6 × Z g × { j } : j ∈ Z 3 }. For each B ∈ F , let Bl = {(a, x + l (mod g), x (mod 3)) : (a, x) ∈ B} where 0 ≤ l ≤ g − 1. Let A = ∪ B∈F {Bl : 0 ≤ l ≤ g − 1}. It is readily checked that A forms a set of base blocks of the resulting HGDD. All blocks are obtained by developing these base blocks by (−, −, +1) mod (−, −, 3).  Lemma 4.6: Let g be a positive integer and t ≥ 3. Then there exists a (g, ht, 1)-cyclic 3-HGDD of type (u, (gh)t ) if u, h, t satisfy one of following conditions. (1) u = 3, h = 2 and t ≡ 0 (mod 2); (2) u ≡ 11 (mod 12), h = 6 and t ≡ 1 (mod 2); (3) u ≡ 11 (mod 12), h = 2 and t ≡ 1 (mod 3); (4) u ≡ 0, 4 (mod 6), h = 1 and t ≡ 1 (mod 2), except for (u, g, t) = (6, 1, 3);

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(5) u ≡ 3, 6, 7, 10 (mod 12), h = 2 and t ≡ 1 (mod 2), and t = 3 when u = 3; (6) u ≡ 5 (mod 6), h = 1 and t ≡ 1 (mod 3) when u = 5, or t ≡ 1, 4, 7 (mod 12) when u > 5, except for (u, t) = (5, 4). Proof: In Condition (4), when (u, t) = (6, 3) and g ≥ 2, the required design can be found in Lemma 4.5. For other values, by Theorem 4.2, there exists a 3-SCHGDD of type (u, h t ), which is also a (1, ht, 1)-cyclic 3-HGDD of type (u, h t ). Inflate it by a 1-cyclic 3-GDD of type g 3 from Corollary 2.7. Hence, by Construction 4.3, we obtain a (g, ht, 1)-cyclic 3-HGDD of type (u, (gh)t ).  Lemma 4.7: Let u ≡ 11 (mod 12) and g ≡ 3 (mod 6). Then there exists a (g, 2t, 1)-cyclic 3-HGDD of type (u, (2g)t ) for any t ≡ 1, 5 (mod 6) and t ≥ 5. Proof: By Theorem 4.2, there exists a 3-SCHGDD of type (u, 6t ). Since (3, 2t) = 1, similar to the proof of Lemma 4.5, we can obtain a (3, 2t, 1)-cyclic 3-HGDD of type (u, 6t ). Take a 1-cyclic 3-GDD of type (g/3)3 from Corollary 2.7. By Construction 4.3, we obtain a (g, 2t, 1)cyclic 3-HGDD of type (u, (2g)t ).  B. h-Cyclic k-IGDDs of Type (gh, th)u We give three constructive methods on h-cyclic k-IGDDs of type (gh, th)u . Construction 4.8 (Inflation-II): Suppose there exist an h-cyclic k-IGDD of type (gh, th)u and a w-cyclic l-GDD of type (vw)k . Then there exists an hw-cyclic l-IGDD of type (ghvw, thvw)u . • Step 1: Start from an h-cyclic k-IGDD of type (gh, th)u (X, Y, G, B) on Iu × Ig × Z h with G = {{i } × Ig × Z h : i ∈ Iu } and Y = Iu × It × Z h . Let F be a set of base blocks of this design. • Step 2: For each block B ∈ F , construct a w-cyclic l-GDD of type (vw)k on B × Iv × Z w with group set {{x} × Iv × Z w : x ∈ B}. Denote the family of base blocks of this design by A B . • Step 3: Let A = ∪ B∈F A B . For each A ∈ A and each (a, b, c, d, e) ∈ A, define a mapping τ : (a, b, c, d, e) → (a, vb + d, c + he). Define τ (A) = {τ (x) : x ∈ A}. Let C = ∪ A∈A τ (A). Then C forms a set of base blocks of the required hw-cyclic l-IGDD of type (ghvw, thvw)u over Iu × Igv × Z hw with group set {{i } × Igv × Z hw : i ∈ Iu } and hole set Iu × It v × Z hw . Construction 4.9 (Weighting Construction-I): Suppose there exist a K -GDD of type m u and an h-cyclic k-IGDD of type (gh, th)n for each n ∈ K . Then there exists an h-cyclic k-IGDD of type (mgh, mth)u . • Step 1: Start from a K -GDD of type m u (X, G, B). • Step 2: For each block B ∈ B, construct an h-cyclic k-IGDD of type (gh, th)|B| on B × Ig × Z h , with group set {{x} × Ig × Z h : x ∈ B}, hole set B × It × Z h . Denote the family of base blocks of this design by A B . • Step 3: Let A = ∪ B∈B A B . Let X  = X × Ig × Z h , G  = {G × Ig × Z h : G ∈ G} and Y  = X × It × Z h . Then A forms a set of base blocks of the required h-cyclic k-IGDD on X  with group set G  and the hole Y  .

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Construction 4.10: Suppose there exist a (g, h, t)-cyclic k-HGDD of type (u, (gh)t ) and an h-cyclic k-IGDD of type (gh + eh, eh)u . Then there exists an h-cyclic k-IGDD of type (gth + eh, eh)u . •

•

•

Step 1: Start from a (g, h, t)-cyclic k-HGDD of type (u, (gh)t ) (X, G, H, B), where X = Iu × Ig × It × Z h , G = {{i } × Ig × It × Z h : i ∈ Iu } and H = {Iu × Ig × {i } × Z h : i ∈ It }. Let F be a set of base blocks. Step 2: Take another point set Y = Iu × Ie × Z h . For each H = Iu × Ig × {i } × Z h ∈ H, construct an h-cyclic k-IGDD of type (gh + eh, eh)u on H ∪ Y with group set {{ j } × (Ig × {i } ∪ Ie ) × Z h : j ∈ Iu } and the hole Y . Denote the family of base blocks of this design by B H . Step 3: Let A = (∪ H ∈H B H )∪F . Then A forms a family of base blocks of an h-cyclic k-IGDD of type (gth + eh, eh)u on X ∪Y with group set {{i }×(Ig × It ∪ Ie )× Z h : i ∈ Iu } and the hole Y .

Applying Constructions 4.8-4.10, we give some results on h-cyclic 3-IGDDs of type (gh, th)t for later use. Lemma 4.11: There exists a 4-cyclic 3-IGDD of type (4v, 4)3 for any odd integer v ≥ 3. Proof: For v ∈ {3, 5}, by Lemma 6.2, there exists a 4-cyclic 3-IGDD of type (4v, 4)3 . For v ≥ 7, apply Construction 4.10 with a (2, 4, (v − 1)/2)-cyclic 3-HGDD of type (3, 8(v−1)/2) from Lemma 4.4 and a 4-cyclic 3-IGDD of type (12, 4)3 to obtain a 4-cyclic 3-IGDD of  type (4v, 4)3 . Lemma 4.12: Let v ≡ 1, 2 (mod 3) and v ≥ 2. There exists a w-cyclic 3-IGDD of type (vw, w)5 for any w ≡ 2 (mod 3). Proof Case 1: w ≡ 5 (mod 6). Start from a 3-IGDD of type (v, 1)5 in Theorem 3.6, which is 1-cyclic. Inflate it by a w-cyclic 3-GDD of type w3 from Corollary 2.7. By Construction 4.8, we obtain a w-cyclic 3-IGDD of type (vw, w)5 . Case 2: w ≡ 2 (mod 6). For v ∈ {2, 4, 7}, the required designs can be found in Example 3.7 and Lemma 6.7. For v = 8, inflate a w-cyclic 3-IGDD of type (2w, w)5 by a 1-cyclic 3-GDD of type 43 from Corollary 2.7 to obtain a w-cyclic 3-IGDD of type (8w, 4w)5 . Filling in the hole with a w-cyclic 3-IGDD of type (4w, w)5 , we obtain a w-cyclic 3-IGDD of type (8w, w)5 . When v ≥ 10 and v ≡ 1 (mod 3), by Lemma 4.4, there exists a (3, w, (v − 1)/3)-cyclic 3-HGDD of type (5, (3w)(v−1)/3). Then apply Construction 4.10 with a w-cyclic 3-IGDD of type (4w, w)5 to obtain a w-cyclic 3-IGDD of type (vw, w)5 . When v = 5 or v ≥ 11 and v ≡ 2 (mod 3), we first construct a w-cyclic 3-IGDD of type (vw, 2w)5 . The required design for v = 5 comes from Lemma 6.6. For v ≥ 11, apply Construction 4.10 with a (3, w, (v − 2)/3)cyclic 3-HGDD of type (5, (3w)(v−2)/3) and a w-cyclic 3-IGDD of type (5w, 2w)5 to obtain a w-cyclic 3-IGDD of type (vw, 2w)5 . Filling in the hole with a w-cyclic 3-IGDD of type (2w, w)5 , we obtain a w-cyclic 3-IGDD of type (vw, w)5 .  Lemma 4.13: There exists a 4-cyclic 3-IGDD of type (4v, 4)5 for any v ≡ 1, 2 (mod 3).

