A New Product Construction for Partial Difference Sets John Polhill Department of Mathematics, Computer Science, and Statistics Bloomsburg University Bloomsburg, PA 17815 email :
[email protected] James A. Davis Department of Mathematics and Computer Science University of Richmond Richmond, VA 23173 email:
[email protected] Ken Smith Department of Mathematics and Statistics Sam Houston State University Huntsville, TX 77341 email :
[email protected] Abstract Relatively few constructions are known of negative Latin square type Partial Difference Sets (PDSs), and most of the known constructions are in elementary abelian groups. We present a product construction that produces negative Latin square type PDSs, and we apply this product construction to generate examples in p-groups of exponent bigger than p.
Keywords: partial difference set, association scheme, difference set
1
Introduction
Let G be a finite group of order v with a subset D of order k such that the differences d1 d2 −1 for d1 , d2 ∈ D, d1 6= d2 represent each nonidentity element of D exactly λ times and the nonidentity element of G − D exactly µ times. Then D is called a (v, k, λ, µ)-partial difference set (PDS) in G. When the identity e 6∈ D and D(−1) = D we call the PDS D regular. The Cayley graph of a PDS is strongly regular [13], a PDS in an elementary abelian group is equivalent to a projective two weight code [2], and a partitioning of a group into certain types of PDSs will correspond to an amorphic association scheme [21]: all of these demonstrate the need to better understand PDSs. A PDS having parameters (n2 , r(n − 1), n + r2 − 3r, r2 − r) is called a Latin square type PDS. Constructions of this sort of PDS can be found for instance in [3], [5], [9], [11], [14]. Similarly, a PDS having parameters (n2 , r(n + 1), −n + r2 + 3r, r2 + r) is called a negative Latin square type PDS. Until the last decade, there were few known constructions of negative Latin square type PDSs, [13]. In [6], Davis and Xiang constructed negative Latin square type PDSs in nonelementary abelian 2-groups. Polhill gave constructions of negative Latin square type PDSs in nonelementary
1
abelian 2-groups and 3-groups, [15] and [16]. More recently, Chen and Polhill generalized to all primes p some of these constructions to certain nonelementary abelian p-groups, [4]. Prior to that, the only known negative Latin square type PDSs in nonelementary abelian groups were reversible Hadamard difference sets and Paley type PDSs, [5] and [12]. Often in the context of the group ring Z[G]. For a subset D in G we write P PDSs are studied P D = d∈D d and D(−1) = d∈D d−1 . This abuse of notation isPwidely accepted; depending on the context D will represent the difference set D or the element d∈D d in the group ring Z[G]. One of the most common tools used to study PDSs is character theory. A character on an abelian group G is a homomorphism from the group to the set of complex numbers having modulus 1 under the operation of multiplication. The principal character sends all group elements to 1. If S ⊂ G and χ is a character of G, then the character sum χ(S) is defined to be Σs∈S χ(s). The following theorem on character sums is quite useful when studying PDSs (see [20] for a proof). Theorem 1.1 The subset D (with e 6∈ D) of the group G is a (n2 , r(n + (−1)δ ), (−1)δ+1 n + r2 + 3(−1)δ r, r2 + (−1)δ )-PDS (δ is either 0 or 1) iff χ0 (D) = r(n + (−1)δ ) for the principal character χ0 and χ(D) = (−1)δ r or (−1)δ (r − n) for every nonprincipal character χ. The PDS D is negative Latin square type when δ = 0 and Latin square type when δ = 1. If χ is a nonprincipal character on a group G then χ(G) = 0, and if χ is principal on G then χ(G) = |G|. This will be used in Section 2 together with Theorem 1.1 to prove that a subset is a