IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 61, NO. 2, FEBRUARY 2015
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(m, n, 3, 1) Optical Orthogonal Signature Pattern Codes With Maximum Possible Size Rong Pan and Yanxun Chang
Abstract— Kitayama proposed a novel code-division multipleaccess (CDMA) network for image transmission called spatial CDMA. Optical orthogonal signature pattern codes (OOSPCs) have attracted wide attention as signature patterns of spatial CDMA. An (m, n, k, λ)-OOSPC is a set of m × n (0, 1)matrices with Hamming weight k and maximum correlation value λ. Let (m, n, k, λ) be the largest possible number of codewords among all (m, n, k, λ)-OOSPCs. In this paper, we concentrate on the calculation of the exact value of (m, n, 3, 1) and the construction of an (m, n, 3, 1)-OOSPC with (m, n, 3, 1) codewords. As a consequence, we show that (m, n, 3, 1) = mn−1 6 − 1 when mn ≡ 14, 20 (mod 24), or mn ≡ 8, 16 (mod 24) and gcd(m, n, 4) = 2, or mn ≡ 2 (mod 6) and gcd(m, n, 4) = 4, and (m, n, 3, 1) = mn−1 6 otherwise. Index Terms— Optical orthogonal signature pattern code, relative difference family, strictly G-invariant packing.
I. I NTRODUCTION
A
N OPTICAL orthogonal signature pattern code (OOSPC) is a family of (0, 1)-matrices with good autoand cross-correlation. Its study has been motivated by an application in a novel code-division multiple-access (CDMA) network for image transmission called spatial CDMA. The spatial CDMA network has promoted the development of high-speed multiple access network applications, especially image applications such as supercomputer visualizations, medical image access, and distribution and digital video broadcasting. Comparing with the traditional CDMA, the spatial CDMA provides higher throughput. For more details, interested readers may refer to [18], [19], and [28]. Let m, n, k and λ be positive integers. An (m, n, k, λ) optical orthogonal signature pattern code (briefly, (m, n, k, λ)-OOSPC) is a family C of m × n (0, 1)-matrices (called codewords) of Hamming weight k satisfying the following two correlation properties: Manuscript received December 22, 2013; revised December 1, 2014; accepted December 2, 2014. Date of publication December 18, 2014; date of current version January 16, 2015. Y. Chang was supported by the NSFC under Grant 11431003. R. Pan is with the Department of Mathematics, Beijing Jiaotong University, Beijing 100044, 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 K. Yang, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2014.2381259
(1) The auto-correlation property: For any (ai j ) ∈ C and every (s, t) ∈ Z m × Z n \ {(0, 0)}, n−1 m−1
ai, j ai⊕s, j ⊕ t ≤ λ;
i=0 j =0
(2) The cross-correlation property: For any distinct (ai j ), (bi j ) ∈ C and every (s, t) ∈ Z m × Z n , n−1 m−1
ai, j bi⊕s, j ⊕ t ≤ λ,
i=0 j =0
where Z l is the additive group of the residual-class ring of are, respectively, integers module l. The additions ⊕ and ⊕ reduced modulo m and n. For any (0, 1)-matrix A = (ai j ) ∈ C, whose rows are indexed by Z m and columns are indexed by Z n , we define X A = {(i, j ) ∈ Z m × Z n : ai j = 1}. Then, F = {X A : A ∈ C} is a set-theoretic representation of the (m, n, k, λ)-OOSPC. Conversely, for any subset X ∈ Z m × Z n , we construct a (0, 1)-matrix A = (ai j ) such that ai j = 1 if and only if (i, j ) ∈ X, where the weight of A equals |X|. Let F be a set of k-subsets of Z m × Z n . We say that F is an (m, n, k, λ)-OOSPC if the following two correlation properties are satisfied: (1’) The auto-correlation property: For any X ∈ F and every (s, t) ∈ Z m × Z n \ {(0, 0)}, |X ∩ (X + (s, t))| ≤ λ; (2’) The cross-correlation property: For any distinct X, Y ∈ F and every (s, t) ∈ Z m × Z n , |X ∩ (Y + (s, t))| ≤ λ, where the addition “+” performs in Z m × Z n . Throughout this paper, we shall always use the set-theoretic notation to list the codewords of a given OOSPC. The number of codewords in an OOSPC is called the size of the OOSPC. For given integers m, n, k and λ, let (m, n, k, λ) be the largest possible size among all (m, n, k, λ)-OOSPCs. An (m, n, k, λ)-OOSPC with size (m, n, k, λ) is said to be maximum. Research on OOSPCs mainly focuses on the calculation of the exact value of (m, n, k, λ) and the construction of an (m, n, k, λ)-OOSPC with maximum size (m, n, k, λ). The most general upper bound on (m, n, k, λ), which is presented on the basis of the Johnson bound [17] for constant weight codes, is (m, n, k, λ) ≤ J (mn, k, λ)
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(1)
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By extensive studies on these combinatorial structures, a wide class of (m, n, k, λ)-OOSPCs can be derived and we will quote J (mn, k, λ) = ··· ··· partially known results when we need. k k−1 k −2 k −λ Now, we give the definition of packings. and x denotes the largest integer not exceeding x. Let v, k and t be positive integers such that t < k < v. When m and n are coprime, it has been shown in [28] that A t-(v, k, 1) packi ng is a pair D = (V, B) where V is a an (m, n, k, λ)-OOSPC is actually a 1-dimensional (mn, k, λ) finite set of v points and B is a collection of k-subsets (called optical orthogonal code (briefly, 1-D (mn, k, λ)-OOC). blocks) of V such that every t-subset of V appears in at most For the formal definition of 1-D OOCs, the reader can refer one block. to [10]. Similarly, a 1-D OOC is said to be maximum if there An automorphism of D is a permutation σ on V such that is no 1-D OOC of larger size with the same parameters. So far, σ keeps B invariant, i.e., B σ = {B σ = {x σ : x ∈ B} : many efforts have been done on constructing 1-D OOCs with maximum size and many results have been made, B ∈ B} = B. The collection of all automorphisms of D for instance, [1], [4]–[10], [12]–[16], [20], [21], [30], [31]. forms a group under composition, called the full automorphism On the other hand, when m and n are not coprime, very group, and any of its subgroups is called an automorphism little has been done about an (m, n, k, λ)-OOSPC with group of D. Throughout this paper when we say that D is maximum size. To our knowledge, the construction of an G-invariant, we mean that D admits G as an automorphism (m, n, k, λ)-OOSPC with maximum size J (mn, k, λ) has been group acting sharply transitively on the points. Suppose that D is G-invariant. For any block B ∈ B, the G-orbit of B settled for the following cases: is the set OrbG (B) of all distinct images of B under G, (1) [28] λ = 1 and m, n are primes such that m = n ≡ 1 that is, OrbG (B) = {B σ : σ ∈ G}. Evidently B can be (mod k(k − 1)); partitioned into some G-orbits. A set of base blocks of D (2) [26] λ = 1, k = 3 and m, n ≡ 1 (mod 2), except for m, is a complete set of representatives for the G-orbits of B. n ≡ 5(mod 6); Moreover, when we say that D is strictly G-invariant, it means (3) [3] λ = 1, k = 3, m = 3 and n ≡ 0 or 6 (mod 24); that D is G-invariant with all G-orbits of its blocks of full (4) [26] λ = 1, k = 3, m = 3 and n is a positive integer; length |G|. (5) [22] λ = 1, k = 4, gcd(m, 18) = 3 and n ≡ 0 As the terminology suggests, given all the base blocks (mod 12); of a strictly Z m × Z n -invariant t-(mn, k, 1) packing ε (6) [25] λ = 2, k = 4, m = 2 x and n = y, where ε ∈ {1, 2}, (t-(m × n, k, 1)-SP in short), we can obtain all its blocks and each factor of x, y is a prime less than 500000 and by successively adding (i, j ) to each base block, where congruent to 53 or 77 modulo 120 or belongs to S = (i, j ) ∈ Z m × Z n . Let A(m × n, k, t) be the largest pos{5, 13,17, 25, 29, 37, 41, 53, 61, 85, 89, 97, 101, 113, 137, sible number of base blocks among all t-(m × n, k, 1)-SPs. 149, 157, 169, 173, 193, 197, 229, 233, 289, 293, 317}. A t-(m × n, k, 1)-SP with A(m × n, k, t) base blocks is said to be maximum. It is obvious that A(m × n, k, t) ≤ A. Equivalent Description J (mn, k, t − 1). When t = 2, 2-(m × n, k, 1)-SP and A(m × An OOSPC is a particular type of 2-dimensional optical n, k, 2) are abbreviated as (m × n, k, 1)-SP and A(m × n, k) orthogonal codes (briefly 2-D OOCs) [29] and closely related respectively. to a 1-D OOC [10]. The following result was shown in [28]. Sawa [25] established the following relation between an Lemma 1.1 [28, Construction 1]: Suppose that there exists (m, n, k, λ)-OOSPC and a (λ + 1)-(m × n, k, 1)-SP. a 1-D (m, k, λ)-OOC with size N. Then for any integer Lemma 1.4 [25]: An (m, n, k, λ)-OOSPC with maximum factorization m = m 1 m 2 satisfying gcd(m 1 , m 2 ) = 1, there size (m, n, k, λ) is equivalent to a strictly Z m × Z n -invariant exists an (m 1 , m 2 , k, λ)-OOSPC with the same size N. (λ + 1)-(mn, k, 1) packing (i.e., (λ + 1)-(m × n, k, 1)-SP) with Remark 1.2: Similar to Lemma 1.1, for any integers A(m × n, k, λ + 1) base blocks. Furthermore, (m, n, k, λ) = m = m 1 m 2 and n = n 1 n 2 satisfying gcd(m 1 n 2 , m 2 n 1 ) = 1, A(m × n, k, λ + 1). an (m, n, k, λ)-OOSPC with size N can be regarded as an Suppose that B is the set of all the base blocks of (m 1 n 1 , m 2 n 2 , k, λ)-OOSPC with the same size N. an (m × n, k, 1)-SP. Without loss of generality we can Based on Lemma 1.1 and Remark 1.2, several known always assume that each base block B ∈ B contains results on 1-D OOCs and OOSPCs are restated as the element (0, 0), i.e., B is of the form {(x 0 , y0 ) = follows. (0, 0), (x 1 , y1 ), . . . , (x k−1 , yk−1 )}. Define Lemma 1.3: There exists an (m, n, 3, 1)-OOSPC with
B = {(x i , yi ) − (x j , y j ) : 0 ≤ i, j ≤ k − 1, i = j } J (mn, 3, 1) codewords for any positive integers m and n satisfying one of the following conditions: as the list differences of B. Then the list differences of B is (1) [1, Th. 1.5] gcd(m, n) = 1 and mn ≡ 14, 20 (mod 24);
B = B∈B B. Note that B covers each non-zero element (2) [26, Th. 3.1] m and n are odd except for m, in Z m × Z n at most once since the (m × n, k, 1)-SP is strictly Z m × Z n -invariant. Define the difference leave of B (briefly, n ≡ 5 (mod 6); DL(B)) as the set of all the non-zero elements in Z m × Z n (3) [26, Lemma 3.1] m = 3m 0 and gcd(m 0 , 3n) = 1. Moreover, an OOSPC is closely related to a certain com- which are not covered by B. B is said to be (s × t)-regular binatorial structure, called a packing. In a particular case, (briefly, (m × n, s × t, k, 1)-SP) if DL(B) along with {(0, 0)} it is nothing else than a relative difference family [2]. forms an additive subgroup S × T of Z m × Z n , where S and T 1 mn − 1 mn − 2
mn − λ
PAN AND CHANG: (m, n, 3, 1) OOSPC WITH MAXIMUM POSSIBLE SIZE
are, respectively, the additive subgroups of order s in Z m and order t in Z n . If an (m, n, k, 1)-OOSPC is equivalent to an (m × n, s × t, k, 1)-SP, then the OOSPC is said to be (s, t)-regular. Evidently, an (s, t)-regular (m, n, k, 1)-OOSPC consists of (mn−st)/(k(k−1)) codewords and all its codewords constitute a (Z m × Z n , ms Z m × nt Z n , k, 1) relative difference family. In particular, a (1 × 1)-regular (m, n, k, 1)-OOSPC is just an (mn, k, 1) difference family on Z m × Z n . The following is a variant of a known result on difference families in [27]. Lemma 1.5 [27, Th. 7]: Let p > 3 be a prime. Then there exists a (1, 1)-regular ( p, p, 3, 1)-OOSPC. B. Main Results In this paper, we are concerned about maximum (m, n, 3, 1)-OOSPCs and determine the exact value of (m, n, 3, 1) for any positive integers m and n. We are to prove the following main result of this paper. Theorem 1.6: Let m and n be positive integers. Then ⎧ J (mn, 3, 1) − 1, ⎪ ⎪ ⎪ ⎪ if mn ≡ 14, 20 (mod 24), ⎪ ⎪ ⎪ ⎪ or mn ≡ 8, 16 (mod 24) ⎨ and gcd(m, n, 4) = 2, (m, n, 3, 1) = ⎪ ⎪ or mn ≡ 2 (mod 6) ⎪ ⎪ ⎪ ⎪ and gcd(m, n, 4) = 4; ⎪ ⎪ ⎩ J (mn, 3, 1), otherwise. With a special attention, the case of k = 3 is the only situation for which the exact value of (m, n, k, 1) is completely determined. While the cases of k ≥ 4 have not been fully figured out. The rest of this paper is organized as follows. In Section II we improve the upper bound on (m, n, 3, 1) which is tighter than the well-known Johnson bound. In Section III an auxiliary design is introduced to quote a recursive construction for (m, n, k, 1)-OOSPCs with general k. In Section IV based on the new upper bound on (m, n, 3, 1), we completely solve the construction problem of (m, n, 3, 1)-OOSPCs with maximum size for any positive integers m and n by using direct constructions and the recursive construction which is given in Section III. Moreover, we give the proof of Theorem 1.6 in this section. Finally, Section V gives a brief conclusion. II. I MPROVED U PPER B OUND ON (m, n, 3, 1) In this section, we shall give a new upper bound on (m, n, 3, 1), which is tighter than the well-known Johnson bound. Lemma 2.1: Let mn ≡ 14, 20 (mod 24) and gcd(m, n, 2) = 1. Then (m, n, 3, 1) ≤ J (mn, 3, 1) − 1. Proof: Without loss of generality, we assume that m ≡ 0 (mod 2) and n ≡ 1 (mod 2). By the Johnson bound, (m, n, 3, 1) ≤ J (mn, 3, 1). Suppose that there exists an (m, n, 3, 1)-OOSPC with size J (mn, 3, 1) = (mn − 2)/6. Such an OOSPC corresponds to an (m × n, 2 × 1, 3, 1)-SP. Let B be the set of all the base blocks of the corresponding packing. Then DL(B) = {(m/2, 0)}. For each base block
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B = {(0, 0), (x 1 , y1 ), (x 2 , y2 )} ∈ B, define B = {0, x 1 (mod 2), x 2 (mod 2)}. Let B = {B : B ∈ B}. Then all the triples in B must be one of the following types: T1 = {0, 0, 0} and T2 = {0, 0, 1}. Moreover, B contains 1 exactly mn−2 or 2 mn times according to whether mn is congruent to 14 or 20 2 modulo 24. On the other hand, let x be the multiplicity of T2 in B. Then B covers 1 exactly 4x times. Hence, we have mn − 2 = 8x ≡ 0 (mod 8) when mn ≡ 14 (mod 24), and mn = 8x ≡ 0 (mod 8) when mn ≡ 20 (mod 24), which are impossible. Therefore, (m, n, 3, 1) ≤ J (mn, 3, 1) − 1. Lemma 2.2: Let b ≥ 2. Let mn ≡ 1 (mod 6) when b is odd, and mn ≡ 5 (mod 6) when b is even. Then (2m, 2b n, 3, 1) ≤ J (2b+1 mn, 3, 1) − 1. Proof: Now we have 2b+1 mn ≡ 4 (mod 6). Suppose that there exists a (2m, 2b n, 3, 1)-OOSPC with size J (2b+1 mn, 3, 1) = (2b mn − 2)/3 and let B be the set of all the base blocks of its corresponding (2m × 2b n, 3, 1)-SP. It is easy to know that DL(B) = {(m, 0), (0, 2b−1 n), (m, 2b−1 n)}. Similar to Lemma 2.1, let B = {B : B ∈ B}. Let x be the multiplicity of T2 in B. By calculating the multiplicity of 1 in B, we have 4x = 2b mn − 2. It is followed by 2x = 2b−1 mn − 1 ≡ 0 (mod 2) which is impossible when b ≥ 2. Therefore, (2m, 2b n, 3, 1) ≤ J (2b+1 mn, 3, 1) − 1. Lemma 2.3: Let a and b be positive integers. Let mn ≡ 1 (mod 6) when a+b is odd, and mn ≡ 5 (mod 6) when a + b is even. Then (2a m, 2b n, 3, 1) ≤ J (2a+b mn, 3, 1) − 1. Proof: Note that there are at least four elements (0, 0), (0, 2b−1 n), (2a−1 m, 0), (2a−1 m, 2b−1 n) not appearing in the list differences of a (2a m×2b n, 3, 1)-SP. By Lemma 1.4, we have (2a m, 2b n, 3, 1) = A(2a m × 2b n, 3) 2a+b mn − 4 ≤ = J (2a+b mn, 3, 1) − 1. 6 Then the conclusion follows. Indeed, an (m, n, 3, 1)-OOSPC can be viewed as an (n, m, 3, 1)-OOSPC. We therefore have (m, n, 3, 1) = (n, m, 3, 1) and get the following theorem by combining the Johnson bound with Lemmas 2.1-2.3. Theorem 2.4: Let m and n be positive integers. Then ⎧ J (mn, 3, 1) − 1, ⎪ ⎪ ⎪ ⎪ if mn ≡ 14, 20 (mod 24), ⎪ ⎪ ⎪ ⎪ or mn ≡ 8, 16 (mod 24) ⎨ and gcd(m, n, 4) = 2, (2) (m, n, 3, 1) ≤ ⎪ ⎪ or mn ≡ 2 (mod 6) ⎪ ⎪ ⎪ ⎪ and gcd(m, n, 4) = 4; ⎪ ⎪ ⎩ J (mn, 3, 1), otherwise. Proof: Based on the Johnson bound, we always have (m, n, 3, 1) ≤ J (mn, 3, 1). Without loss of generality, we assume that m = 2a m and n = 2b n , where 0 ≤ a ≤ b and m n ≡ 1 (mod 2). When mn ≡ 14, 20 (mod 24) and gcd(m, n, 2) = 1, the conclusion follows immediately from Lemma 2.1.