Proof: For v = 2, there exists a 4-cyclic 3-IGDD of type (8, 4)5 from Lemma 6.4. For v ∈ {4, 8}, by Example 3.7 and Lemma 4.12, there exists a 2-cyclic 3-IGDD of type (v, 2)5 . Inflate it by a 2-cyclic 3-GDD of type 43 from Corollary 2.7. By Construction 4.8, we obtain a 4-cyclic 3-IGDD of type (4v, 8)5 . For v ∈ {5, 7}, a 4-cyclic 3-IGDD of type (4v, 8)5 exists by Lemma 6.3. Then filling in the hole with a 4-cyclic 3-IGDD of type (8, 4)5 to obtain a 4-cyclic 3-IGDD of type (4v, 4)5 for each v ∈ {4, 5, 7, 8}. When v ≥ 10 and v ≡ 1 (mod 3), by Lemma 4.4, there exists a (3, 4, (v −1)/3)-cyclic 3-HGDD of type (5, 12(v−1)/3). Then apply Construction 4.10 with a 4-cyclic 3-IGDD of type (16, 4)5 to obtain a 4-cyclic 3-IGDD of type (4v, 4)5 . When v ≥ 11 and v ≡ 2 (mod 3), apply Construction 4.10 with a (3, 4, (v − 2)/3)-cyclic 3-HGDD of type (5, 12(v−2)/3) and a 4-cyclic 3-IGDD of type (20, 8)5 to obtain a 4-cyclic 3-IGDD of type (4v, 8)5 . Then fill in the hole with a 4-cyclic 3-IGDD of type (8, 4)5 to obtain a 4-cyclic 3-IGDD  of type (4v, 4)5 . Lemma 4.14: There exists a 2-cyclic 3-IGDD of type (2v, 2)u if u, v satisfy one of the following conditions. (1) u ≡ 0, 1 (mod 3), u ≥ 3 and v ≡ 1 (mod 2); (2) u ≡ 5 (mod 6), u ≥ 11 and v ≡ 1, 5 (mod 6). Proof Condition (1): When v ∈ {3, 5}, by Lemma 6.1, there exists a 2-cyclic 3-IGDD of type (2v, 2)n for n ∈ {3, 4, 6}. Then apply Construction 4.9 with a PBD(u, {3, 4, 6}) from Lemma 2.2 to obtain a 2-cyclic 3-IGDD of type (2v, 2)u . When v ≥ 7, by Lemma 4.4, there exists a (2, 2, (v −1)/2)cyclic 3-HGDD of type (u, 4(v−1)/2). Then apply Construction 4.10 with a 2-cyclic 3-IGDD of type (6, 2)u to obtain a 2-cyclic 3-IGDD of type (2v, 2)u . Condition (2): By Lemma 2.1, there is a 3-GDD of type 1u−5 51 , which is also a PBD(u, {3, 5}). Give weight 2v to each point of this PBD and input a 2-cyclic 3-IGDD of type (2v, 2)k for each k ∈ {3, 5} from Lemma 4.12 and Condition (1). By Construction 4.9, we obtain a 2-cyclic  3-IGDD of type (2v, 2)u . Lemma 4.15: Suppose u ≡ 5 (mod 6). Then there exists a 6-cyclic 3-IGDD of type (6v, 6)u for any v ≡ 1 (mod 2). Proof: For u = 3, by Lemma 4.14, there exists a 2-cyclic 3-IGDD of type (2v, 2)3 . Inflate it by a 3-cyclic 3-GDD of type 33 from Corollary 2.7. By Construction 4.8, we obtain a 6-cyclic 3-IGDD of type (6v, 6)3 . For u = 5, let v = 3x r , r ≡ 1, 5 (mod 6). We use induction on x. By Lemma 4.14, there is a 2-cyclic 3-IGDD of type (2r, 2)5 . Inflate it by a 3-cyclic 3-GDD of type 33 to obtain a 6-cyclic 3-IGDD of type (6r, 6)5 . When x ≥ 1, assume that there exists a 6-cyclic 3-IGDD of type (6 · 3x−1r, 6)5 . Apply Construction 4.8 with a 1-cyclic 3-GDD of type 33 to obtain a 6-cyclic 3-IGDD of type (6v, 18)5 . Filling in the hole with a 6-cyclic 3-IGDD of type (18, 6)5 from Lemma 6.5, we obtain a 6-cyclic 3-IGDD of type (6v, 6)5 . Further apply Construction 4.9 with a PBD(u, {3, 5}), we obtain the required 6-cyclic 3-IGDD of type (6v, 6)u for any u ≡ 5 (mod 6) and u ≥ 11. 

WANG AND CHANG: COMBINATORIAL CONSTRUCTIONS OF OPTIMAL 3-D OOCs

Corollary 4.16: Let u ≡ 5 (mod 6). Then there exists a 2-cyclic 3-IGDD of type (2v, 6)u for any v ≡ 3 (mod 6). Proof: By Lemma 4.15, there exists a 6-cyclic 3-IGDD of type (2v, 6)u . Hence, we have a 2-cyclic 3-IGDD of type (2v, 6)u .  C. h-Cyclic k-IGDDs of Type (ht)(u,e) We give three constructive methods for h-cyclic k-IGDDs of type (ht)(u,e) . The algorithms of Constructions 4.17 and 4.18 are omitted, which are similar to that of Constructions 4.8 and 4.9, respectively. Construction 4.17 (Inflation-III): If there exist an h-cyclic k-IGDD of type (gh)(n,t ) and a w-cyclic l-GDD of type (vw)k , then there exists an hw-cyclic l-IGDD of type (ghvw)(n,t ) . Construction 4.18 (Weighting Construction-II): Suppose there exists a K -GDD of type g1 u 1 · · · gr ur . If there exists an h-cyclic l-GDD of type (ht)k for each k ∈ K , then there exists an h-cyclic l-GDD of type (g1 ht)u 1 · · · (gr ht)ur . Construction 4.19 (Filling Construction-IV): Suppose that the following designs exist: (1) an h-cyclic k-GDD of type {gi ht : i = 1, 2, . . . , r }; (2) an h-cyclic k-IGDD of type (ht)(gi +e,e) for each 1 ≤ i ≤ r − 1. (u+e,gr +e) , Then there r exists an h-cyclic k-IGDD of type (ht) u = i=1 gi . Furthermore, if an h-cyclic k-IGDD of type (ht)(gr +e,e) exists, then an h-cyclic k-IGDD of type (ht)(u+e,e) exists. • Step 1: Start with an h-cyclic k-GDD of type {gi ht : i = 1, 2, . . . , r } (X, G, B) on X = ∪ri=1 (X i × It × Z h ) with G = {X i × It × Z h : 1 ≤ i ≤ r }, where |X i | = gi . Let F be a set of base blocks. • Step 2: Let Y = Y0 ×It ×Z h be an eht-set disjoint from X. For each group G ∈ G of size gi ht, 1 ≤ i ≤ r − 1, we construct an h-cyclic k-IGDD of type (ht)(gi +e,e) on G ∪ Y with group set {{x} × It × Z h : x ∈ X i ∪ Y0 } and hole set Y . Denote by AG the family of base blocks of this design. • Step 3: Let G r be the group of size htgr and A = ∪G∈G \{G r } AG ∪ F . Then A forms a set of base blocks of an h-cyclic k-IGDD of type (ht)(u+e,gr +e) on X ∪ Y with group set {{x} × It × Z h : x ∈ (∪ri=1 X i ) ∪ Y0 } and the hole G r ∪ Y . • Step 4: Further construct an h-cyclic k-IGDD of type (ht)(gr +e,e) on G r ∪Y with group set {{x}× It × Z h : x ∈ X r ∪ Y0 } and hole set Y . Denote the family of base blocks of this design by AG r . Then A∪AG r forms a set of base blocks of an h-cyclic k-IGDD of type (ht)(u+e,e) on X ∪ Y with group set {{x} × It × Z h : x ∈ (∪ri=1 X i ) ∪ Y0 } and the hole Y . Construction 4.20: Suppose there exists a PBD (u, K ∪ {k ∗ }). If there exist an m-cyclic l-IGDD of type m (u,k) and an l-SCHGDD of type (n, m t ) for each n ∈ K , then there exists an mt-cyclic l-IGDD of type (mt)(u,k) . • Step 1: Start with a PBD(u, K ∪ {k ∗ }) (X, B) on X = Iu with a single block B ∗ of size k. • Step 2: For each B ∈ B\{B ∗ }, construct an l-SCHGDD of type (|B|, m t ) on B × Z mt , with group set {{x} × Z mt : x ∈ B} and hole set {B × {i, t + i, . . . ,

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(m − 1)t + i } : 0 ≤ i ≤ t − 1}. Let A B be the base block set and A = ∪ B∈B\{B ∗} A B . Step 3: Construct an m-cyclic l-IGDD of type m (u,k) on Iu × Z m with group set {{i } × Z m : i ∈ Iu } and hole set B ∗ × Z m . Denote the set of base blocks by F . For each B = {(ai , bi ) : 1 ≤ i ≤ l} ∈ F , let t B = {(ai , tbi ) : 1 ≤ i ≤ l}. Let C = ∪ B∈F t B. Step 4: A ∪ C forms a set of base blocks of the required mt-cyclic l-IGDD of type (mt)(u,k) with group set {{i } × Z mt : i ∈ Iu } and hole set B ∗ × Z mt .

We summarize our results on h-cyclic 3-IGDDs of type (gh)(u,t ) as follows. Lemma 4.21 [31]: For 1 ≤ d ≤ 4, if (n, k) ≡ (0, 1), (1, d), (2, 0), (3, d + 1) (mod (4, 2)) such that n ≥ 2d − 3 and (n/2)(2d − 1 − n) + 1 ≤ k ≤ (n/2)(n − 2d + 5) + 1, then [d, d + 3n] \ {k + d + n − 1} can be partitioned into triples {ai , bi , ci }, 1 ≤ i ≤ n such that ai + bi = ci . Lemma 4.22: There exists a 2-cyclic 3-IGDD of type 2(u,t ) if u, t satisfy one of the following conditions. (1) u = 30 and t = 10; (2) u ≡ 3, 7 (mod 12), u ≥ 7 and t = 3; (3) u ≡ 10 (mod 12), u ≥ 22 and t = 7; (4) u ≡ 6 (mod 12), u ≥ 18, u = 30, and t = 6; (5) u ≡ 5 (mod 12), u ≥ 17 and t = 5; (6) u ≡ 11 (mod 12), and t = 2. Proof Condition (1): There is a 2-cyclic 3-IGDD of type 2(30,10) in Lemma 6.9. Condition (2): By Example 3.10, a 2-cyclic 3-IGDD of type 2(7,3) exists. When u ≥ 15, start with a 2-cyclic 3-GDD of type 8(u−3)/4 from Corollary 2.7. Applying Construction 4.19 with a 2-cyclic 3-IGDD of type 2(7,3), we obtain a 2-cyclic 3-IGDD of type 2(u,3). Condition (3): Start from a 4-GDD of type 3(u−7)/361 in Lemma 2.3. Take a 2-cyclic 3-GDD of type 24 from Corollary 2.7, and apply Construction 4.18 to obtain a 2-cyclic 3-GDD of type 6(u−7)/3121 . Filling in the groups with a 2-cyclic 3-IGDD of type 2(4,1), which is also a 2-cyclic 3-GDD of type 24 in Corollary 2.7, we obtain a 2-cyclic 3-IGDD of type 2(u,7) by Construction 4.19. Condition (4): For u = 18, by Lemma 6.9, there exists a 2-cyclic 3-IGDD of type 2(18,6). For u ≥ 42, by Corollary 2.7, there exists a 2-cyclic 3-GDD of type 24(u−6)/12. Then apply Construction 4.19 with a 2-cyclic 3-IGDD of type 2(18,6) to obtain a 2-cyclic 3-IGDD of type 2(u,6). Condition (5): A 2-cyclic 3-IGDD of type 2(17,5) exists in Lemma 6.8. When u = 29, by Lemma 2.1, there exists a 3-GDD of type 43 21 . Apply Construction 4.18 with a 2-cyclic 3-GDD of type 43 to obtain a 2-cyclic 3-GDD of type 163 81 . Filling in the groups with a 2-cyclic 3-IGDD of type 2(9,1), which is also a 2-cyclic 3-GDD of type 29 in Corollary 2.7, we obtain a 2-cyclic 3-IGDD of type 2(29,5) by Construction 4.19. For u ≥ 41, start from a 2-cyclic 3-GDD of type 24(u−5)/12 from Corollary 2.7. Filling in its groups by a 2-cyclic 3-IGDD of type 2(17,5), we obtain a 2-cyclic 3-IGDD of type 2(u,5) by Construction 4.19.