PDS. For the rest of the paper we assume G is an abelian group with identity element e. We write G∗ to represent the nonidentity elements of G. In certain groups we can partition G∗ into PDSs. If the PDSs are either Latin square type or negative Latin square type, then we can determine the character sums on the PDSs a bit more precisely, which we do in the following theorem. Theorem 1.2 1 If G is an abelian group so that G∗ can be partitioned into f PDSs D1 , D2 , . . . , Df each with the parameters (n2 , r(n+1), r2 +3r −n, r2 +r), and if χ is a nonprincipal character on G, then χ(Dj ) = r − n for precisely one j, 1 ≤ j ≤ f and χ(Di ) = r for i 6= j. 2 If G is an abelian group so that G∗ can be partitioned into f PDSs D1 , D2 , . . . , Df each with the parameters (n2 , r(n − 1), r2 − 3r + n, r2 − r), and if χ is a nonprincipal character on G, then χ(Dj ) = n − r for precisely one j, 1 ≤ j ≤ f and χ(Di ) = −r for i 6= j. 3 Suppose G is an abelian group so that G∗ can be partitioned into f PDSs L01 , L02 , L3 , . . . , Lf where Li has parameters (n2 , r(n + 1), r2 + 3r − n, r2 + r), 3 ≤ i ≤ f and L0i has parameters (n2 , (r − 1)(n + 1), (r − 1)2 + 3(r − 1) − n, (r − 1)2 − (r − 1)), i = 1, 2. Define L1 = L01 ∪ {e} and L2 = L02 ∪ {e}. If χ is a nonprincipal character on G, then χ(Lj ) = r − n for precisely one j, 1 ≤ j ≤ f and χ(Li ) = r for i 6= j. 4 Suppose G is an abelian group so that G∗ can be partitioned into f PDSs L01 , L02 , L3 , . . . , Lf where Li has parameters (n2 , r(n − 1), r2 − 3r + n, r2 − r), 3 ≤ i ≤ f and L0i has parameters (n2 , (r + 1)(n − 1), (r + 1)2 − 3(r + 1) + n, (r + 1)2 − (r + 1)), i = 1, 2. Define L1 = L01 ∪ {e} and L2 = L02 ∪ {e}. If χ is a nonprincipal character on G, then χ(Lj ) = n − r for precisely one j, 1 ≤ j ≤ f and χ(Li ) = −r for i 6= j. Proof: Suppose χ is a nonprincipal character on the group G in statement 1. In the remarks before this theorem we have that χ(G) = 0 and hence χ(G∗ ) = −1. By Theorem 1.1, we know that all character values for each of the Di will be either r or r − n, so suppose we have x of the Di with character sum r − n and hence f − x of the Di with character sum r. This gives −1 = χ(G∗ ) = Σfi=1 χ(Di ) = x(r − n) + (f − x)(r) = f r − xn = (n − 1) − xn. Adding xn + 1 to both sides yields n = xn, and hence x = 1. Thus, there is exactly one j satisfying χ(Di ) = r − n and χ(Di ) = r, i 6= j. The proof of statements 2, 3, and 4 are similar.
2
2
The Product Theorems
This section contains two theorems that apply to abelian groups G and G0 of order n2 for any positive integer n. We will demonstrate the usefulness of the theorems for the case when n is a prime power. The theorems require G∗ to have a partition into Latin square type PDSs and G0∗ to have a partition into negative Latin square type PDSs. The result is a single PDS. In [16] and [4], product theorems are given in which the output is a partition of the group into PDSs of the Latin or negative Latin square type which can then be used recursively with the same product construction. In the theorems below, since the identity element appears twice in one of the decompositions, we cannot get the needed partition into partial difference sets as ouput in the product group. We remark that this notion of partitioning the group into partial difference sets fits well into the context of amorphic association schemes, see for instance [7] or [21]. We will elaborate on this relationship in the last section of this paper. The proof of the second theorem is omitted due to its similarity to that of the first. Theorem 2.1 Let G and G0 be groups of order n2 . Suppose that G∗ has a partition into f disjoint ∗ (n2 , r(n+1), r2 +3r−n, r2 +r)- negative Latin square type PDSs, which we call Di . Suppose that G0 has a partition into f Latin square type PDSs, denoted