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When mn ≡ 20 (mod 24) and gcd(m, n, 2) = 2, which corresponds to a = b = 1 and m n ≡ 5 (mod 6), the conclusion follows from Lemma 2.3. When mn ≡ 8, 16 (mod 24) and gcd(m, n, 4) = 2, which corresponds to b > a = 1 and m n ≡ 1, 5 (mod 6), the conclusion follows from Lemmas 2.2 and 2.3. When mn ≡ 2 (mod 6) and gcd(m, n, 4) = 4, which corresponds to m n ≡ 1 (mod 6) and a + b ≡ 1 (mod 2), or m n ≡ 5 (mod 6) and a + b ≡ 0 (mod 2), the conclusion follows from Lemma 2.3. III. AUXILIARY C OMBINATORIAL T OOLS Recall the introduction in Section I-A, an (s, t)-regular (m, n, k, 1)-OOSPC is actually a (Z m × Z n , ms Z m × nt Z n , k, 1) relative difference family. In view of design theory, the following results are straightforward. Lemma 3.1: If 1 ≤ st ≤ k(k − 1), then there exists an (s, t)-regular (m, n, k, 1)-OOSPC (or an (m × n, s × t, k, 1)-SP) attaining the Johnson bound. Lemma 3.2 (Filling Construction): Suppose that there exists: (1) an (s, t)-regular (m, n, k, 1)-OOSPC with b1 codewords; (2) an (s, t, k, 1)-OOSPC with b2 codewords. Then there exists an (m, n, k, 1)-OOSPC with b1 + b2 codewords. Moreover, if a (g, h)-regular (s, t, k, 1)-OOSPC with b2 codewords is given, then a (g, h)-regular (m, n, k, 1)OOSPC with b1 + b2 codewords is obtained. In order to quote more recursive constructions for OOSPCs, we first introduce the notion of difference matrices. Let G be an additive group (not necessarily abelian) of order v. A (G, k, λ) difference matrix (briefly, (G, k, λ)-DM) is a k ×λv matrix D = (di j ) with entries from G such that for any distinct rows x and y, the multiset {dxi −d yi : 1 ≤ i ≤ λv} contains each element of G exactly λ times. If G = Z v , the difference matrix is called cyclic and denoted by (v, k, λ)-CDM. Difference matrices play an important role in our constructions of (m, n, 3, 1)-OOSPCs. Much work on difference matrices has been done (see [11]). Here we only list partially known results on difference matrices for later use. Lemma 3.3 [2, Th. 2.13]: Let G be a finite group and let k be the least prime factor dividing |G|. Then there exists a (G, k, 1)-DM. Lemma 3.4 [23, Lemma 3.6]: Let x and y be positive integers. Then there exists a (Z 2x × Z 2 y , 3, 1)-DM. In the rest of this paper, the (Z m × Z n , 3, 1)-DM is used many times, and its existence is guaranteed by Lemma 3.3 or Lemma 3.4. Hence, we will not repeatedly say that “the needed (Z m × Z n , 3, 1)-DM exists by Lemma 3.3 (or Lemma 3.4)”. The following result in [23], which is actually an immediate corollary of [2, Corollary 5.8], can be used to improve the previous results on (s, t)-regular (m, n, k, 1)-OOSPCs. Theorem 3.5: Let m, n, x and y be positive integers. Suppose that there exists: (1) an (s, t)-regular (m, n, k, 1)-OOSPC; (2) a (Z x × Z y , k, 1)-DM. Then there exists an (sx, t y)-regular (mx, ny, k, 1)-OOSPC.
Sketch of Proof: • Step 1: Start from an (s, t)-regular (m, n, k, 1)OOSPC C, which is constructed on Z m × Z n . Let D = (γi j )i=1,...,k; j =1,...,x y be a (Z x × Z y , k, 1)-DM, where each γi j = (ai j , bi j ) ∈ Z x × Z y . • Step 2: For any codeword C = {α1 , . . . , αk } ∈ C, let AC consist of the following codewords: {α1 + (a1 j m, b1 j n), . . . , αk + (akj m, bkj n)}
•
with j = 1, . . . , x y, where the addition “+” performs in Z mx × Z ny . Step 3: Take A= AC . C∈C
Then A forms the desired (sx, t y)-regular (mx, ny, k, 1)OOSPC on Z mx × Z ny . Furthermore, as a corollary of Theorem 3.5, we have the following result. Corollary 3.6: Let x and y be positive integers. If an (s, t)-regular (m, n, 3, 1)-OOSPC exists, then so does a (2x s, 2 y t)-regular (2x m, 2 y n, 3, 1)-OOSPC. IV. E XISTENCE S PECTRUM FOR M AXIMUM (m, n, 3, 1)-OOSPC S In this section, the main purpose is to construct an (m, n, 3, 1)-OOSPC attaining the bound in (2). The construction of a maximum (m, n, 3, 1)-OOSPC in the case of mn ≡ 0 (mod 2) is more difficult than that in the case of mn ≡ 1 (mod 2). To simplify the construction, we give some (2a 3α , 2b 3β , 3, 1)-OOSPCs firstly. Lemma 4.1: There exists an (s, t)-regular (m, n, 3, 1)OOSPC for each (m, n, s, t) ∈ {(4, 4, 2, 2), (4, 8, 2, 4), (8, 8, 4, 4), (8, 16, 4, 8)}. Proof: By computer search, there is a (2, 2)-regular (4, 4, 3, 1)-OOSPC with 2 codewords: {(0, 0), (0, 1), (1, 0)}, {(0, 0), (1, 1), (2, 3)}, and a (2, 4)-regular (4, 8, 3, 1)-OOSPC with 4 codewords: {(0, 0), (0, 1), (1, 0)}, {(0, 0), (0, 3), (1, 1)}, {(0, 0), (1, 2), (2, 5)}, {(0, 0), (1, 4), (2, 1)}. Moreover, start from such a (2, 2)-regular (4, 4, 3, 1)-OOSPC. Applying Corollary 3.6 with (x, y) = (1, 1) and (1, 2) respectively, we obtain a (4, 4)-regular (8, 8, 3, 1)-OOSPC and a (4, 8)-regular (8, 16, 3, 1)-OOSPC. Lemma 4.2: Let integers a, b and j satisfy one of the following conditions: (1) a = 0, b ≥ 3, b > j > 0 and b ≡ j (mod 2); (2) a = 1, b ≥ 4, b > j ≥ 2 and b ≡ j (mod 2). Then there exists a (2a , 2 j )-regular (2a , 2b , 3, 1)-OOSPC. Proof: For Condition (1), use induction on b−2 j . When b− j = 1 (i.e., j = b − 2), we construct the desired 2 (1, 2 j )-regular (1, 2b , 3, 1)-OOSPC on Z 2b which consists of the following codewords: {0, 2i + 1, 2b−1 − 2i − 1} with 0 ≤ i < 2b−3 . When b−2 j > 1, suppose that a
PAN AND CHANG: (m, n, 3, 1) OOSPC WITH MAXIMUM POSSIBLE SIZE
(1, 2 j +2 )-regular (1, 2b , 3, 1)-OOSPC and a (1, 2 j )-regular (1, 2 j +2 , 3, 1)-OOSPC exist. Then the desired (1, 2 j )-regular (1, 2b , 3, 1)-OOSPC is obtained by Lemma 3.2. For Condition (2), there is a (1, 2 j −1 )-regular (1, 2b−1 , 3, 1)-OOSPC from Condition (1). We therefore obtain the desired (2, 2 j )-regular (2, 2b , 3, 1)-OOSPC by applying Corollary 3.6 with (x, y) = (1, 1). Lemma 4.3: Let b ≥ a ≥ 2 and i ∈ {1, 2}. Then there exists a (2i , 2 j )-regular (2a , 2b , 3, 1)-OOSPC, where i + j ≡ a +b (mod 2) and j ∈ {1, 2} or {2, 3} according to whether i = 1 or 2. Proof: Consider the case of i = 2. If a ∈ {2, 3}, use induction on b. It is trivial when (a, b) = (2, 2), (2, 3). When (a, b) = (2, 4), there is a (1, 2)-regular (1, 8, 3, 1)-OOSPC from Lemma 4.2. We therefore obtain the desired (4, 4)-regular (4, 16, 3, 1)-OOSPC by applying Corollary 3.6 with (x, y) = (2, 1). When (a, b) = (3, 3), (3, 4), the conclusion holds by Lemma 4.1. When b > 4, start from a (1, 2b −2 )-regular (1, 2b−2 , 3, 1)OOSPC, which exists by Lemma 4.2, where b ∈ {3, 4} and b ≡ b (mod 2). Applying Corollary 3.6 with (x, y) = (a, 2), we obtain a (2a , 2b )-regular (2a , 2b , 3, 1)-OOSPC. Finally the conclusion holds by applying Lemma 3.2 with the conclusion of the case of a ∈ {2, 3} and b ∈ {3, 4}. If a ≥ 4, there is a (2a −1 , 1)-regular (2a−1 , 1, 3, 1)OOSPC, which is from Lemma 4.2, where a ∈ {2, 3} and a ≡ a (mod 2). Then the conclusion holds by applying Corollary 3.6 with (x, y) = (1, b) and Lemma 3.2 with the conclusions of the cases of a = 2, 3. For the case of i = 1, there is a (2, 2)-regular (4, 4, 3, 1)-OOSPC and a (2, 4)-regular (4, 