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Condition (6): Construct the required design on Z 2u−4 ∪ {x 0 , x 1 , y0 , y1 } with group set {{i, u − 2 + i } : 0 ≤ i ≤ u − 3}∪ {{x 0 , x 1 }, {y0 , y1 }} and hole set {x 0 , x 1 , y0 , y1 }. Let (d, n) = (2, (u − 2)/3 − 1). Obviously, n ≡ 2 (mod 4). We can always find a even k so that (n/2)(2d − 1 − n) + 1 ≤ k ≤ (n/2)(n − 2d + 5) + 1. Since (n, k) ≡ (2, 0) (mod (4, 2)), by Lemma 4.21, the set [2, u−3]\{k+n+1} can be partitioned into triples {ai , bi , ci }, 1 ≤ i ≤ n such that ai + bi = ci . Then, the blocks {0, 1, x 0 }, {0, k + n + 1, y0 }, {0, ai , ci }, 1 ≤ i ≤ n, form a set of base blocks of the required design. All blocks are obtained by developing the base blocks +1 (mod 2u − 4). Here, x i + 1 = x j , yi + 1 = y j , where j ≡ i + 1 (mod 2). Clearly, this 3-IGDD of type 2(u,2) is 2-cyclic.  Lemma 4.23: Let u ≡ 5 (mod 6) and u ≥ 11. Then there exists a w-cyclic 3-IGDD of type (vw)(u,5) if v, w satisfy one of the following conditions. (1) v ≡ 1, 2 (mod 3) and w ≡ 5 (mod 6); (2) v ≡ 2, 4 (mod 6) and w ≡ 2, 8, 10 (mod 12). Proof: By Lemma 2.1, there is a 3-GDD of type 1u−5 51 . Give weight vw to each point of this GDD and input a w-cyclic 3-GDD of type (vw)3 from Corollary 2.7. By Construction 4.18, we obtain a w-cyclic 3-GDD of type (vw)u−5 (5vw)1 , which is also a w-cyclic 3-IGDD of  type (vw)(u,5) . Lemma 4.24: Let u ≡ 5 (mod 12). Then there exists a w-cyclic 3-IGDD of type (vw)(u,5) for any v ≡ 1, 5 (mod 6) and w ≡ 2, 8, 10 (mod 12). Proof: Start from a 3-GDD of type 1u−5 51 in Lemma 2.1, which is also a PBD(u, {3, 5∗ }). Applying Construction 4.20 with a 3-SCHGDD of type (3, 2w/2 ) from Theorem 4.2 and a 2-cyclic 3-IGDD of type 2(u,5) from Lemma 4.22, we obtain a w-cyclic 3-IGDD of type w(u,5) . Further apply Construction 4.17 with a 3-GDD of type v 3 to obtain a w-cyclic 3-IGDD of type (vw)(u,5) .  Lemma 4.25: Let u ≡ 11 (mod 12). Then there exists a 6-cyclic 3-IGDD of type 6(u,3). Proof: A 6-cyclic 3-IGDD of type 6(11,3) exists by Lemma 6.10. When u ≥ 23, start from a 6-cyclic 3-GDD of type 24(u−3)/4 in Corollary 2.7. Filling in the groups with a 6-cyclic 3-IGDD of type 6(7,3), by Construction 4.19, we obtain a 6-cyclic 3-IGDD of type 6(u,3). The required 6-cyclic 3-IGDD of type 6(7,3) can by obtained by applying Construction 4.17 with a 2-cyclic 3-IGDD of type 2(7,3) and a 3-cyclic 3-GDD of type 33 .  Corollary 4.26: There exists a 2-cyclic 3-IGDD of type 6(u,3) for any u ≡ 11 (mod 12). Proof: By Lemma 4.25, there exists a 6-cyclic 3-IGDD of type 6(u,3) for any u ≡ 11 (mod 12). Hence, we have a 2-cyclic 3-IGDD of type 6(u,3).  V. M AIN R ESULT In this section, applying the filling constructions established in above section, we obtain the main result of this paper. A. The Cases of u ≡ 0, 1 (mod 3) We first determine the size of an optimal w-cyclic 3-GDP of type (vw)u whose parameters are not covered by Corollary 2.7 and Theorem 2.12 for u ≡ 0, 1 (mod 3).

Lemma 5.1: There exists a w-cyclic 3-GDP of type (vw)u with J ∗ (u ×v ×w, 3, 1) base blocks for any v, w ≡ 1 (mod 2) and u ≡ 0, 4 (mod 6). Proof: When w = 1, a 1-cyclic 3-GDP of type v u with ∗ J (u × v × 1, 3, 1) base blocks exists by Theorem 2.12. When (u, v, w) = (6, 1, 3), the required design is constructed on I6 × Z 3 with group set {{i } × Z 3 : i ∈ I6 }. The base blocks are listed as follows: {(0, 0), (1, 0), (2, 0)}, {(0, 0), (3, 0), (4, 0)}, {(0, 0), (5, 0), (1, 1)}, {(0, 0), (2, 1), (3, 1)}, {(0, 0), (4, 1), (5, 1)}, {(0, 0), (1, 2), (3, 2)}, {(0, 0), (2, 2), (4, 2)}, {(1, 0), (4, 0), (2, 1)}, {(1, 0), (5, 0), (3, 1)}, {(1, 0), (4, 1), (3, 2)}, {(1, 0), (5, 1), (4, 2)}, {(2, 0), (5, 0), (3, 2)}, {(2, 0), (3, 1), (5, 1)}, {(2, 0), (4, 1), (5, 2)}. It is readily checked that J ∗ (6 × 1 × 3, 3, 1) = 14. Hence, it is an optimal 3-cyclic 3-GDP of type 36 . For other values, by Lemma 4.6, there is a (v, w, 1)-cyclic 3-HGDD of type (u, v w ), which has u(u −1)v 2 (w −1)/6 base blocks. Filling the holes with a 1-cyclic 3-GDP of type v u with J ∗ (u × v × 1, 3, 1) = uv/3(u − 1)v/2 base blocks, then we obtain a w-cyclic 3-GDP of type (vw)u with u(u − 1)v 2 (w −1)/6+uv/3(u −1)v/2 = uv/3(u −1)vw/2 base blocks, by Construction 3.4. This number achieves the upper bound in Lemma 2.11. Thus it is an optimal w-cyclic 3-GDP  of type (vw)u . Lemma 5.2: Let u ≡ 3, 6, 7, 10 (mod 12) and v ≡ 1 (mod 2). Then there exists a w-cyclic 3-GDP of type (vw)u with J ∗ (u × v × w, 3, 1) base blocks for any w ≡ 0 (mod 2) when u = 3, and w ≡ 2 (mod 4) when u ≥ 4. Proof Case 1: w = 2. We divide this situation into two subcases. Subcase 1: v = 1. When u ∈ {3, 7}, 2-cyclic 3-GDPs of type 2u with J ∗ (u × 1 × 2, 3, 1) base blocks have been constructed in Examples 3.5 and 3.10, respectively. For each u ∈ {6, 10}, we construct a 2-cyclic 3-GDP of type 2u on Iu × Z 2 with the group set {{i } × Z 2 : i ∈ Iu }. Only base blocks are listed below. u = 6 : {(0, 0), (1, 0), (2, 1)}, {(1, 0), (3, 0), (0, 1)}, {(2, 0), (3, 1), (4, 1)}, {(3, 0), (1, 1), (5, 1)}, {(4, 0), (0, 1), (3, 1)}, {(5, 0), (0, 1), (2, 1)}, {(0, 1), (4, 1), (5, 1)}, {(1, 1), (2, 1), (4, 1)}, {(2, 1), (3, 1), (5, 1)}. u = 10 : {(0, 0), (1, 0), (2, 0)}, {(0, 0), (3, 0), (4, 0)}, {(0, 0), (5, 0), (6, 0)}, {(0, 0), (7, 0), (8, 0)}, {(0, 0), (9, 0), (1, 1)}, {(0, 0), (2, 1), (3, 1)}, {(0, 0), (4, 1), (5, 1)}, {(0, 0), (6, 1), (7, 1)}, {(0, 0), (8, 1), (9, 1)}, {(1, 0), (3, 0), (5, 0)}, {(1, 0), (4, 0), (6, 0)}, {(1, 0), (7, 0), (9, 0)}, {(1, 0), (8, 0), (2, 1)}, {(1, 0), (3, 1), (6, 1)}, {(1, 0), (4, 1), (7, 1)}, {(1, 0), (5, 1), (8, 1)}, {(2, 0), (4, 0), (8, 0)}, {(2, 0), (5, 0), (3, 1)},

WANG AND CHANG: COMBINATORIAL CONSTRUCTIONS OF OPTIMAL 3-D OOCs

{(2, 0), (6, 0), (7, 1)}, {(2, 0), (7, 0), (5, 1)}, {(2, 0), (9, 0), (4, 1)}, {(2, 0), (6, 1), (9, 1)}, {(3, 0), (7, 0), (8, 1)}, {(3, 0), (8, 0), (6, 1)}, {(3, 0), (9, 0), (7, 1)}, {(3, 0), (4, 1), (9, 1)}, {(4, 0), (5, 1), (7, 1)}, {(4, 0), (6, 1), (8, 1)}, {(5, 0), (9, 0), (6, 1)}. It is readily checked that the number of base blocks of these designs achieve the upper bound in Lemma 2.11. Hence, they are optimal. For each other u, take a 2-cyclic 3-IGDD of type 2(u,t ) from Lemma 4.22, which has (u(u − 1) − t (t − 1))/3 base blocks. Filling in its hole with above 2-cyclic 3-GDP of type 2t for some t ∈ {3, 6, 7, 10}, which has t (t −1)/3−1 base blocks, we then obtain a 2-cyclic 3-GDP of type 2u by Construction 3.11, which has u(u − 1)/3 − 1 base blocks. This number achieves the upper bound in Lemma 2.11. Hence, it is optimal. Subcase 2: v > 1. By Lemma 4.14, there exists a 2-cyclic 3-IGDD of type (2v, 2)u , which has u(u−1)(v 2 −1)/3 base blocks. Filling in the hole with above 2-cyclic 3-GDP of type 2u with u(u − 1)/3 − 1 base blocks, by Construction 3.8, we obtain a 2-cyclic 3-GDP of type (2v)u with u(u − 1)v 2 / 3 − 1 = J ∗ (u × v × 2, 3, 1) base blocks. Case 2: (u, w) = (3, 4). We first construct a 4-cyclic 3-GDP of type 43 on I3 × Z 4 with the group set {{i } × Z 4 : i ∈ I3 }, which has J ∗ (3 × 1 × 4, 3, 1) = 3 base blocks. Only base blocks are listed as follows: {(0, 0), (1, 0), (2, 0)}, {(0, 0), (1, 1), (2, 2)}, {(0, 0), (2, 1), (1, 2)}. When v ≥ 3, by Lemma 4.11, there exists a 4-cyclic 3-IGDD of type (4v, 4)3 , which has 4(v 2 − 1) base blocks. Apply Construction 3.8 with above 4-cyclic 3-GDP of type 43 to obtain a 4-cyclic 3-GDP of type (4v)3 with 4v 2 − 1 = J ∗ (3 × v × 4, 3, 1) base blocks. Case 3: (u, w) = (3, 6). A 6-cyclic 3-GDP of type 63 with ∗ J (3 × 1 × 6, 3, 1) = 5 base blocks exists by Example 2.4. When v ≥ 3, in the proof of Lemma 4.15, there exists a 6-cyclic 3-IGDD of type (6v, 6)3 , which has 6(v 2 − 1) base blocks. Filling the hole with above 6-cyclic 3-GDP of type 63 , we obtain a 6-cyclic 3-GDP of type (6v)3 with 6v 2 − 1 = J ∗ (3 × v × 6, 3, 1) base blocks. Case 4: Other values. By Lemma 4.6, there exists a (v, w, 1)-cyclic 3-HGDD of type (u, (2v)w/2 ), which has u(u − 1)v 2 (w − 2)/6 base blocks. Filling in the hole with a 2-cyclic 3-GDP of type (2v)u with u(u−1)v 2 /3−1 base blocks from Case 1, we obtain a w-cyclic 3-GDP of type (vw)u , which has u(u − 1)v 2 w/6 − 1 = J ∗ (u × v × w, 3, 1) base blocks.  B. The Case of u ≡ 5 (mod 6) At the beginning, we give a direct construction for an optimal w-cyclic 3-GDP of type w5 when w ≡ 2 (mod 3). Lemma 5.3: There exists a w-cyclic 3-GDP of type w5 with J ∗ (5 × 1 × w, 3, 1) base blocks for any w ≡ 2 (mod 3) and w ≥ 5.