by Li , such that for i > 2, Li has parameters 2 (n2 , r(n − 1), r2 − 3r + n, r2 − r) while L1 and L2 have parameters (n2 , (r + 1)(n − 1), (r + 1) − Sf 2 3(r + 1) + n, (r + 1) − (r + 1)). Then P = (D1 × (L1 ∪ {e})) ∪ (D2 × (L2 ∪ {e})) ∪ ( i=3 Di × Li ) is an (n4 , `(n2 + 1), −n + `2 + 3`, `2 + `)-negative Latin square type PDS in G × G0 where ` = r(n + 1). Pf Proof: We first check the cardinality. |P | = |D1 |(|L1 | + 1) + |D2 |(|L2 | + 1) + i=3 |Di ||Li | = Pf 2(r(n + 1))[(r + 1)(n − 1) + 1] + i=3 (r(n + 1))(r(n − 1)) = f r2 (n2 − 1) + 2nr(n + 1). At this point we use the fact that f r = n − 1, so our sum becomes: r(n − 1)(n2 − 1) + 2nr(n + 1) = r(n + 1)(n2 + 1) = `(n2 + 1). Let φ be a nonprincipal character on G × G0 , so that φ = χ ⊗ ψ where χ is a character on G and ψ is a character on G0 . By Theorem 1.1, we need to show that φ(P ) = ` or ` − n2 . Case 1: χ is principal on G, ψ is nonprincipal on G0 . In this case we have φ(P ) = |D1 |(ψ(L1 ) + Pf 1) + |D2 |(ψ(L2 ) + 1) + i=3 |Di |ψ(Li ) = r(n + 1)(ψ(G0 ) + 1) = ` since ψ(G0 ) = 0. Case 2: χ is nonprincipal on G, ψ is principal on G0 . By Theorem 1.2, for exactly one Dm we will have that ψ(Dm ) = r − n and for all i 6= m we have ψ(Di ) = r. If m = 1 we have the Pf following: φ(P ) =P(r − n)(|L1 | + 1) + r(|L2 | + 1) + i=3 r|Li | = (r − n)(r(n − 1) + n) + e r(r(n − 1) + n) + i=3 r(r(n − 1)) = r(n + 1) − n2 = ` − n2 . The m = 2 case is similar and also results in φ(P ) = ` − n2 . If m 6∈ {1, 2}, a similar calculation yields φ(P ) = `. Case 3: χ is nonprincipal on G, ψ is nonprincipal on G0 . Again by Theorem 1.2, we know that for one Dm we will have that ψ(Dm ) = r − n and for all i 6= m we have ψ(Di ) = r, and similarly for exactly one Lh we will have that ψ(Lh ) = n − r and for all j 6= h we have Pf ψ(Lj ) = −r. If h = m, we get the following: φ(P ) = (r − n)(n − r) + i=2 r(−r) = −f r2 − n2 + 2rn = −r(n − 1) + 2rn − n2 = r(n + 1) − n2 = ` − n2 . If h 6= m, we get Pf φ(P ) = (r − n)(−r) + (n − r)(r) + i=3 r(−r) = f r2 + 2rn = `. Thus all nonprincipal characters φ satisfy φ(P ) = ` − n2 or φ(P ) = `, and P is a negative Latin square type PDS.
Theorem 2.2 Let G and G0 be groups of order n2 . Suppose that G∗ has a partition into f disjoint ∗ (n2 , r(n − 1), r2 − 3r + n, r2 − r)- Latin square type PDSs, denoted Di . Suppose that G0 has a partition into f negative Latin square type PDSs, denoted Li , such that for i > 2, Li has parameters 3
2
(n2 , r(n + 1), r2 + 3r − n, r2 + r) while L1 and L2 have parameters (n2 , (r − 1)(n + 1), (r − 1) + Sf 2 3(r − 1) − n, (r − 1) + (r − 1)). Then P = (D1 × (L1 ∪ {e})) ∪ (D2 × (L2 ∪ {e})) ∪ ( i=3 Di × Li ) is an (n4 , `(n2 − 1), n + `2 − 3`, `2 − `)-Latin square type PDS in G × G0 . where ` = r(n − 3).
3
Some constructions of PDSs in 2-groups
In this section, we will give various constructions of PDSs in 2-groups that we will use with the product theorems to generate families of PDSs in the next section. In each case character theoretic arguments will verify the claims in the examples. Example 3.1 Denniston [8] constructed a (64, 18, 2, 6) negative Latin square type PDS in the group (Z2 )6 .The group (Z2 )6 also has a (64, 28, 12, 12)-Latin square type PDS that come from reversible Hadamard difference sets containing the identity, and removing the identity yields a (64, 27, 10, 12) negative Latin square type PDS. We can partition ((Z2 )6 )∗ into two (64, 18, 2, 6)-negative Latin square type PDSs, E1 and E2 and one (64, 27, 10, 12)-negative Latin square type PDS, E3 . Let G = (Z2 )6 = hx, y, z, w, u, v|x2 = y 2 = z 2 = w2 = u2 = v 2 = 1i. We use the methods of [6], being careful to keep E1 and E2 disjoint: E1
= {x, y, xy, xu, yv, xyuv, xz, yw, xyzw, xzv, ywuv, xyzwu, xwu, yzwv, xyzuv, xwv, yzwuv, xyzu}
E2
= {z, w, zw, zuv, wu, zwv, yzv, xywuv, xzwu, yzuv, xywu, xzwv, xyz, xw, yzw, xyzv, xwuv, yzwu}