8, 3, 1)OOSPC from Lemma 4.1. Hence, the conclusion holds by applying Lemma 3.2 with the assertion of the case of i = 2. Lemma 4.4: Let g ∈ {1, 2, 4}. There exists an (s, tg)regular (m, ng, 3, 1)-OOSPC for each (m, n, s, t) ∈ {(1, 27, 1, 3), (3, 9, 1, 3), (9, 3, 1, 3)}. Proof: All the codewords of the desired (s, tg)-regular (m, ng, 3, 1)-OOSPC are listed in Appendix A. Lemma 4.5: Let α ≥ 0 and β > 0. There exists an (s, t)-regular (3α u, 3β v, 3, 1)-OOSPC for the following types of parameters: (1) u = 1, v ∈ {1, 2, 4} and (s, t) belongs to {(1, 3v), (1, 9v), (3, 3v)}; (2) u = 2, v ∈ {2, 4, 8} and (s, t) belongs to {(2, 6), (2, 12), (2, 18), (2, 24), (2, 36)}. Proof: For Type (1), use induction on α. When α = 0, use induction on β. It is trivial for β = 1, 2. For β = 3, the conclusion follows from Lemma 4.4. For β ≥ 4, start from a (1, 3v)- or (1, 9v)-regular (1, 3β−1 v, 3, 1)OOSPC, which exists by the induction hypothesis. Applying Theorem 3.5 with (x, y) = (1, 3), we obtain a (1, 9v)or (1, 27v)-regular (1, 3β v, 3, 1)-OOSPC. Finally, the desired (1, 3v)- or (1, 9v)-regular (1, 3β v, 3, 1)-OOSPC is obtained by Lemma 3.2. When α ≥ 1, it is trivial for (α, β) = (1, 1). For (α, β) = (1, 2), (2, 1), the conclusion follows from Lemma 4.4. For the remaining cases, start from an
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(s, t)-regular (3α−1 , 3β v, 3, 1)-OOSPC, which exists by the induction hypothesis, where (s, t) belongs to {(1, 3v), (1, 9v), (3, 3v)}. Applying Theorem 3.5 with (x, y) = (3, 1) and Lemma 3.2 with the conclusion of the case of (α, β) ∈ {(1, 1), (1, 2), (2, 1)}, we then obtain the desired (s, t)-regular (3α u, 3β v, 3, 1)-OOSPC. For Type (2), it is trivial for (α, β) = (0, 1), or (α, β) = (0, 2) with v = 2, 4. For (α, β) = (0, 2) with v = 8, or (α, β) = (1, 1), all the codewords of the desired (s, t)regular (3α u, 3β v, 3, 1)-OOSPC are listed in Appendix B. For the remaining cases, there is an (s , t )-regular (3α , 3β , 3, 1)-OOSPC, whose existence is guaranteed by Type (1), where (s , t ) belongs to {(1, 3), (1, 9), (3, 3)}. Hence, we obtain the desired (s, t)-regular (3α u, 3β v, 3, 1)OOSPC by applying Theorem 3.5 with (x, y) = (u, v) and Lemma 3.2 with the conclusion of the case of (α, β) ∈ {(0, 1), (0, 2), (1, 1)}. Lemma 4.6: Let g ∈ {2, 4}. Let mn ≡ 3 (mod 6) and gcd(m, n) = 3i , where integer i ≥ 0. Then there exists an (m, gn, 3, 1)-OOSPC with J (gmn, 3, 1) codewords. Proof: Without loss of generality, set m = 3α m and n = 3β n , where β = 0 and gcd(m n , 6) = gcd(m , n ) = 1. Start from an (s, gt)-regular (3α , 3β g, 3, 1)-OOSPC, which is from Lemma 4.5, where (s, t) belongs to {(1, 3), (1, 9), (3, 3)}. Applying Theorem 3.5 with (x, y) = (m , n ), we obtain an (m s, gn t)-regular (m, gn, 3, 1)-OOSPC. Note that, for each (s, t) ∈ {(1, 3), (1, 9), (3, 3)}, there exists an (m s, gn t, 3, 1)-OOSPC with J (gm n st, 3, 1) codewords by Lemma 1.3 (1) and (3). Hence, the desired (m, gn, 3, 1)-OOSPC with J (gmn, 3, 1) codewords exists by Lemma 3.2. Next, according to the values of m and n, we distinguish four cases to consider the constructions of maximum (m, n, 3, 1)-OOSPCs.
A. mn ≡ 1 (mod 2) In this subsection, to solve the construction problem of an (m, n, 3, 1)-OOSPC attaining the bound in (2), our main task is to construct it in the cases of m, n ≡ 5 (mod 6). Lemma 4.7: Let m, n ≡ 5 (mod 6) be integers. Then there exists an (m, n, 3, 1)-OOSPC with J (mn, 3, 1) codewords. Proof: Set m = pm and n = qn , where p, q ≡ 5 (mod 6) are primes and m , n ≡ 1 (mod 6). If m = n = 1, i.e., both m and n are primes, by Lemma 1.3 (1) and Lemma 1.5, there exists a (1, 1)-regular (m, n, 3, 1)-OOSPC with J (mn, 3, 1) codewords. Otherwise, start from a (1, 1)-regular ( p, q, 3, 1)-OOSPC, which is from the first case. Applying Theorem 3.5 with (x, y) = (m , n ), we obtain an (m , n )-regular (m, n, 3, 1)-OOSPC. Since an (m , n , 3, 1)-OOSPC with J (m n , 3, 1) codewords exists by Lemma 1.3 (2), we obtain the desired (m, n, 3, 1)-OOSPC with J (mn, 3, 1) codewords by Lemma 3.2. Combining Lemma 1.3 (2) with Lemma 4.7, the following theorem is straightforward. Theorem 4.8: Let mn ≡ 1 (mod 2). Then there exists an (m, n, 3, 1)-OOSPC attaining the bound in (2).
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B. mn ≡ 0 (mod 2) and gcd(m, n, 2) = 1 In this subsection, we shall create an infinite family of maximum (m, n, 3, 1)-OOSPCs in which one of integers m and n is even and the another is odd. To achieve our purpose successfully, we first give some (s, t)-regular (m, n, 3, 1)-OOSPCs with special parameters by using direct constructions. Lemma 4.9: Let p > 3 be a prime. Then there exists a (1, v)-regular ( p, pv, 3, 1)-OOSPC for each v ∈ {2, 4, 8, 16}. Proof: We construct the desired (1, v)-regular ( p, pv, 3, 1)-OOSPC on (Fp2 , +) × Z v , which is isomorphic to Z p × Z pv , where (Fp2 , +) is the additive group of the finite field Fp2 . When v = 2, 4, let ω and ε be a primitive element and a cube primitive root of Fp2 respectively. Then the desired (1, v)-regular ( p, pv, 3, 1)-OOSPC consists of the following codewords: v=2:
{(0, 0), (1, 0), (ε + 1, 0)} · (ωi , 1), i ∈ {1, 3, . . . , ( p 2 − 7)/6}, {(0, 0), ((ω − 1)2 , 0), (ω(ω − 1), 1)} · (ω j , 1), j ∈ {0, 2, . . . , ( p 2 − 5)/2};
v =4:
{(0, 0), (1, 0), (ε + 1, 0)} · (ωi , 1), {(0, 0), (1, 1), (ε + 2, 2)} · (ωi , 1), {(0, 0), (ε, 1), (ε − 1, 2)} · (ωi , 1), {(0, 0), (ε2 , 1), (ε2 − ε, 2)} · (ωi , 1), i ∈ {0, 1 . . . , ( p 2 − 7)/6}.
When v = 8, 16, all the codewords of the desired (1, v)-regular ( p, pv, 3, 1)-OOSPC can be divided into two parts C1 and C2 , where C1 is the set of all the codewords of a (1, v4 )-regular ( p, v4p , 3, 1)-OOSPC on (Fp2 , +) × 4Z v which has just been constructed directly,
is {(0, 0), (s, 1), (2s, 3)} : s ∈ F or and C 2 \ {0} 2 p {(0, 0), (s, 1), (2s, 7)}, {(0, 0), (s, 2), (2s, 5)} : s ∈ Fp2 \ {0} according to whether v = 8 or 16. Next, applying Theorem 3.5 or Lemma 3.2 with Lemma 4.9, we give the main result of this subsection step by step. Lemma 4.10: Let d be a positive integer such that gcd(d, 6) = 1. Then there exists a (1, v)-regular (d, dv, 3, 1)OOSPC for each v ∈ {2, 4, 8, 16}. Proof: Let d = p1α1 p2α2 · · · prαr be the factorization of d, where pi > 3 is a prime and αi ≥ 1 is an integer for each i . For each prime pi , there is a (Z pi × Z pi , 3, 1)-DM from Lemma 3.3 and a (1, v)regular ( pi , pi v, 3, 1)-OOSPC from Lemma 4.9. Start from a (1, v)-regular ( p1 , p1 v, 3, 1)-OOSPC. Applying Theorem 3.5 with (x, y) = ( pi , pi ), we obtain a ( pi , pi v)-regular ( p1 pi , p1 pi v, 3, 1)-OOSPC. Then applying Lemma 3.2 with a (1, v)-regular ( pi , pi v, 3, 1)-OOSPC, we obtain a (1, v)-regular ( p1 pi , p1 pi v, 3, 1)-OOSPC. Hence, the desired (1, v)-regular (d, dv, 3, 1)-OOSPC can be obtained by repeating this process. Lemma 4.11: Let g ∈ {2, 4} and m 0 n 0 ≡ 1 (mod 2). Then there exists an (m 0 , gn 0 , 3, 1)-OOSPC attaining the bound in (2).