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Proof: The required designs are constructed on I5 × Z w with group set {{i } × Z w : i ∈ I5 }. Only base blocks are listed as follows: w = 5 : {(0, 0), (1, 0), (2, 0)}, {(0, 0), (3, 0), (4, 0)}, {(0, 0), (1, 1), (3, 1)}, {(2, 0), (3, 3), (4, 1)}, {(0, 0), (2, 1), (4, 1)}, {(0, 0), (1, 2), (4, 2)}, {(0, 0), (2, 2), (3, 2)}, {(0, 0), (1, 3), (2, 4)}, {(0, 0), (2, 3), (3, 4)}, {(0, 0), (3, 3), (4, 4)}, {(0, 0), (1, 4), (4, 3)}, {(1, 0), (3, 1), (4, 3)}, {(1, 0), (2, 4), (4, 1)}, {(1, 0), (2, 2), (3, 4)}, {(1, 0), (2, 3), (3, 2)}, {(1, 0), (3, 3), (4, 2)}. w ≥ 8 : {(0, 0), (1, 0), (3, (w − 5)/3)}, {(0, 0), (2, (w − 5)/3), (4, 0)}, {(0, 0), (1, 1), (2, (w + 1)/3)}, {(0, 0), (1, (w − 2)/3), (4, 1)}, {(1, 0), (2, 1), (3, (w + 4)/3)}, {(0, 0), (2, (w + 4)/3), (4, 2)}, {(2, 0), (3, 1), (4, (w + 1)/3)}, {(2, 0), (3, w − 1), (4, (w − 2)/3)}, {(0, 0), (1, (w + 1)/3), (3, (2w − 1)/3)}, {(1, 0), (2, w − 1), (4, (w + 1)/3)}, {(0, 0), (2, 2(w + 1)/3), (3, (w + 1)/3)}, {(1, 0), (2, w − 2), (3, (w − 8)/3)}, {(0, 0), (3, 2(w + 4)/3), (4, (w + 4)/3)}, {(0, 0), (1, w − 2), (3, 2(w − 2)/3)}, {(0, 0), (1, 2(w − 2)/3), (2, (w − 8)/3)}, {(0, 0), (1, w − 1), (4, (2w − 1)/3)}, {(1, 0), (2, (2w − 1)/3), (4, (w − 2)/3)}, {(0, 0), (2, (2w + 5)/3), (4, w − 1)}, {(1, 0), (3, (2w + 5)/3), (4, 2(w + 4)/3)}, {(0, 0), (2, (w − 2)/3), (3, (w − 2)/3)}, {(1, 0), (3, (2w − 1)/3), (4, (2w − 1)/3)}, {(1, 0), (3, (w + 1)/3), (4, (w − 5)/3)}, {(0, 0), (1, (2w − 1)/3), (2, (2w − 1)/3)}, {(0, 0), (3, 2(w + 1)/3), (4, (w + 1)/3)}, {(0, 0), (3, (2w + 5)/3), (4, 2(w + 1)/3)}, {(1, 0), (2, (w + 1)/3), (4, 2(w − 2)/3)}, {(l, 0), (l + 1, j ), (l + 2, 2(w − 2)/3 + 2 j )}, {(l, 0), (l + 1, (w − 2)/3 + j ), (l + 3, 2(w − 2)/3 − j + 1)}, where j ∈ [2, (w − 5)/3], and l runs over I5 and the first components of elements are reduced modulo 5. When w = 8, j ∈ ∅.  Lemma 5.4: There exists a w-cyclic 3-GDP of type (vw)5 with J ∗ (5 ×v ×w, 3, 1) base blocks for any v ≡ 1, 2 (mod 3) and w ≡ 1, 2 (mod 3). Proof Case 1: w ≡ 2 (mod 3). Start with a w-cyclic 3-IGDD of type (vw, w)5 from Lemma 4.12. Take a w-cyclic 3-GDP of type w5 with J ∗ (5 × 1 × w, 3, 1) base blocks from

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Example 3.9 and Lemma 5.3. By Construction 4.10, we obtain the required w-cyclic 3-GDP of type (vw)5 with J ∗ (5 × v × w, 3, 1) base blocks. Case 2: w ≡ 1 (mod 3). When w = 1, the required designs exist by Theorem 2.12. When w = 4 and v = 1, we construct a 4-cyclic 3-GDP of type 45 on I5 × Z 4 with group set {{i }× Z 4 : i ∈ I5 }. The J ∗ (5 × 1 × 4, 3, 1) base blocks of are listed as follows: {(0, 0), (1, 0), (2, 0)}, {(0, 0), (3, 0), (4, 0)}, {(0, 0), (1, 1), (3, 1)}, {(0, 0), (2, 1), (4, 1)}, {(0, 0), (1, 2), (4, 2)}, {(0, 0), (2, 2), (1, 3)}, {(0, 0), (3, 2), (4, 3)}, {(0, 0), (2, 3), (3, 3)}, {(1, 0), (2, 1), (3, 2)}, {(1, 0), (3, 1), (4, 3)}, {(1, 0), (4, 1), (2, 2)}, {(1, 0), (4, 2), (3, 3)}. When v ≥ 2, by Lemma 4.13, there exists a 4-cyclic 3-IGDD of type (4v, 4)5 . Then apply Construction 3.8 with above 4-cyclic 3-GDP of type 45 to obtain a 4-cyclic 3-GDP of type (4v)5 with J ∗ (5 × v × 4, 3, 1) base blocks. When w ≥ 7, by Lemma 4.6, there exists a (v, w, 1)-cyclic 3-HGDD of type (5, v w ). Then filling in the holes with a 1-cyclic 3-GDP of type v 5 with J ∗ (5 × v × 1, 3, 1) base blocks from Theorem 2.12, by Construction 3.4, we obtain a w-cyclic 3-GDP of type (vw)5 with J ∗ (5 × v × w, 3, 1) base blocks.  Lemma 5.5: Let u > 5. A w-cyclic 3-GDP of type (vw)u with J ∗ (u × v × w, 3, 1) base blocks exists if one of the following conditions holds. (1) u ≡ 5 (mod 6), v ≡ 1, 2 (mod 3) and w ≡ 5 (mod 6); (2) u ≡ 5 (mod 6), v ≡ 2, 4 (mod 6) and w ≡ 2, 8, 10 (mod 12); (3) u ≡ 5 (mod 12), v ≡ 1, 5 (mod 6) and w ≡ 2, 8, 10 (mod 12). Proof: By Lemmas 4.23 and 4.24, there exists a w-cyclic 3-IGDD of type (vw)(u,5) . Take a w-cyclic 3-GDP of type (vw)5 with J ∗ (5 × v × w, 3, 1) base blocks from Lemma 5.4 to fill in the hole. By Construction 3.11, we obtain a w-cyclic 3-GDP of type (vw)u with J ∗ (u ×v ×w, 3, 1) base blocks.  Lemma 5.6: Let u ≡ 11 (mod 12). Then there exists a w-cyclic 3-GDP of type (vw)u with J ∗ (u × v × w, 3, 1) base blocks if one of following conditions holds. (1) v ≡ 1, 5 (mod 6) and w ≡ 2 (mod 6); (2) v ≡ 3 (mod 6) and w ≡ 2, 10 (mod 12). Proof Case 1: w = 2. When v = 1, by Lemma 4.22, there exists a 2-cyclic 3-IGDD of type 2(u,2) , which is also a 2-cyclic 3-GDP of type 2u with J ∗ (u ×1×2, 3, 1) base blocks. When v ≡ 1, 5 (mod 6) and v ≥ 5, by Lemma 4.14, there exists a 2-cyclic 3-IGDD of type (2v, 2)u . Filling in the hole with above 2-cyclic 3-GDP of type 2u , we obtain a 2-cyclic 3-GDP of type (2v)u with J ∗ (u × v × 2, 3, 1) base blocks. When v = 3, by Corollary 4.26, there exists a 2-cyclic 3-IGDD of type 6(u,3). Filling in the hole with a 2-cyclic 3-GDP of type 63 with J ∗ (3 × 3 × 2, 3, 1) base blocks from Lemma 5.2, we then obtain a 2-cyclic 3-GDP of type 6u with J ∗ (u × 3 × 2, 3, 1) base blocks. When v ≡ 3 (mod 6) and

v ≥ 9, by Corollary 4.16, there exists a 2-cyclic 3-IGDD of type (2v, 6)u . Filling in the hole with above 2-cyclic 3-GDP of type 6u , we obtain a 2-cyclic 3-GDP of type (2v)u with J ∗ (u × v × 2, 3, 1) base blocks. Case 2: w > 2. For each condition, by Lemma 4.7, there exists a (v, w, 1)-cyclic 3-HGDD of type (u, (2v)w/2 ). Then filling in the holes with above 2-cyclic 3-GDP of type (2v)u , by Construction 3.4, we obtain a w-cyclic 3-GDP  of type (vw)u with J ∗ (u × v × w, 3, 1) base blocks. Lemma 5.7: Let u ≡ 5 (mod 6) and u ≥ 11. Then there exists a w-cyclic 3-GDP of type (vw)u with J ∗ (u × v × w, 3, 1) base blocks for any w ≡ 1, 4, 7 (mod 12) and v ≡ 1, 2 (mod 3). Proof: Start with a (v, w, 1)-cyclic 3-HGDD of type (u, v w ) from Lemma 4.6. Filling the holes with above 1-cyclic 3-GDP of type v u from Theorem 2.12, we obtain a w-cyclic 3-GDP of type (vw)u with J ∗ (u × v × w, 3, 1) base blocks.  Lemma 5.8: Let u ≡ 11 (mod 12). Then there exists a w-cyclic 3-GDP of type (vw)u with J ∗ (u × v × w, 3, 1) base blocks for any w ≡ 6 (mod 12) and v ≡ 1 (mod 2). Proof: When w = 6 and v = 1, start with a 6-cyclic 3-IGDD of type 6(u,3) from Lemma 4.25. Then filling in the hole with the 6-cyclic 3-GDP of type 63 in Example 2.4, we obtain a 6-cyclic 3-GDP of type 6u with J ∗ (u ×1×6, 3, 1) base blocks. When v > 1, by Lemma 4.15, there exists a 6-cyclic 3-IGDD of type (6v, 6)u . Apply Construction 3.8 with above 6-cyclic 3-GDP of type 6u to obtain a 6-cyclic 3-GDP of type (6v)u with J ∗ (u × v × 6, 3, 1) base blocks. When w ≥ 18, start with a (v, w, 1)-cyclic 3-HGDD of type (u, (6v)w/6 ) from Lemma 4.6. Filling the holes with above 6-cyclic 3-GDP of type (6v)u , we obtain a w-cyclic 3-GDP of type (vw)u with J ∗ (u × v × w, 3, 1) base blocks.  Now we are in a position to prove our main result of this paper. Theorem 5.9: There exists an optimal w-cyclic 3-GDP of type (vw)u with J ∗ (u × v × w, 3, 1) base blocks for any positive integers v, w and u ≡ 2 (mod 6) with the possible exceptions of u ≡ 11 (mod 12), v ≡ 1, 5 (mod 6) and w ≡ 10 (mod 12). Proof: Combining Corollary 2.7, Theorem 2.12, and Lemmas 5.1, 5.2, and 5.4-5.8, the conclusion then holds.  Equivalently, the size of an optimal AM-OPP 3-D (u×v×w, 3, 1)-OOC is determined for the same parameters by Theorem 2.5. Theorem 5.10: There exists an optimal AM-OPP 3-D (u × v × w, 3, 1)-OOC with J ∗ (u × v × w, 3, 1) codewords for any positive integers v, w and u ≡ 2 (mod 6) with the possible exceptions of u ≡ 11 (mod 12), v ≡ 1, 5 (mod 6) and w ≡ 10 (mod 12). Recall that an AM-OPP 3-D (u × 1 × w, k, 1)-OOC is also referred to as an AM-OPPW 2-D (u ×w, k, 1)-OOC [22]. The known result on optimal AM-OPPW 2-D (u × w, 3, 1)-OOCs is improved. Theorem 5.11: There exists an optimal AM-OPPW 2-D (u × w, 3, 1)-OOC with J ∗ (u × 1 × w, 3, 1) codewords for any positive integers w and u ≡ 2 (mod 6) with the possible exceptions of u ≡ 11 (mod 12) and w ≡ 10 (mod 12).

WANG AND CHANG: COMBINATORIAL CONSTRUCTIONS OF OPTIMAL 3-D OOCs

VI. C ONCLUDING R EMARKS In this paper, we have established the equivalence between optimal 3-D OOCs with the AM-OPP restriction and optimal w-cyclic GDPs, and determined the size of an optimal AM-OPP 3-D (u × v × w, 3, 1)-OOC for all v, w and all u ≡ 2 (mod 6) with some possible exceptions. Most of the progress results from two aspects. The first is the direct constructions of the ingredient designs, in particular for w-cyclic 3-GDPs of type w5 for any w ≡ 2 (mod 3) and w-cyclic 3-IGDDs of type (vw, w)5 for small values v and w ≡ 2 (mod 6). The second is the use of “Filling Construction” shown in Constructions 3.4, 3.8 and 3.11. However, at present we cannot completely settle the problem for the remaining cases. The main problem is that we need to find some new auxiliary designs with the prescribed permutation group for the recursion. A PPENDIX A Lemma 6.1: There exists a 2-cyclic 3-IGDD of type (2h, 2)u for each (h, u) ∈ {(3, 3), (3, 4), (3, 6), (5, 3), (5, 4), (5, 6)}. Proof: These designs are constructed on I2hu . For each (h, u) ∈ {(3, 3), (3, 6)}, the group set is {{i, u +i, . . . , (2h −1) u + i } : 0 ≤ i ≤ u − 1}, and the hole is {x : x ≡ 0 (mod h), x ∈ I2hu }. For (h, u) = (3, 3), the group set is {{i, 3 + i, . . . , 15 + i } : 0 ≤ i ≤ 2}, and the hole is {x : x ≡ 0, 1, 2 (mod 9), x ∈ I18 }. For (h, u) = (3, 6), the group set is {{i, 5 + i, . . . , 25+i } : 0 ≤ i ≤ 4}∪{{x : x ≥ 30}}, and the hole is {x : x ≡ 0 (mod 3), x ∈ I36 }. The base blocks are listed as follows: (h, u) = (3, 3) : {3, 7, 17}, {0, 4, 5}, {0, 13, 17}, {0, 14, 16}, {2, 4, 15}, {2, 7, 12}, {1, 3, 5}, {5, 15, 16}, {3, 4, 14}, {2, 6, 16}, {2, 3, 13}, {0, 7, 8}, {1, 12, 17}, {1, 14, 15}, {4, 6, 17}, {1, 6, 8}. (h, u) = (3, 4) : {0, 10, 17}, {0, 11, 22}, {0, 13, 23}, {0, 1, 2}, {0, 5, 7}, {0, 14, 19}, {1, 4, 10}, {2, 5, 11}. (h, u) = (3, 6) : {0, 16, 22}, {0, 19, 31}, {0, 23, 32}, {0, 28, 35}, {0, 4, 7}, {0, 14, 26}, {1, 10, 30}, {1, 13, 29}, {0, 29, 34}, {1, 8, 14}, {1, 20, 34}, {1, 23, 35}, {2, 11, 30}, {0, 13, 17}, {0, 1, 2}, {0, 8, 11}. (h, u) = (5, 3) : {1, 6, 17}, {0, 13, 14}, {3, 14, 19}, {0, 16, 23}, {0, 7, 11}, {2, 7, 24}, {0, 17, 19}, {0, 22, 29}, {0, 26, 28}, {1, 9, 11}, {2, 3, 13}, {1, 14, 24}, {1, 18, 23}, {2, 12, 28}, {0, 1, 2}, {0, 4, 8}. (h, u) = (5, 4) : {0, 14, 17}, {0, 18, 19}, {0, 21, 26}, {0, 22, 23}, {0, 1, 2}, {0, 11, 13}, {1, 16, 39}, {0, 31, 37}, {0, 34, 39},

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{0, 3, 6}, {1, 14, 32}, {1, 22, 27}, {1, 23, 28}, {2, 12, 33}, {0, 7, 9}, {0, 29, 38}, {2, 19, 29}, {3, 14, 28}, {1, 8, 18}, {3, 9, 24}, {0, 27, 33}, {1, 12, 19}, {2, 13, 27}, {1, 4, 11}. (h, u) = (5, 6) : {0, 13, 16}, {0, 14, 17}, {0, 19, 23}, {0, 21, 26}, {0, 1, 2}, {0, 28, 32}, {0, 31, 33}, {0, 37, 39}, {0, 38, 43}, {0, 7, 8}, {0, 41, 44}, {0, 46, 53}, {0, 47, 57}, {0, 51, 59}, {0, 3, 4}, {0, 52, 56}, {1, 11, 22}, {1, 14, 21}, {1, 16, 32}, {2, 9, 29}, {1, 24, 34}, {1, 27, 42}, {1, 28, 44}, {1, 29, 48}, {0, 9, 11}, {1, 38, 47}, {1, 39, 52}, {2, 18, 37}, {2, 19, 48}, {1, 18, 26}, {2, 23, 34}, {2, 24, 39}, {2, 28, 54}, {3, 18, 38}, {0, 22, 27}, {0, 29, 34}, {0, 49, 58}, {1, 23, 33}, {2, 13, 42}, {3, 24, 49}.  For (h, u) = (3, 3), let α = 8i=0 (i 9 + i ) be a permutation  on I18 . For (h, u) = (3, 6), let α = 2i=0 (i 3 + i · · · 27 + i ) 2 permutation on I36 . For each (h, u) ∈ i=0 (30+i 33+i ) be a {(3, 3), (3, 6)}, let α = h−1 i=0 (i h + i · · · (2u − 1)h + i ) be a permutation on I2hu . Let G be the group generated by α. All blocks are obtained by developing these base blocks under the action of G. Obviously, each of these designs is isomorphic  to a 2-cyclic 3-IGDD of type (2h, 2)u . Lemma 6.2: There exists a 4-cyclic 3-IGDD of type (4h, 4)3 for each h ∈ {3, 5}. Proof: This designs are constructed on I12h with group set {{i, 3 + i, . . . , 3(4h − 1) + i } : 0 ≤ i ≤ 2}. For h = 3, the hole is {x : x ≤ 2 (mod 9), x ∈ I36 }. For h = 5, the hole is {x : x ≡ 0 (mod 5), x ∈ I60 }. The base blocks are listed as follows: h = 3 : {0, 4, 5}, {0, 13, 17}, {0, 14, 16}, {0, 22, 32}, {0, 23, 31}, {0, 7, 8}, {1, 12, 17}, {1, 14, 15}, {1, 21, 32}, {1, 23, 30}, {1, 3, 5}, {2, 12, 22}, {2, 13, 24}, {2, 15, 31}, {2, 16, 30}, {1, 6, 8}, {3, 17, 34}, {3, 22, 35}, {3, 26, 31}, {4, 23, 33}, {2, 3, 4}, {0, 25, 35}, {0, 26, 34}, {2, 21, 34}, {5, 16, 33}, {2, 6, 7}, {5, 24, 34}, {1, 24, 35}, {1, 26, 33}, {2, 25, 33}, {3, 7, 23}, {4, 6, 26}. h = 5 : {0, 13, 14}, {0, 16, 23}, {0, 17, 19}, {0, 22, 29}, {0, 26, 28}, {0, 32, 37}, {0, 34, 44}, {0, 38, 43}, {0, 46, 59}, {0, 47, 58}, {0, 52, 53}, {1, 18, 26}, {1, 21, 47}, {1, 24, 32}, {1, 29, 42}, {1, 38, 48}, {1, 39, 59}, {2, 12, 34}, {2, 18, 37}, {2, 19, 42}, {2, 33, 58}, {3, 19, 44}, {3, 29, 34}, {1, 9, 23}, {2, 28, 48}, {0, 31, 41}, {0, 49, 56}, {1, 33, 44}, {0, 7, 11}, {1, 6, 17}, {0, 4, 8}, {0, 1, 2}.