Proof: Let m 0 = dm and n 0 = dn , where gcd(d, 6) = 1, gcd(m , n ) = 3i and integer i ≥ 0. Start from a (1, g)regular (d, dg, 3, 1)-OOSPC, which exists by Lemma 4.10. Applying Theorem 3.5 with (x, y) = (m , n ), we obtain an (m , gn )-regular (m 0 , gn 0 , 3, 1)-OOSPC. Note that there exists an (m , gn , 3, 1)-OOSPC with J (gm n , 3, 1) − 1 codewords if m n ≡ 7 (mod 12) and g = 2, or m n ≡ 5 (mod 6) and g = 4 by the construction of a 1-D (n, 3, 1)-OOC in [10, Sec. IV-E], or with J (gm n , 3, 1) codewords otherwise by Lemma 4.6. Hence, by Lemma 3.2, we obtain the desired (m 0 , gn 0 , 3, 1)-OOSPC with J (gm 0 n 0 , 3, 1) − 1 codewords if m 0 n 0 ≡ 7 (mod 12) and g = 2, or m 0 n 0 ≡ 5 (mod 6) and g = 4, or with J (gm 0 n 0 , 3, 1) codewords otherwise. Theorem 4.12: Let mn ≡ 0 (mod 2) and gcd(m, n, 2) = 1. Then there exists an (m, n, 3, 1)-OOSPC attaining the bound in (2). Proof: Without loss of generality, we assume that m is odd and n = 2b n 0 where n 0 is odd and b ≥ 1. When b ∈ {1, 2}, the conclusion holds by Lemma 4.11. When b = 3, 4, we have two cases. Case 1: mn 0 ≡ 7 (mod 12) with b = 3, or mn 0 ≡ 5 (mod 6) with b = 4. Let m = dm and n 0 = dn , where gcd(m , n ) = 1. Start from a (1, 2b )-regular (d, 2b d, 3, 1)-OOSPC, which exists by Lemma 4.10. Applying Theorem 3.5 with (x, y) = (m , n ), we obtain an (m , 2b n )-regular (m, n, 3, 1)-OOSPC. Note that an (m , 2b n , 3, 1)-OOSPC with J (2b m n , 3, 1) codewords exists by Lemma 1.3 (1). We therefore obtain the desired (m, n, 3, 1)-OOSPC with J (mn, 3, 1) codewords by Lemma 3.2. Case 2: mn 0 ≡ 7 (mod 12) with b = 3, or mn 0 ≡ 5 (mod 6) with b = 4. Start from a (1, 2b−2 )-regular (1, 2b , 3, 1)-OOSPC, which exists by Lemma 4.2. We then obtain the desired (m, n, 3, 1)OOSPC with J (mn, 3, 1) codewords by applying Theorem 3.5 with (x, y) = (m, n 0 ) and applying Lemma 3.2 with Lemma 4.11. When b ≥ 5, start from a (1, 2i )-regular (1, 2b , 3, 1)OOSPC, which exists by Lemma 4.2, where i ∈ {3, 4} and i ≡ b (mod 2). Then the desired (m, n, 3, 1)-OOSPC with J (mn, 3, 1) codewords is obtained by applying Theorem 3.5 with (x, y) = (m, n 0 ) and Lemma 3.2 with the assertion of the case of b ∈ {3, 4}. C. mn ≡ 0 (mod 4) and gcd(m, n, 4) = 2 In this subsection, we discuss the construction of an (m, n, 3, 1)-OOSPC with maximum size for any even integers m and n such that gcd(m, n, 4) = 2. To our knowledge, very little has been done on maximum (m, n, 3, 1)-OOSPCs with m, n ≡ 0 (mod 2). Lemma 4.13: Let g ∈ {6, 12, 18, 24, 36}. Let m be an odd integer such that gcd(m, 3) = 1. Then there exists a (2, gm, 3, 1)-OOSPC with J (2gm, 3, 1) codewords. Proof: When m = 1, the desired (2, gm, 3, 1)OOSPC with J (2gm, 3, 1) codewords exists for each g ∈ {6, 12, 18, 24, 36}, which lists as follows.
PAN AND CHANG: (m, n, 3, 1) OOSPC WITH MAXIMUM POSSIBLE SIZE
t 6 12 18
24
36
Codewords {(0, 0), (0, 1), (1, 2)}; {(0, 0), (0, 1), (0, 3)}, {(0, 0), (0, 4), (1, 1)}, {(0, 0), (0, 5), (1, 7)}; {(0, 0), (0, 1), (0, 3)}, {(0, 0), (0, 4), (1, 1)}, {(0, 0), (0, 5), (1, 7)}, {(0, 0), (0, 6), (1, 10)}, {(0, 0), (0, 7), (1, 12)}; {(0, 0), (0, 1), (0, 3)}, {(0, 0), (0, 4), (0, 9)}, {(0, 0), (0, 6), (1, 1)}, {(0, 0), (0, 7), (1, 3)}, {(0, 0), (0, 8), (1, 15)}, {(0, 0), (0, 10), (1, 16)}, {(0, 0), (0, 11), (1, 13)}; {(0, 0), (0, 1), (1, 16)}, {(0, 0), (0, 2), (0, 6)}, {(0, 0), (0, 3), (0, 11)}, {(0, 0), (0, 5), (0, 12)}, {(0, 0), (0, 9), (1, 1)}, {(0, 0), (0, 10), (1, 4)}, {(0, 0), (0, 13), (1, 22)}, {(0, 0), (0, 14), (1, 3)}, {(0, 0), (0, 15), (1, 5)}, {(0, 0), (0, 16), (1, 29)}, {(0, 0), (0, 17), (1, 34)}.
When m > 1 and gcd(m, 6) = 1, there exists a (1, 3)regular (1, 3m, 3, 1)-OOSPC and a (1, 3)-regular (1, 9m, 3, 1)OOSPC which are equivalent to a cyclic STS(3m) and a cyclic STS(9m) in [24]. Applying Corollary 3.6 with appropriate (x, y), we can obtain a (2, t)-regular (2, gm, 3, 1)-OOSPC for each g ∈ {6, 12, 18, 24, 36}, where t belongs to {6, 12, 24}. Then, by applying Lemma 3.2 with the conclusion of the case of m = 1, we obtain the desired (2, gm, 3, 1)-OOSPC with J (2gm, 3, 1) codewords. Lemma 4.14: Let m ≡ 1, 5 (mod 6). Then there exists a (2m, 2, 3, 1)-OOSPC attaining the bound in (2). Proof: The desired (2m, 2, 3, 1)-OOSPC is constructed on Z m × Z 2 × Z 2 . Let C1 be the set of all the J (m, 3, 1) codewords of a maximum (m, 1, 3, 1)-OOSPC on Z m ×{0}×{0}, which exists by Lemma 1.3 (1). And let C2 consist of the following codewords: {(0, 0, 0), (i, 0, 1), (2i, 1, 0)} with i ∈ {1, 2, . . . , (m−1)/2}. It is readily checked that C1 ∪C2 forms the desired (2m, 2, 3, 1)-OOSPC, where |C1 ∪ C2 | = J (4m, 3, 1) or J (4m, 3, 1) − 1 according to whether m ≡ 1 or 5 (mod 6). Lemma 4.15: Let m ≡ 1, 5 (mod 6). Then there exists a (2m, 4, 3, 1)-OOSPC attaining the bound in (2). Proof: We construct the desired (2m, 4, 3, 1)-OOSPC with J (8m, 3, 1) − 1 codewords on Z m × Z 2 × Z 4 . Let N = {0, 2} be the additive subgroup of order 2 in Z 4 . When m ≡ 1 (mod 6), start from a (1, 1)-regular (m, 1, 3, 1)-OOSPC, which exists by Lemma 1.3 (1). Applying Corollary 3.6 with (x, y) = (1, 2), we obtain a (2, 4)-regular (2m, 4, 3, 1)-OOSPC. Since there is a trivial (2, 4, 3, 1)-OOSPC without codeword, we then obtain a (2m, 4, 3, 1)-OOSPC with J (8m, 3, 1) − 1 codewords by Lemma 3.2. When m ≡ 5 (mod 6), let C1 be the set of the J (2m, 3, 1) codewords of a maximum (m, 2, 3, 1)-OOSPC on Z m × {0} × N, which exists by Lemma 1.3 (1). And let C2 consist of the following codewords: {(0, 0, 0), (2i, 0, 1), (3i, 1, 2)}, {(0, 0, 0), (−2i, 0, 1), (−i, 1, 1)} with i ∈ {1, 2, . . . , (m − 1)/2}. It is readily checked that C1 ∪ C2 forms the desired (2m, 4, 3, 1)-OOSPC with J (8m, 3, 1) − 1 codewords.