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 1, JANUARY 2015

 For h = 3, let α = 8i=0 (i 9+i · · · 27+i ) be a permutation  on I36 . For h = 5, let α = 4i=0 (i 5 + i · · · 55 + i ) be a permutation on I60 . Let G be the group generated by α. All blocks are obtained by developing above base blocks under the action of G. Obviously, each of this design is isomorphic  to a 4-cyclic 3-IGDD of type (4h, 4)3 . Lemma 6.3: There exists a 4-cyclic 3-IGDD of type (4h, 8)5 for each h ∈ {5, 7}. Proof: This designs are constructed on I20h with group set {{i, 5 + i, . . . , 5(4h − 1) + i } : 0 ≤ i ≤ 4} and hole set {x : x ≥ 20(h − 2), x ∈ I20h }. The base blocks are listed as follows: h = 5 : {0, 11, 72}, {0, 12, 81}, {0, 13, 77}, {0, 14, 76}, {0, 2, 61}, {0, 17, 74}, {0, 24, 96}, {0, 16, 73}, {0, 23, 91}, {0, 1, 62}, {0, 18, 82}, {0, 19, 83}, {0, 21, 87}, {0, 22, 88}, {0, 6, 64}, {0, 26, 28}, {0, 27, 56}, {0, 31, 34}, {0, 32, 38}, {0, 8, 71}, {0, 43, 49}, {0, 59, 98}, {1, 20, 73}, {0, 53, 97}, {0, 4, 67}, {0, 44, 52}, {0, 46, 93}, {0, 47, 94}, {0, 58, 99}, {0, 3, 66}, {1, 10, 68}, {1, 13, 67}, {1, 14, 70}, {1, 17, 69}, {1, 8, 72}, {1, 22, 80}, {1, 23, 74}, {1, 25, 93}, {1, 28, 84}, {0, 9, 68}, {1, 32, 94}, {2, 20, 69}, {0, 37, 41}, {1, 19, 90}, {0, 7, 69}, {1, 35, 98}, {1, 38, 47}, {1, 44, 95}, {1, 50, 83}, {2, 5, 91}, {2, 23, 80}, {2, 26, 98}, {2, 29, 83}, {1, 29, 77}, {1, 2, 60}, {2, 14, 68}. h = 7 : {0, 11, 122}, {0, 1, 102}, {0, 3, 111}, {0, 2, 106}, {0, 13, 139}, {0, 4, 103}, {0, 28, 61}, {0, 6, 113}, {0, 12, 136}, {0, 31, 63}, {0, 18, 44}, {0, 24, 71}, {0, 16, 128}, {0, 7, 116}, {0, 8, 121}, {0, 9, 123}, {0, 17, 114}, {0, 21, 59}, {0, 22, 58}, {0, 23, 57}, {0, 14, 112}, {0, 19, 46}, {0, 49, 127}, {0, 48, 137}.  For h = 5, let α = 2i=0 (i 3 +i · · · 57 +i )(60 63 · · · 78 61 64 · · · 79 62 65 · · · 77)(80 83 · · · 98 81 84 · · · 99 82 85 · · · 97) be a permutation on I100 . For h = 7, let α = (0 1 · · · 99)(100 101 · · · 119)(120 121 · · · 139) be a permutation on I140 . Let G be the group generated by α. All blocks are obtained by developing the base blocks under the action of G. Clearly, each of this design is isomorphic to  a 4-cyclic 3-IGDD of type (4h, 8)5 . Lemma 6.4: There exists a 4-cyclic 3-IGDD of type (8, 4)5 . Proof: This design is constructed on I40 with group set {{i, 5 + i, . . . , 35 + i } : 0 ≤ i ≤ 4} and the hole set {x : x ≡ 0 (mod 2), x ∈ I40 }. The base blocks are listed as follows:

Obviously, this design is isomorphic to a 2-cyclic 3-IGDD  of type (8, 4)5 . Lemma 6.5: There exists a 6-cyclic 3-IGDD of type (18, 6)5. Proof: The desired design is constructed on I90 with group set {{i, 5 + i, . . . , 85 + i } : 0 ≤ i ≤ 4} and hole set {x : x ≡ 0 (mod 3), x ∈ I90 }. The base blocks are listed as follows: {0, 13, 17}, {0, 14, 16}, {0, 19, 26}, {0, 22, 28}, {0, 23, 29}, {0, 31, 43}, {0, 32, 41}, {0, 49, 77}, {0, 52, 74}, {0, 68, 86}, {0, 34, 47}, {0, 37, 53}, {0, 38, 67}, {0, 44, 71}, {0, 46, 73}, {0, 56, 79}, {0, 58, 89}, {0, 59, 76}, {0, 61, 82}, {0, 62, 83}, {1, 10, 49}, {1, 19, 65}, {1, 20, 53}, {1, 34, 83}, {1, 35, 59}, {0, 64, 88}, {2, 14, 50}, {0, 1, 2}, {0, 4, 7}, {0, 8, 11}, {1, 37, 80}, {1, 38, 77}. 2 Let α = i=0 (i 3 + i · · · 87 + i ) be a permutation on I90 . Let G be the group generated by α. All blocks are obtained by developing above base blocks under the action of G. Clearly, this design is isomorphic to a 2-cyclic 3-IGDD  of type (18, 6)5. Lemma 6.6: Let w ≡ 2 (mod 6). Then there exists a w-cyclic 3-IGDD of type (5w, 2w)5 . Proof: The required designs are constructed on Z 5 × I5 × Z w with the group set {{i } × I5 × Z w : i ∈ Z 5 } and hole set Z 5 ×I2 ×Z w . Only initial base blocks are listed below. {(0, 0, 0), (1, 2, 0), (2, 2, 0)}, {(0, 0, 0), (3, 2, 0), (1, 3, 0)}, {(0, 0, 0), (4, 2, 0), (3, 3, 0)}, {(0, 0, 0), (2, 3, 0), (3, 4, 0)}, {(0, 0, 0), (4, 3, 0), (1, 3, 1)}, {(0, 0, 0), (1, 4, 0), (4, 4, 0)}, {(0, 0, 0), (2, 4, 0), (3, 4, 1)}, {(0, 0, 0), (2, 2, 1), (4, 4, 1)}, {(0, 0, 0), (3, 3, 1), (4, 3, 1)}, {(0, 1, 0), (1, 2, 0), (4, 3, 1)}, {(0, 1, 0), (2, 2, 0), (3, 2, 1)}, {(0, 2, 0), (2, 3, 0), (4, 4, 1)}, {(0, 1, 0), (4, 2, 0), (1, 4, 1)}, {(0, 1, 0), (1, 4, 0), (4, 4, 1)}, {(0, 1, 0), (2, 3, 0), (3, 4, 1)}, {(0, 1, 0), (1, 3, 0), (4, 2, w/2)}, {(0, 1, 0), (4, 3, 0), (2, 4, 1)}, {(0, 2, 0), (1, 4, 0), (3, 2, w − 1)}, {(0, 2, 0), (4, 3, 1), (1, 4, 1)}, {(0, 1, 0), (3, 3, 0), (2, 2, w/2)}, {(0, 1, 0), (3, 4, 0), (4, 4, 0)}, {(0, 3, 0), (3, 4, 0), (4, 3, w − 1)}, {(0, 1, 0), (3, 2, 0), (2, 4, 0)}, {(0, 0, 0), (1, 2, w/2), (3, 2, w/2)}, {(0, 1, 0), (1, 2, w/2), (2, 3, w/2)}, {(0, 1, 0), (1, 3, w/2), (3, 3, w/2)}, {(0, 0, 0), (4, 2, w/2), (2, 4, w/2)}, {(0, 0, 0), (2, 3, w/2), (1, 4, w/2)}, {(0, 0, 0), (1, 2, i), (2, 2, 2i)}, {(0, 0, 0), (1, 2, i + w/2), (2, 2, 2i + 1)}, {(0, 0, 0), (3, 2, i), (1, 3, 2i)}, {(0, 0, 0), (3, 2, i + w/2), (1, 3, 2i + 1)}, {(0, 0, 0), (4, 2, i), (3, 3, 2i)}, {(0, 0, 0), (4, 2, i + w/2), (3, 3, 2i + 1)}, {(0, 0, 0), (2, 3, i), (4, 3, 2i)}, {(0, 0, 0), (2, 3, i + w/2), (4, 3, 2i + 1)}, {(0, 0, 0), (1, 4, i), (4, 4, 2i)}, {(0, 0, 0), (1, 4, i + w/2), (4, 4, 2i + 1)},

{0, 19, 31}, {0, 21, 39}, {0, 9, 17}, {0, 1, 3},

{(0, 0, 0), (2, 4, i), (3, 4, 2i)}, {(0, 0, 0), (2, 4, i + w/2), (3, 4, 2i + 1)},

{0, 23, 37}, {0, 27, 33}, {0, 7, 11}, {0, 13, 29}.  Let α = 1i=0 (i 2 + i · · · 38 + i ) be a permutation on I40 . Let G be the group generated by α. All blocks are obtained by developing above base blocks under the action of G.

{(0, 1, 0), (1, 2, i), (3, 2, 2i)}, {(0, 1, 0), (1, 2, i + w/2), (3, 2, 2i + 1)}, {(0, 1, 0), (2, 2, i), (3, 4, 2i)}, {(0, 1, 0), (2, 2, i + w/2), (3, 4, 2i + 1)}, {(0, 1, 0), (4, 2, i), (2, 4, 2i)}, {(0, 1, 0), (4, 2, i + w/2), (2, 4, 2i + 1)}, {(0, 1, 0), (1, 3, i), (4, 4, 2i)}, {(0, 1, 0), (1, 3, i + w/2), (4, 4, 2i + 1)},

WANG AND CHANG: COMBINATORIAL CONSTRUCTIONS OF OPTIMAL 3-D OOCs

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{(0, 1, 0), (2, 3, i), (1, 4, 2i)}, {(0, 1, 0), (2, 3, i + w/2), (1, 4, 2i + 1)},

{(0, 0, 0), (2, 5, i), (3, 6, 2i)}, {(0, 0, 0), (2, 5, i + w/2), (3, 6, 2i + 1)},

{(0, 1, 0), (3, 3, i), (4, 3, 2i)}, {(0, 1, 0), (3, 3, i + w/2), (4, 3, 2i + 1)},

{(0, 0, 0), (1, 2, i), (2, 3, 2i)}, {(0, 0, 0), (1, 2, i + w/2), (2, 3, 2i + 1)},

{(0, 2, 0), (1, 3, i), (2, 4, 2i)}, {(0, 2, 0), (1, 3, i + w/2), (2, 4, 2i + 1)},

{(0, 0, 0), (3, 4, i), (4, 5, 2i)}, {(0, 0, 0), (3, 4, i + w/2), (4, 5, 2i + 1)},

{(0, 2, 0), (2, 3, i), (4, 4, 2i)}, {(0, 2, 0), (2, 3, i + w/2), (4, 4, 2i + 1)}.

{(0, 0, 0), (2, 1, i), (4, 3, 2i)}, {(0, 0, 0), (2, 1, i + w/2), (4, 3, 2i + 1)},

Here, i ∈ ∅ when w = 2, w/2 + 1 ≤ i ≤ w − 1 when w ≥ 8. All base blocks are obtained by (+1, −, −) mod (5, −, −).  Lemma 6.7: Let w ≡ 2 (mod 6). Then there exists a w-cyclic 3-IGDD of type (vw, w)5 for each v ∈ {2, 4, 7} and (v, w) = (2, 2). Proof: The required designs are constructed on Z 5 × Iv × Z w with the group set {{i } × Iv × Z w : i ∈ Z 5 } and hole set Z 5 × {0} × Z w . Only initial base blocks are listed below. v = 2, w ≥ 8 : {(0, 0, 0), (1, 1, i), (4, 1, 2i)}, {(0, 0, 0), (1, 1, i + w/2), (4, 1, 2i + 1)}, {(0, 0, 0), (3, 1, i), (2, 1, 2i)}, {(0, 0, 0), (3, 1, i + w/2), (2, 1, 2i + 1)}, {(0, 0, 0), (3, 1, 0), (4, 1, 1)}, {(0, 0, 0), (2, 1, w − 2), (4, 1, w − 1)}, {(0, 0, 0), (4, 1, 0), (2, 1, 1)}, {(0, 0, 0), (4, 1, w − 2), (3, 1, w − 1)}, {(0, 0, 0), (1, 1, 0), (2, 1, 0)}, {(0, 0, 0), (3, 1, w/2 − 1), (1, 1, w − 1)}, {(0, 0, 0), (1, 1, w/2), (3, 1, w/2)}, {(0, 0, 0), (1, 1, w/2−1), (2, 1, w − 1)}. v = 4: {(0, 0, 0), (1, 1, 0), (2, 2, 0)}, {(0, 0, 0), (3, 3, 0), (2, 3, 0)}, {(2, 2, 0), (4, 3, 0), (0, 2, 1)}, {(3, 3, 0), (1, 3, 0), (0, 3, 1)}, {(2, 2, 0), (3, 3, 0), (1, 2, 1)}, {(0, 0, 0), (1, 2, 0), (2, 3, w/2)}, {(0, 0, 0), (2, 2, 1), (3, 2, 1)}, {(0, 0, 0), (1, 2, 1), (4, 3, 1)}, {(1, 1, 0), (4, 2, 0), (2, 3, 1)}, {(0, 0, 0), (3, 2, 0), (4, 1, w/2)},

{(0, 0, 0), (3, 2, i), (1, 5, 2i)}, {(0, 0, 0), (3, 2, i + w/2), (1, 5, 2i + 1)}, {(0, 0, 0), (2, 6, i), (1, 3, 2i)}, {(0, 0, 0), (2, 6, i + w/2), (1, 3, 2i + 1)}, {(0, 0, 0), (4, 1, i), (3, 5, 2i)}, {(0, 0, 0), (4, 1, i + w/2), (3, 5, 2i + 1)}, {(0, 0, 0), (2, 4, i), (4, 6, 2i)}, {(0, 0, 0), (2, 4, i + w/2), (4, 6, 2i + 1)}, {(1, 1, 0), (4, 4, i), (3, 1, 2i)}, {(1, 1, 0), (4, 4, i + w/2), (3, 1, 2i + 1)}, {(1, 1, 0), (4, 2, i), (3, 6, 2i)}, {(1, 1, 0), (4, 2, i + w/2), (3, 6, 2i + 1)}, {(1, 1, 0), (0, 3, i), (2, 5, 2i)}, {(1, 1, 0), (0, 3, i + w/2), (2, 5, 2i + 1)}, {(1, 1, 0), (0, 1, i), (3, 5, 2i)}, {(1, 1, 0), (0, 1, i + w/2), (3, 5, 2i + 1)}, {(1, 1, 0), (2, 3, i), (4, 3, 2i)}, {(1, 1, 0), (2, 3, i + w/2), (4, 3, 2i + 1)}, {(1, 1, 0), (3, 4, i), (0, 4, 2i)}, {(1, 1, 0), (3, 4, i + w/2), (0, 4, 2i + 1)}, {(1, 1, 0), (0, 6, i), (2, 6, 2i)}, {(1, 1, 0), (0, 6, i + w/2), (2, 6, 2i + 1)}, {(1, 1, 0), (3, 2, i), (0, 2, 2i)}, {(1, 1, 0), (3, 2, i + w/2), (0, 2, 2i + 1)}, {(2, 2, 0), (0, 3, i), (3, 6, 2i)}, {(2, 2, 0), (0, 3, i + w/2), (3, 6, 2i + 1)}, {(2, 2, 0), (1, 4, i), (3, 2, 2i)}, {(2, 2, 0), (1, 4, i + w/2), (3, 2, 2i + 1)}, {(2, 2, 0), (3, 4, i), (4, 3, 2i)}, {(2, 2, 0), (3, 4, i + w/2), (4, 3, 2i + 1)}, {(2, 2, 0), (4, 5, i), (3, 5, 2i)}, {(2, 2, 0), (4, 5, i + w/2), (3, 5, 2i + 1)}, {(2, 2, 0), (0, 6, i), (1, 3, 2i)}, {(2, 2, 0), (0, 6, i + w/2), (1, 3, 2i + 1)}, {(2, 2, 0), (1, 5, i), (4, 6, 2i)}, {(2, 2, 0), (1, 5, i + w/2), (4, 6, 2i + 1)}, {(3, 3, 0), (1, 4, i), (4, 3, 2i)}, {(3, 3, 0), (1, 4, i + w/2), (4, 3, 2i + 1)}, {(3, 3, 0), (2, 5, i), (4, 5, 2i)}, {(3, 3, 0), (2, 5, i + w/2), (4, 5, 2i + 1)}, {(3, 3, 0), (0, 6, i), (1, 5, 2i)}, {(3, 3, 0), (0, 6, i + w/2), (1, 5, 2i + 1)}, {(4, 4, 0), (2, 5, i), (0, 4, 2i)}, {(4, 4, 0), (2, 5, i + w/2), (0, 4, 2i + 1)},

{(1, 1, 0), (2, 2, 1), (4, 2, 1)}, {(1, 1, 0), (4, 3, 0), (2, 1, w/2)},

{(4, 4, 0), (3, 6, i), (2, 6, 2i)}, {(4, 4, 0), (3, 6, i + w/2), (2, 6, 2i + 1)},

{(1, 1, 0), (0, 1, 0), (4, 3, 1)}, {(1, 1, 0), (0, 2, 0), (3, 1, w − 1)},

{(4, 4, 0), (0, 6, i), (3, 5, 2i)}, {(4, 4, 0), (0, 6, i + w/2), (3, 5, 2i + 1)},

{(0, 0, 0), (1, 3, 1), (2, 1, 1)}, {(0, 0, 0), (3, 3, w/2), (4, 2, w/2)},

{(2, 2, 0), (1, 2, 0), (3, 2, 1)}, {(2, 2, 0), (0, 5, 1), (3, 5, 1)},

{(0, 0, 0), (4, 1, 0), (1, 3, 0)}, {(0, 0, 0), (3, 1, 0), (4, 3, 0)},

{(4, 4, 0), (0, 5, 1), (1, 6, 1)}, {(1, 1, 0), (4, 2, 0), (2, 6, 0)},

{(0, 0, 0), (4, 2, 0), (2, 1, 0)}, {(0, 0, 0), (1, 1, w/2), (3, 1, w/2)},

{(2, 2, 0), (0, 3, 0), (1, 3, 1)}, {(0, 0, 0), (4, 4, 0), (3, 1, 0)},

{(0, 0, 0), (1, 1, i), (2, 2, 2i)}, {(0, 0, 0), (1, 1, i + w/2), (2, 2, 2i + 1)},

{(4, 4, 0), (3, 4, 0), (0, 4, 1)}, {(1, 1, 0), (0, 4, 0), (2, 2, 1)},

{(0, 0, 0), (3, 3, i), (1, 2, 2i)}, {(0, 0, 0), (3, 3, i + w/2), (1, 2, 2i + 1)},

{(3, 3, 0), (1, 6, 1), (4, 5, 1)}, {(0, 0, 0), (4, 5, 0), (2, 1, 0)},

{(0, 0, 0), (2, 3, i), (1, 3, 2i)}, {(0, 0, 0), (2, 3, i + w/2), (1, 3, 2i + 1)},

{(0, 0, 0), (1, 5, 1), (4, 6, 1)}, {(0, 0, 0), (3, 6, 1), (4, 5, 1)},

{(0, 0, 0), (4, 1, i), (2, 1, 2i)}, {(0, 0, 0), (4, 1, i + w/2), (2, 1, 2i + 1)},

{(0, 0, 0), (4, 4, 1), (1, 4, 1)}, {(2, 2, 0), (4, 2, 0), (1, 3, 0)},

{(0, 0, 0), (3, 1, i), (4, 3, 2i)}, {(0, 0, 0), (3, 1, i + w/2), (4, 3, 2i + 1)},

{(0, 0, 0), (2, 2, 1), (3, 1, 1)}, {(0, 0, 0), (3, 3, 0), (1, 6, 0)},

{(0, 0, 0), (4, 2, i), (3, 2, 2i)}, {(0, 0, 0), (4, 2, i + w/2), (3, 2, 2i + 1)},

{(2, 2, 0), (1, 6, 0), (3, 6, 1)}, {(2, 2, 0), (0, 4, 0), (3, 3, 1)},

{(1, 1, 0), (3, 3, i), (4, 2, 2i)}, {(1, 1, 0), (3, 3, i + w/2), (4, 2, 2i + 1)},

{(1, 1, 0), (4, 4, 0), (2, 5, 0)}, {(3, 3, 0), (1, 4, 0), (0, 6, w/2)},

{(1, 1, 0), (0, 1, i), (4, 3, 2i)}, {(1, 1, 0), (0, 1, i + w/2), (4, 3, 2i + 1)},

{(1, 1, 0), (0, 5, 0), (2, 3, 0)}, {(1, 1, 0), (4, 3, 0), (0, 1, w/2)},

{(1, 1, 0), (0, 2, i), (3, 2, 2i)}, {(1, 1, 0), (0, 2, i + w/2), (3, 2, 2i + 1)},

{(2, 2, 0), (1, 5, 0), (4, 6, 1)}, {(1, 1, 0), (0, 1, 0), (3, 2, w/2)},

{(2, 2, 0), (3, 3, i), (0, 3, 2i)}, {(2, 2, 0), (3, 3, i + w/2), (0, 3, 2i + 1)}.