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Lemma 4.16: Let m ≡ 1 (mod 6). Then there exists a (2, 8)-regular (2m, 8, 3, 1)-OOSPC. Moreover, there exists a (2m, 8, 3, 1)-OOSPC attaining the bound in (2). Proof: Start from a (1, 1)-regular (m, 1, 3, 1)-OOSPC, which exists by Lemma 1.3 (1). We then get the desired (2, 8)regular (2m, 8, 3, 1)-OOSPC by applying Corollary 3.6 with (x, y) = (1, 3). Moreover, there is a (2, 8, 3, 1)-OOSPC with 1 codeword: {(0, 0), (0, 1), (0, 3)}. Hence, we get a (2m, 8, 3, 1)-OOSPC with J (16m, 3, 1) − 1 codewords by Lemma 3.2. Lemma 4.17: Let m ≡ 5 (mod 6). Then there exists a (2m, 8, 3, 1)-OOSPC attaining the bound in (2). Proof: We first prove the existence of (2 p, 8, 3, 1)OOSPC with J (16 p, 3, 1) − 1 codewords by constructing its corresponding (2 p × 8, 3, 1)-SP with J (16 p, 3, 1) − 1 base blocks for any prime p ≡ 5 (mod 6). By Lemmas 1.3 (1) and 1.4, let B1 be the set of all the J ( p, 3, 1) base blocks of a maximum ( p × 1, 3, 1)-SP on Z p × {0} × {0} with {±(( p − 1)/2, 0, 0)} ⊆ DL(B1 ). And construct B2 on Z p × Z 2 × Z 8 as follows: {(0, 0, 0), (2i, 0, 1), (4i + 1, 0, 2)}, {(0, 0, 0), (4i − 1, 0, 2), (2i, 1, 1)}, {(0, 0, 0), (i, 0, 4), (2i, 1, 0)}, i ∈ {1, 2, 3, . . . , ( p − 1)/2}, {(0, 0, 0), ( j, 0, 3), (2 j, 1, 5)}, j ∈ {2, 3, . . . , p − 1} \ {( p − 1)/2, ( p + 1)/2}, {(0, 0, 0), (( p + 1)/2, 0, 0), (0, 0, 3)}, {(0, 0, 0), (( p + 1)/2, 0, 3), (0, 1, 5)}, {(0, 0, 0), (1, 0, 1), (1, 1, 2)}, {(0, 0, 0), (1, 0, 2), (2, 1, 5)}, {(0, 0, 0), (1, 0, 3), (1, 1, 5)}. It is readily verified that B1 ∪ B2 forms a (2 p × 8, 3, 1)-SP with J (16 p, 3, 1) − 1 base blocks. Next, set m = pm , where m ≡ 1 (mod 6) and p ≡ 5 (mod 6) is a prime. Start from a (2, 8)-regular (2m , 8, 3, 1)-OOSPC, which exists by Lemma 4.16. Applying Theorem 3.5 with (x, y) = ( p, 1), we have a (2 p, 8)regular (2m, 8, 3, 1)-OOSPC. Then applying Lemma 3.2 with the claim, we obtain the desired (2m, 8, 3, 1)-OOSPC with J (16m, 3, 1) − 1 codewords. Lemma 4.18: Let g ∈ {2, 4, 8} and mn ≡ 1 (mod 2). Then there exists a (2m, gn, 3, 1)-OOSPC attaining the bound in (2). Proof: Let m = 3α dm and n = 3β dn , where α, β ≥ 0 and gcd(m , n ) = gcd(dm n , 6) = 1. Start from a (1, 1)regular (d, d, 3, 1)-OOSPC, which exists by Theorem 4.8. When α = β = 0, applying Theorem 3.5 twice with (x, y) = (2, g) and (x, y) = (m , n ), we obtain a (2m , gn )regular (2m, gn, 3, 1)-OOSPC. Note that a (2m , gn , 3, 1)OOSPC attaining the bound in (2) is equivalent to a (2m n , g, 3, 1)-OOSPC attaining the bound in (2), which exists by Lemmas 4.14-4.17. We therefore obtain the desired (2m, gn, 3, 1)-OOSPC attaining the bound in (2) by Lemma 3.2.
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When (α, β) = (0, 0), without loss of generality, we can assume that β = 0. We first obtain a (2 · 3α , 3β g)regular (2 · 3α d, 3β dg, 3, 1)-OOSPC by applying Theorem 3.5 twice with (x, y) = (2, g) and (x, y) = (3α , 3β ). From Lemma 4.5, there is a (2, h)-regular (2·3α , 3β g, 3, 1)-OOSPC, where h belongs to {6, 12, 18, 24, 36}. Then, a (2, h)-regular (2 · 3α d, 3β dg, 3, 1)-OOSPC exists by Lemma 3.2. Next, applying Theorem 3.5 with (x, y) = (m , n ), we have a (2m , hn )-regular (2m, gn, 3, 1)-OOSPC. Finally, the desired (2m, gn, 3, 1)-OOSPC with J (2gmn, 3, 1) codewords can be obtained by applying Lemma 3.2 with Lemma 4.13. The following is a theorem in this subsection. Theorem 4.19: Let mn ≡ 0 (mod 4) and gcd(m, n, 4) = 2. Then there exists an (m, n, 3, 1)-OOSPC attaining the bound in (2). Proof: Without loss of generality, we can assume that m = 2m and n = 2b n , where m , n are odd and b ≥ 1. For each b ∈ {1, 2, 3}, the conclusion immediately follows from Lemma 4.18. For the case b ≥ 4, there is a (2, 2i )-regular (2, 2b , 3, 1)-OOSPC from Lemma 4.2, where i ∈ {2, 3} and i ≡ b (mod 2). We then obtain the desired (m, n, 3, 1)-OOSPC attaining the bound in (2) by applying Theorem 3.5 with (x, y) = (m , n ) and Lemma 3.2 with Lemma 4.18. D. m, n ≡ 0 (mod 4) By the bound in (2), we observe that a maximum (m 0 , n 0 , 3, 1)-OOSPC can not attain the Johnson bound for each (m 0 , n 0 ) ∈ {(4, 8), (1, 4 p), (2, 2 p), (2, 4 p)}, but a maximum (4, 8 p, 3, 1)-OOSPC can do that, where p ≡ 5 (mod 6) is a prime. Hence, in the subsection, we can not complete the existence of (m, n, 3, 1)OOSPCs attaining the bound in (2) readily by using Theorem 3.5 and the difficulty naturally concentrates on the construction of a (4, 8 p, 3, 1)-OOSPC with J (32 p, 3, 1) codewords. First of all, to obtain the desired (4, 8 p, 3, 1)-OOSPC, we introduce a property of a maximum (2 × p, 3, 1)-SP. Lemma 4.20: Let p ≡ 5 (mod 6) be a prime. Then there exists a (2 × p, 3, 1)-SP with J (2 p, 3, 1) base blocks whose difference leave contains {±(0, 1)} or {±(1, 1)} according to whether p ≡ 5 or 11 (mod 12). Proof: The existence of a (2× p, 3, 1)-SP with J (2 p, 3, 1) base blocks is guaranteed by Lemmas 1.3 (1) and 1.4. Denoted by B the set of all its base blocks, we just need to point out the structure of DL(B). It is readily checked that DL(B) must be {(1, 0), ±(i, α)}, where i = 0 or 1 and α = 0. Since there exists an element β ∈ Z p such that αβ = 1, without loss of generality we can assume that DL(B) = {(1, 0), ±(i, 1)}. Let x be the number of the base blocks of the form {(0, 0), (0, ∗), (1, ∗)} in B and k be the number of the elements in DL(B) which is of the form (1, ∗). We then have p − k = 4x by calculating the number of the differences of the form (1, ∗). It is easy to know that k must be 1 if p ≡ 5 (mod 12), or 3 if p ≡ 11 (mod 12). That is, DL(B) contains {±(0, 1)} if p ≡ 5 (mod 12), or {±(1, 1)} if p ≡ 11 (mod 12).