{(0, 0, 0), (2, 3, 1), (1, 3, 1)}, {(3, 3, 0), (2, 6, 0), (1, 4, w/2)},

v = 7:

{(0, 0, 0), (4, 3, 1), (3, 5, 1)}, {(2, 2, 0), (3, 4, 0), (1, 5, w/2)},

{(0, 0, 0), (1, 1, i), (2, 2, 2i)}, {(0, 0, 0), (1, 1, i + w/2), (2, 2, 2i + 1)},

{(1, 1, 0), (0, 3, 0), (2, 5, 1)}, {(2, 2, 0), (4, 5, 0), (0, 3, w/2)},

{(0, 0, 0), (3, 3, i), (4, 4, 2i)}, {(0, 0, 0), (3, 3, i + w/2), (4, 4, 2i + 1)},

{(1, 1, 0), (3, 4, 0), (4, 3, 1)}, {(2, 2, 0), (1, 4, w/2), (0, 6, w/2)},

{(0, 0, 0), (1, 6, i), (3, 1, 2i)}, {(0, 0, 0), (1, 6, i + w/2), (3, 1, 2i + 1)},

{(1, 1, 0), (4, 5, 0), (3, 5, 1)}, {(4, 4, 0), (2, 6, 0), (1, 5, w − 1)},

{(0, 0, 0), (4, 2, i), (1, 4, 2i)}, {(0, 0, 0), (4, 2, i + w/2), (1, 4, 2i + 1)},

{(0, 0, 0), (1, 1, 0), (2, 2, 0)}, {(0, 0, 0), (1, 1, w/2), (3, 3, w/2)},

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 1, JANUARY 2015

{(0, 0, 0), (4, 2, 0), (1, 4, 0)}, {(0, 0, 0), (1, 6, w/2), (4, 2, w/2)}, {(0, 0, 0), (3, 6, 0), (2, 3, 0)}, {(1, 1, 0), (3, 2, 0), (4, 1, w − 1)}, {(0, 0, 0), (3, 4, 0), (4, 3, 0)}, {(0, 0, 0), (1, 2, w/2), (2, 6, w/2)}, {(0, 0, 0), (2, 6, 0), (4, 1, 0)}, {(0, 0, 0), (2, 1, w/2), (4, 1, w/2)}, {(0, 0, 0), (1, 3, 0), (3, 5, 0)}, {(3, 3, 0), (4, 5, 0), (1, 3, w − 1)}, {(0, 0, 0), (2, 5, 0), (1, 2, 0)}, {(0, 0, 0), (2, 5, w/2), (3, 4, w/2)}, {(0, 0, 0), (3, 2, 0), (1, 5, 0)}, {(0, 0, 0), (3, 2, w/2), (2, 4, w/2)}, {(0, 0, 0), (2, 4, 0), (4, 6, 0)}, {(3, 3, 0), (0, 3, 0), (4, 6, w − 1)}, {(1, 1, 0), (3, 3, 1), (0, 4, 1)}, {(1, 1, 0), (4, 4, w/2), (0, 6, w/2)}, {(2, 2, 0), (3, 3, 0), (4, 4, 1)}, {(1, 1, 0), (3, 6, 0), (4, 6, w − 1)}, {(1, 1, 0), (3, 6, 1), (2, 6, 1)}, {(1, 1, 0), (2, 3, w/2), (3, 4, w/2)}, {(2, 2, 0), (1, 6, 1), (4, 3, 1)}, {(2, 2, 0), (3, 4, w/2), (4, 5, w/2)}, {(4, 4, 0), (1, 5, 0), (3, 5, 1)}, {(1, 1, 0), (0, 6, 0), (2, 4, w − 1)}, {(1, 1, 0), (0, 5, 1), (4, 5, 1)}, {(0, 5, 0), (2, 6, w − 1), (4, 6, w − 1)}.

Here, w/2 + 1 ≤ i ≤ w − 2 when v = 2 and w ≥ 8. For v ∈ {4, 7}, i ∈ ∅ when w = 2, w/2 + 1 ≤ i ≤ w − 1 when w ≥ 8. All base blocks are obtained by (+1, −, −) mod (5, −, −).  A PPENDIX B Lemma 6.8: There exists a 2-cyclic 3-IGDD of type 2(17,5). Proof: The required design is constructed on I34 with group set {{i, 12+i } : 0 ≤ i ≤ 11}∪{{ j, j +5} : 24 ≤ j ≤ 28} and hole set {x : x ≥ 24, x ∈ I34 }. The base blocks are listed as follows: {0, 10, 25}, {0, 11, 26}, {0, 13, 27}, {0, 1, 2}, {2, 6, 24}, {0, 15, 29}, {0, 17, 30}, {0, 18, 31}, {0, 3, 4}, {0, 9, 24}, {0, 19, 32}, {0, 21, 33}, {1, 11, 25}, {0, 5, 7}, {1, 5, 10}, {1, 15, 26}, {1, 18, 27}, {1, 19, 29}, {0, 6, 8}, {2, 3, 10}, {2, 11, 30}, {2, 23, 32}, {0, 14, 28}, {1, 7, 9}{3, 7, 28}, {1, 14, 31}, {1, 22, 33}, {2, 7, 15}.   Let α = 3i=0 (i 4 + i · · · 20 + i ) 5i=0 (24 + i 29 + i ) be a permutation on I34 . Let G be the group generated by α. All blocks are obtained by developing the base blocks under the action of G. Clearly, this design is isomorphic to a 2-cyclic  3-IGDD of type 2(17,5). Lemma 6.9: There exists a 2-cyclic 3-IGDD of type 2(u,u/3) for each u ∈ {18, 30}. Proof: The desired designs are constructed on I2u . The group set is {{i, u + i } : 0 ≤ i ≤ u − 1}, and the hole set is {x : x ≡ 0 (mod 3), x ∈ I2u }. The base blocks are listed as follows: u = 18 : {0, 10, 13}, {0, 11, 14}, {0, 16, 19}, {0, 17, 20}, {0, 1, 2}, {0, 22, 25}, {1, 14, 15}, {2, 14, 21}, {5, 15, 34}, {1, 3, 7}, {0, 29, 34}, {0, 31, 35}, {1, 10, 20}, {1, 12, 13}, {0, 7, 8}, {1, 31, 33}, {3, 14, 25}, {4, 25, 34}, {2, 17, 25}, {0, 4, 5}, {1, 23, 25}, {1, 24, 29}, {1, 26, 32}, {1, 30, 35}, {1, 6, 8}, {2, 22, 31}, {2, 23, 32}, {2, 24, 34}, {3, 13, 23}, {2, 6, 13}, {3, 17, 26}, {4, 15, 26},

{4, 16, 24}, {4, 17, 32}, {2, 8, 15}, {5, 24, 25}, {0, 28, 32}, {1, 17, 22}, {2, 16, 26}, {2, 3, 11}, {3, 16, 35}, {0, 23, 26}, {1, 16, 21}, {5, 17, 33}, {3, 5, 22}, {2, 4, 12}. u = 30 : {0, 10, 14}, {0, 11, 13}, {0, 16, 22}, {0, 17, 23}, {0, 1, 2}, {0, 29, 46}, {0, 31, 47}, {0, 32, 50}, {0, 34, 49}, {0, 4, 7}, {0, 19, 26}, {0, 20, 25}, {0, 35, 55}, {0, 37, 58}, {0, 5, 8}, {0, 41, 53}, {0, 43, 56}, {0, 44, 59}, {1, 10, 32}, {1, 11, 37}, {1, 19, 38}, {1, 29, 50}, {2, 11, 35}, {0, 28, 40}, {0, 38, 52}, {1, 26, 34}.  For u = 18, let α = 8i=0 (i 9 + i 18 + i 27 + i ) be a  permutation on I36 . For u = 30, let α = 2i=0 (i 3+i · · · 57+i ) be a permutation on I60 . Let G be the group generated by α. All blocks are obtained by developing above base blocks under the action of G. Obviously, each of these designs is isomorphic  to a 2-cyclic 3-IGDD of type 2(u,u/3). Lemma 6.10: There exists a 6-cyclic 3-IGDD of type 6(11,3). Proof: The desired design is constructed on I66 with the group set {i, 11 + i, . . . , 55 + i : 0 ≤ i ≤ 10} and hole set {x : x ≤ 2 (mod 11), x ∈ I66 }. The base blocks are listed as follows: {0, 14, 17}, {0, 15, 18}, {0, 16, 20}, {0, 19, 25}, {2, 4, 17}, {0, 27, 32}, {0, 41, 48}, {2, 18, 32}, {0, 21, 26}, {1, 9, 15}, {0, 28, 36}, {0, 29, 37}, {0, 30, 38}, {0, 31, 39}, {2, 6, 15}, {0, 42, 49}, {0, 43, 50}, {0, 51, 60}, {0, 52, 61}, {2, 5, 19}, {0, 54, 63}, {0, 53, 62}, {1, 19, 29}, {0, 40, 47}, {3, 5, 37}, {0, 58, 64}, {0, 59, 65}, {1, 16, 26}, {1, 17, 27}, {2, 7, 20}, {1, 20, 25}, {1, 21, 31}, {1, 30, 42}, {1, 32, 36}, {4, 6, 29}, {1, 38, 50}, {1, 18, 28}, {1, 49, 62}, {2, 31, 52}, {2, 9, 21}, {1, 39, 51}, {1, 40, 52}, {1, 43, 61}, {1, 48, 60}, {4, 5, 39}, {1, 53, 63}, {1, 58, 65}, {1, 59, 64}, {2, 10, 16}, {5, 7, 32}, {2, 25, 38}, {2, 27, 42}, {2, 28, 47}, {2, 30, 53}, {7, 9, 43}, {2, 36, 59}, {2, 37, 51}, {2, 39, 60}, {2, 40, 64}, {0, 7, 10}, {2, 43, 58}, {1, 37, 47}, {1, 41, 54}, {3, 19, 51}, {5, 6, 40}, {2, 49, 65}, {2, 50, 63}, {2, 54, 62}, {3, 17, 43}, {1, 6, 10}, {3, 20, 48}, {3, 21, 40}, {3, 27, 52}, {3, 28, 42}, {1, 5, 14}, {3, 31, 49}, {4, 27, 53}, {2, 14, 26}, {6, 41, 43}, {6, 7, 42}, {3, 32, 53}, {3, 39, 41}, {4, 19, 40}, {4, 20, 43}, {1, 3, 7}, {4, 28, 52}, {4, 30, 54}, {4, 31, 50}, {4, 32, 49}, {0, 3, 4}, {5, 29, 52}, {6, 31, 51}, {2, 29, 48}, {4, 21, 41}, {0, 6, 9}, {5, 41, 42}, {8, 42, 43}, {2, 41, 61}, {5, 28, 43}, {0, 5, 8}, {3, 29, 30}, {3, 18, 38}, {1, 4, 8}, {2, 3, 8}.  Let α = 10 i=0 (i 11 + i . . . 55 + i ) be a permutation on I66 . Let G be the group generated by α. All blocks are obtained by developing above base blocks under the action of G. Obviously, this design is isomorphic to a 6-cyclic 3-IGDD  of type 6(11,3). ACKNOWLEDGMENT The authors would like to thank the anonymous referees and Prof. Tor Helleseth, the Associate Editor for Sequences, for their helpful comments and valuable suggestions to improve the readability of this paper.

WANG AND CHANG: COMBINATORIAL CONSTRUCTIONS OF OPTIMAL 3-D OOCs

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Lidong Wang was born in Hebei, China, in 1981. He received the B.S. degree and the M.S. degree from Hebei Normal University, China, in 2005 and 2008 respectively. He is currently a Ph.D. student at Beijing Jiaotong University, China. His research interests include Combinatorial Design Theory and Coding Theory. Since 2009, he has been a Lecturer at the Department of Basic Courses, Chinese People’s Armed Police Force Academy, Langfang, China.

Yanxun Chang was born in Hebei, China, in 1962. He received the B.S. degree and the M.S. degree from Hebei Normal University, China, in 1985 and 1988 respectively, and the Ph.D. degree from Suzhou University, China, in 1995, all in mathematics. From 1988 to 1995, he was with Hebei Normal University as a TeachingResearch Assistant (1988-1990), a Lecturer (1990-1992) and then an Associate Professor (1993-1995). From 1996 to 1997, he was an Associate Professor at the Department of Mathematics, Beijing Jiaotong University, China. Since 1998, he has been a Full Professor at the Department of Mathematics, Beijing Jiaotong University, China. His research interests include Combinatorial Design Theory, Coding Theory, Cryptography, and their interactions. Dr. Chang received several national and provincial awards from China.