Next, we give a direct construction to create a (2, 2)-regular (4, 8 p, 3, 1)-OOSPC, which attains the bound in (2). Lemma 4.21: Let p ≡ 5 (mod 6) be a prime. Then there exists a (2, 2)-regular (4, 8 p, 3, 1)-OOSPC. Proof: We obtain the desired (2, 2)-regular (4, 8 p, 3, 1)OOSPC by constructing its corresponding (4×8 p, 2×2, 3, 1)SP on Z 4 × Z 8 × Z p . First, by Lemma 4.20, let B1 be the set of the ( p−2)/3 base blocks of a (2× p, 3, 1)-SP on {0}×{0, 4}× Z p with DL(B1 ) = {(0, 4, 0), ±(0, 0, 1)} if p ≡ 5 (mod 12), or DL(B1 ) = {(0, 4, 0), ±(0, 4, 1)} if p ≡ 11 (mod 12). Next, let B2 consist of {(0, 0, 0), (0, 0, 1), (1, 5, 0)}, {(0, 0, 0), (1, 1, 0), (2, 2, 1)}, {(0, 0, 0), (1, 2, −1), (2, 6, 1)} if p ≡ 5 (mod 12), or {(0, 0, 0), (0, 4, 1), (1, 1, 1)}, {(0, 0, 0), (1, 1, 0), (2, 6, −1)}, {(0, 0, 0), (1, 2, −1), (2, 6, 1)} if p ≡ 11 (mod 12). Finally, let B3 consist of {(0, 0, 0), (0, 1, 1), (1, 0, 2)} · (1, 1, s), {(0, 0, 0), (0, 2, 1), (2, 3, 2)} · (1, 1, s), {(0, 0, 0), (0, 3, 1), (1, 6, 2)} · (1, 1, s), s ∈ Z p , {(0, 0, 0), (1, 1, i ), (2, 6, 2i − 1)}, i ∈ Z p \ {0, 1}, {(0, 0, 0), (1, 2, j ), (2, 4, 2 j + 2)}, j ∈ {4t, 4t + 1 : t = 0, 1, . . . , ( p − 5)/4} i f p ≡ 5 (mod 12), j ∈ {4t + 1, 4t + 2 : t = 0, 1, . . . , ( p − 7)/4} ∪ { p − 2} i f p ≡ 11 (mod 12), {(0, 0, 0), (1, 4, k), (2, 0, 2k + 4)}, k ∈ {8t + a : t = 0, 1, . . . ( p − 13)/8, a = 1, 4, 6, 7} ∪ { p − 4, p − 1} i f p ≡ 5 (mod 24), k ∈ {8t + a : t = 0, 1, . . . , ( p − 9)/8, a = 3, 4, 5, 6} i f p ≡ 17 (mod 24), k ∈ {8t + a : t = 0, 1, . . . ( p − 11)/8, a = 0, 3, 5, 6} ∪ { p − 3} i f p ≡ 11 (mod 24), k ∈ {8t + a : t = 0, 1, . . . ( p − 7)/8, a = −1, 0, 1, 6} \ { p − 1} i f p ≡ 23 (mod 24). Then B1 ∪ B2 ∪ B3 forms the (4 × 8 p, 2 × 2, 3, 1)-SP. Finally, we establish the main theorem in this subsection. Theorem 4.22: Let m, n ≡ 0 (mod 4). Then there exists an (m, n, 3, 1)-OOSPC attaining the bound in (2). Proof: Without loss of generality, let m = 2a m and n = 2b n where m n ≡ 1 (mod 2) and b ≥ a ≥ 2. We then have two cases. Case 1: m n ≡ 5 (mod 6) and a + b ≡ 1 (mod 2). If m ≡ 1 (mod 6) and n ≡ 5 (mod 6), set n = pn 0 , where n 0 ≡ 1 (mod 6) and p ≡ 5 (mod 6) is a prime. Start from a (4, 8)-regular (2a , 2b , 3, 1)-OOSPC which exists by Lemma 4.3. We then obtain a (2, 2)-regular (2a , 2b p, 3, 1)OOSPC by applying Theorem 3.5 with (x, y) = (1, p) and
PAN AND CHANG: (m, n, 3, 1) OOSPC WITH MAXIMUM POSSIBLE SIZE
Lemma 3.2 with Lemma 4.21. Finally, applying Theorem 3.5 with (x, y) = (m , n 0 ) and Lemma 3.2 with Lemma 4.18, we obtain the desired (m, n, 3, 1)-OOSPC attaining the bound in (2). If m ≡ 5 (mod 6) and n ≡ 1 (mod 6), set m = pm 0 , where m 0 ≡ 1 (mod 6) and p ≡ 5 (mod 6) is a prime. The proof is similar to that of the first case. Case 2: m n ≡ 1, 3 (mod 6) and a + b ≡ 1 (mod 2), or m n ≡ 1 (mod 2) and a + b ≡ 0 (mod 2). Start from a (2, 2 j )-regular (2a , 2b , 3, 1)-OOSPC where j ∈ {1, 2} and j ≡ a + b − 1 (mod 2), which exists by Lemma 4.3. We then obtain a (2m , 2 j n )regular (m, n, 3, 1)-OOSPC by applying Theorem 3.5 with (x, y) = (m , n ). Finally, the desired (m, n, 3, 1)-OOSPC is obtained by Lemma 3.2, where the needed (2m , 2 j n , 3, 1)OOSPC attaining the bound in (2) is from Lemma 4.18 for each j ∈ {1, 2}. Proof of Theorem 1.6: Combining Theorems 4.8, 4.12, 4.19 and 4.22, we know that (m, n, 3, 1) attains the bound in (2). Hence, the conclusion follows.
V. C ONCLUSION In this paper, we have completely determined the exact value of the size of a maximum (m, n, 3, 1)-OOSPC (say (m, n, 3, 1)) for any positive integers m and n. In conclusion, we point out that (m, n, 3, 1) equals J (mn, 3, 1) − 1 if mn ≡ 14, 20 (mod 24), or mn ≡ 8, 16 (mod 24) and gcd(m, n, 4) = 2, or mn ≡ 2 (mod 6) and gcd(m, n, 4) = 4, or J (mn, 3, 1) otherwise. Now the reader may want to know the gap between the maximum sizes of 1-D OOCs and OOSPCs. Let (m, k, 1) be the largest possible size among all 1-D (m, k, 1)-OOCs by using the notation in [10]. From [1, Th. 1.5] and [10, Sec. IV-E], we know that (m, 3, 1) is m−1 6 −1 if m ≡ 14, 20 (mod 24), or m−1 otherwise. In comparison with the exact value of 6 (m, n, 3, 1) in Theorem 1.6, it is observed that the following formula is a gap between the maximum sizes of 1-D OOCs and OOSPCs. ⎧
(mn, 3, 1) − 1, ⎪ ⎪ ⎪ ⎪ ⎪ if mn ≡ 8, 16 (mod 24) ⎪ ⎪ ⎪ ⎨ and gcd(m, n, 4) = 2, (m, n, 3, 1) = ⎪ or mn ≡ 2 (mod 6) ⎪ ⎪ ⎪ ⎪ ⎪ and gcd(m, n, 4) = 4, ⎪ ⎪ ⎩
(mn, 3, 1), otherwise. On the other hand, a natural question is why we are concerned about maximum optical orthogonal signature pattern codes with weight 3. We know that research on OOSPCs mainly focuses on calculating the exact value of (m, n, k, λ) and constructing an (m, n, k, λ)-OOSPC with maximum size (m, n, k, λ). Generally speaking, calculating the exact value of (m, n, k, λ) is difficult and hitherto it has only been completely settled for k = 3 and λ = 1. Although the case of k = 3 is too small for practical application, it is significant in theory and it may help to study the other larger cases.
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A PPENDIX A Some (s, tg)-regular (m, ng, 3, 1)-OOSPCs of Lemma 4.4. g (m, n, s, t)
Codewords
{(0, 0), (0, 1), (0, 3)}, {(0, 0), (0, 5), (0, 15)}, {(0, 0), (0, 1), (1, 0)}, 1 (3, 9, 1, 3) {(0, 0), (0, 4), (1, 2)}, {(0, 0), (1, 0), (2, 2)}, (9, 3, 1, 3) {(0, 0), (2, 0), (5, 1)},
{(0, 0), (0, 4), (0, 11)}, {(0, 0), (0, 6), (0, 14)}; {(0, 0), (0, 2), (1, 6)}, {(0, 0), (1, 1), (2, 4)}; {(0, 0), (1, 1), (4, 1)}, {(0, 0), (2, 1), (5, 0)};
{(0, 0), (0, 1), (0, 3)}, {(0, 0), (0, 5), (0, 16)}, {(0, 0), (0, 8), (0, 25)}, {(0, 0), (0, 13), (0, 28)}, {(0, 0), (0, 1), (0, 5)}, {(0, 0), (0, 7), (1, 1)}, 2 (3, 9, 1, 3) {(0, 0), (1, 2), (2, 7)}, {(0, 0), (1, 7), (2, 3)}, {(0, 0), (1, 0), (2, 4)}, {(0, 0), (1, 2), (4, 0)}, (9, 3, 1, 3) {(0, 0), (2, 1), (5, 3)}, {(0, 0), (2, 3), (4, 2)},
{(0, 0), (0, 4), (0, 10)}, {(0, 0), (0, 7), (0, 30)}, {(0, 0), (0, 12), (0, 32)}, {(0, 0), (0, 14), (0, 35)}; {(0, 0), (0, 2), (1, 0)}, {(0, 0), (0, 8), (1, 3)}, {(0, 0), (1, 4), (2, 10)}, {(0, 0), (1, 9), (2, 1)}; {(0, 0), (1, 1), (2, 0)}, {(0, 0), (1, 3), (5, 1)}, {(0, 0), (2, 2), (5, 5)}, {(0, 0), (3, 0), (6, 1)};
{(0, 0), (0, 1), (0, 5)}, {(0, 0), (0, 3), (0, 59)}, {(0, 0), (0, 7), (0, 46)}, {(0, 0), (0, 10), (0, 38)}, (1, 27, 1, 3) {(0, 0), (0, 14), (0, 30)}, {(0, 0), (0, 21), (0, 43)}, {(0, 0), (0, 25), (0, 51)}, {(0, 0), (0, 32), (0, 66)}, {(0, 0), (0, 1), (0, 8)}, {(0, 0), (0, 4), (0, 14)}, {(0, 0), (0, 11), (1, 34)}, {(0, 0), (0, 16), (1, 1)}, 4 (3, 9, 1, 3) {(0, 0), (1, 4), (2, 9)}, {(0, 0), (1, 7), (2, 3)}, {(0, 0), (1, 11), (2, 23)}, {(0, 0), (1, 15), (2, 5)}, {(0, 0), (1, 0), (2, 1)}, {(0, 0), (1, 3), (5, 5)}, {(0, 0), (1, 5), (6, 1)}, (9, 3, 1, 3) {(0, 0), (1, 7), (3, 1)}, {(0, 0), (1, 10), (3, 2)}, {(0, 0), (2, 0), (4, 10)}, {(0, 0), (2, 5), (4, 0)}, {(0, 0), (2, 11), (5, 4)},
{(0, 0), (0, 2), (0, 15)}, {(0, 0), (0, 6), (0, 50)}, {(0, 0), (0, 8), (0, 19)}, {(0, 0), (0, 12), (0, 53)}, {(0, 0), (0, 17), (0, 37)}, {(0, 0), (0, 23), (0, 47)}, {(0, 0), (0, 29), (0, 60)}, {(0, 0), (0, 33), (0, 68)}; {(0, 0), (0, 2), (2, 0)}, {(0, 0), (0, 5), (1, 25)}, {(0, 0), (0, 13), (2, 33)}, {(0, 0), (0, 17), (2, 18)}, {(0, 0), (1, 6), (2, 14)}, {(0, 0), (1, 9), (2, 19)}, {(0, 0), (1, 14), (2, 6)}, {(0, 0), (1, 19), (2, 7)}; {(0, 0), (1, 2), (4, 5)}, {(0, 0), (1, 4), (5, 1)}, {(0, 0), (1, 6), (2, 3)}, {(0, 0), (1, 8), (3, 4)}, {(0, 0), (1, 11), (4, 6)}, {(0, 0), (2, 2), (5, 11)}, {(0, 0), (2, 9), (5, 9)}, {(0, 0), (3, 6), (6, 2)}.
(1, 27, 1, 3)
(1, 27, 1, 3)
A PPENDIX B Some (s, t)-regular (m, n, 3, 1)-OOSPCs of Lemma 4.5 (2). For each (m, n, s, t) ∈ {(2, 72, 2, 12), (6, 6, 2, 6), (6, 12, 2, 12), (6, 24, 2, 12)}, an (s, t)-regular (m, n, 3, 1)-OOSPC is constructed by listing all its codewords as follows: (m, n, s, t)
Codewords
{(0, 0), (0, 1), (0, 23)}, {(0, 0), (0, 3), (0, 38)}, {(0, 0), (0, 5), (0, 32)}, {(0, 0), (0, 8), (1, 11)}, {(0, 0), (0, 10), (1, 19)}, (2, 72, 2, 12) {(0, 0), (0, 13), (1, 26)}, {(0, 0), (0, 15), (1, 29)}, {(0, 0), (0, 17), (1, 34)}, {(0, 0), (0, 20), (1, 45)}, {(0, 0), (0, 31), (1, 32)},
{(0, 0), (0, 2), (0, 28)}, {(0, 0), (0, 4), (0, 29)}, {(0, 0), (0, 7), (1, 2)}, {(0, 0), (0, 9), (1, 16)}, {(0, 0), (0, 11), (1, 21)}, {(0, 0), (0, 14), (1, 22)}, {(0, 0), (0, 16), (1, 31)}, {(0, 0), (0, 19), (1, 39)}, {(0, 0), (0, 21), (1, 44)}, {(0, 0), (0, 33), (1, 37)};
(6, 6, 2, 6)
{(0, 0), (1, 0), (2, 1)}, {(0, 0), (1, 3), (2, 2)},
{(0, 0), (1, 2), (2, 0)}, {(0, 0), (2, 3), (4, 1)};
(6, 12, 2, 12)
{(0, 0), (1, 0), (2, 7)}, {(0, 0), (1, 2), (2, 10)}, {(0, 0), (1, 4), (2, 9)}, {(0, 0), (2, 0), (4, 4)},
{(0, 0), (1, 1), (2, 11)}, {(0, 0), (1, 3), (2, 2)}, {(0, 0), (1, 6), (2, 3)}, {(0, 0), (2, 1), (4, 6)};
{(0, 0), (0, 1), (1, 17)}, {(0, 0), (0, 5), (5, 10)}, {(0, 0), (0, 9), (5, 11)}, {(0, 0), (1, 0), (5, 13)}, {(0, 0), (1, 2), (2, 23)}, (6, 24, 2, 12) {(0, 0), (1, 4), (2, 3)}, {(0, 0), (1, 6), (3, 11)}, {(0, 0), (1, 8), (3, 17)}, {(0, 0), (1, 10), (2, 1)}, {(0, 0), (2, 0), (4, 8)},
{(0, 0), (0, 3), (2, 18)}, {(0, 0), (0, 7), (4, 17)}, {(0, 0), (0, 11), (2, 13)}, {(0, 0), (1, 1), (2, 21)}, {(0, 0), (1, 3), (3, 1)}, {(0, 0), (1, 5), (3, 9)}, {(0, 0), (1, 7), (3, 3)}, {(0, 0), (1, 9), (3, 19)}, {(0, 0), (1, 12), (2, 6)}, {(0, 0), (2, 12), (4, 5)}.
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ACKNOWLEDGMENT The authors would like to thank the anonymous referees and Prof. Kyeongcheol Yang, the Associate Editor for Sequences, for their constructive comments and suggestions that greatly improve the quality of this paper. R EFERENCES [1] R. J. R. Abel and M. Buratti, “Some progress on (v, 4, 1) difference families and optical orthogonal codes,” J. Combinat. Theory, Ser. A, vol. 106, no. 1, pp. 59–75, 2004. [2] M. Buratti, “Recursive constructions for difference matrices and relative difference families,” J. Combinat. Designs, vol. 6 no. 3, pp. 165–182, 1998. [3] M. Buratti, “1-rotational Steiner triple systems over arbitrary groups,” J. Combinat. Designs, vol. 9 no. 3, pp. 215–226, 2001. [4] M. Buratti, “Cyclic designs with block size 4 and related optimal optical orthogonal codes,” Designs, Codes Cryptograph., vol. 26, nos. 1–3, pp. 111–125, 2002. [5] M. Buratti and A. Pasotti, “Further progress on difference families with block size 4 or 5,” Designs, Codes Cryptograph., vol. 56, no. 1, pp. 1–20, 2010. [6] Y. Chang, R. Fuji-Hara, and Y. Miao, “Combinatorial constructions of optimal optical orthogonal codes with weight 4,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1283–1292, May 2003. [7] Y. Chang and L. Ji, “Optimal (4up, 5, 1) optical orthogonal codes,” J. Combinat. Designs, vol. 12, no. 5, pp. 346–361, 2004. [8] Y. Chang and Y. Miao, “Constructions for optimal optical orthogonal codes,” Discrete Math., vol. 261, nos. 1–3, pp. 127–139, Jan. 2003. [9] Y. Chang and J. Yin, “Further results on optimal optical orthogonal codes with weight 4,” Discrete Math., vol. 279, nos. 1–3, pp. 135–151, Mar. 2004. [10] F. Chung, J. A. Salehi, and V. K. Wei, “Optical orthogonal codes: Design, analysis and applications,” IEEE Trans. Inf. Theory, vol. 35, no. 3, pp. 595–604, May 1989. [11] C. J. Colbourn, “Difference matrices,” in CRC Handbook of Combinatorial Designs, C. J. Colbourn and J. H. Dinitz, Eds. Boca Raton, FL, USA: CRC Press, 2007, pp. 411–419. [12] T. Feng, Y. Chang, and L. Ji, “Constructions for rotational Steiner quadruple systems,” J. Combinat. Designs, vol. 17, no. 5, pp. 353–368, Sep. 2009. [13] T. Feng, Y. Chang, and L. Ji, “Constructions for strictly cyclic 3-designs and applications to optimal OOCs with λ = 2,” J. Combinat. Theory, Ser. A, vol. 115, no. 8, pp. 1527–1551, Nov. 2008. [14] R. Fuji-Hara and Y. Miao, “Optical orthogonal codes: Their bounds and new optimal constructions,” IEEE Trans. Inf. Theory, vol. 46, no. 7, pp. 2396–2406, Nov. 2000. [15] R. Fuji-Hara, Y. Miao, and J. Yin, “Optimal (9v, 4, 1) optical orthogonal codes,” SIAM J. Discrete Math., vol. 14, no. 2, pp. 256–266, 2001. [16] G. Ge and J. Yin, “Constructions for optimal (v, 4, 1) optical orthogonal codes,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2998–3004, Nov. 2001. [17] S. Johnson, “A new upper bound for error-correcting codes,” IRE Trans. Inf. Theory, vol. 8, no. 3, pp. 203–207, Apr. 1962. [18] K. Kitayama, “Novel spatial spread spectrum based fiber optic CDMA networks for image transmission,” IEEE J. Sel. Areas Commun., vol. 12, no. 4, pp. 762–772, May 1994.
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Rong Pan was born in Yunnan, China, in 1988. She is currently a Ph.D. student at Beijing Jiaotong University, China. Her research interests include Combinatorial Design Theory and Coding Theory.
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 Teaching- Research Assistant (1988-1990), a Lecturer (1990-1992) and then an Associated Professor (1993-1995). From 1996 to 1997, he was an Associated 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.