E3
= G − E1 − E2
Similar methods will allow a partition of ((Z2 )6 )∗ into 3 (64, 21, 8, 6)-Latin square type PDSs. Example 3.2 By using the rational idempotent method found in [19], we show that G = (Z8 )2 = hx, y|x8 = y 8 = 1i can also be partitioned into two (64, 18, 2, 6)-negative Latin square type PDSs, P1 and P2 , and one (64, 27, 10, 12)-negative Latin square type PDS, P3 . P1
=
[(hyi ∪ hxi ∪ hxyi) ∩ (G − 2G)] ∪ [(hx4 y 2 i ∪ hx2 y 4 i ∪ hx2 y 6 i) ∩ (2G − 4G)]
P2
=
[(hx2 yi ∪ hxy 2 i ∪ hxy 3 i) ∩ (G − 2G)] ∪ [(hy 2 i ∪ hx2 i ∪ hx2 y 2 >) ∩ (2G − 4G)]
P3
=
G − P1 − P2 − {e}
Also in Z8 × Z8 we can form 3 disjoint Latin square type PDSs with parameters (64, 21, 8, 6) by using the framework from [14]. The PDSs are as follows:
D1
=
[(hyi ∪ hx2 yi) ∩ (G − 2G)] ∪ [(hxy 4 i ∪ hxy 5 i) ∩ (G − 4G)] ∪ {y 4 }
D2
=
[(hxi ∪ hxy 2 i) ∩ (G − 2G)] ∪ [(hx4 yi ∪ hxy 7 i) ∩ (G − 4G)] ∪ {x4 }
D3
=
[(hxyi ∪ hxy 3 i) ∩ (G − 2G)] ∪ [(hx6 yi ∪ hxy 6 i) ∩ (G − 4G)] ∪ {x4 y 4 }
Example 3.3 We have not found a partition of Z16 × Z16 into three negative Latin square type PDSs of cardinality 85 each, but we have a partition of Z16 × Z16 into three Latin square type PDSs of size 90, 90, and 75 that will be useful in conjunction with Theorem 2.1:
4
Da
=
[((
13 [
hxy i i) ∪ hx8 yi ∪ hx10 yi ∪ hx12 yi) ∩ (G − 2G)] ∪
i=8 2 4
[(hx y i ∪ hx2 y 6 i ∪ hx4 y 2 i) ∩ (2G − 4G)] ∪ [(hx4 i ∪ hy 4 i ∪ hx4 y 4 i) ∩ (4G − 8G)] Db
=
[((
7 [
hxy i i) ∪ hx2 yi ∪ hx4 yi ∪ hx6 yi) ∩ (G − 2G)] ∪
i=2
[(hx2 y 8 i ∪ hx2 y 10 i ∪ hx8 y 2 i) ∩ (2G − 4G)] ∪ [(hx4 y 8 i ∪ hx8 y 4 i ∪ hx4 y 12 i) ∩ (4G − 8G)] Dc
= G − Da − Db − {1}
We will use the following theorem that appears in [14] as Theorem 4.3 of that paper. 2t
Theorem 3.4 Let G = (Zpr ) . Then there exists a Latin square type partial difference set P with 2 2 parameters (p2rt , (ept n + f )(prt − 1), prt + (ept n + f ) − 3(ept n + f ), (ept n + f ) − (ept n + f )) for (r−1)t all integers e and f such that 0 ≤ e < pt and 0 ≤ f ≤ pt + 1, where n = p pt −1−1 . 2t
Theorem 3.5 Let G = (Zpr ) . There exists pt − 1 partial difference sets of the Latin square type Di for 1 ≤ i ≤ pt − 1 that form a partition of the nonidentity elements of G with |D1 | = |D2 | = Pr−1 (x + 1)(prt − 1) and |Di | = x(prt − 1) for i 6= 1, 2, x = j=0 pjt . The parameters of D1 and D2 2 2 are (p2rt , (x + 1)(prt − 1), (x + 1) − 3(x + 1) + prt , (x + 1) − (x + 1)), and for i 6= 1, 2, Di has parameters (p2rt , x(prt − 1), x2 − 3x + prt , x2 − x). Proof: If we let e = f = 1 ∀i 6= 1, 2 and e = 1, f = 2 for i = 1, 2 in the previous theorem, the construction from [14] allows us to select the partial difference sets so that they are disjoint and partition the nonidentity elements of G. The result follows.
4
Constructions and Questions
The following Theorem uses Theorem 2.1 in combination with Theorem 3.5 and other known PDSs to construct new negative Latin square type PDSs. Pr−1 2rt 2t Theorem 4.1 Let G = (Zp ) , G0 = (Zpr ) , and x = ( j=0 pjt ) for r, t positive integers. Then there exist (p4rt , x(prt + 1)(p2rt + 1), −prt + (x(prt + 1))2 + 3x(prt + 1), (x(prt + 1))2 + x(prt + 1))negative Latin square type partial difference sets in both G × G and G × G0 . Proof: The partitioning (prt , x(prt + 1), x2 + 3x − prt , x2 + x) negative Latin square type PDSs in G exist from well known cyclotomy constructions [13]. The groups G and G0 both have the required partitioning (p2rt , x(prt − 1), x2 − 3x + prt , x2 − x) Latin square type PDSs (together with the two larger PDSs). In the former case, we can use a partial congruence partition [13]. In the latter case, we can use Theorem 3.5. The theorem follows from Theorem 2.1. The following examples use Theorems 2.1 and 2.2 in combination with Examples 3.1, 3.2 (both for the first example below), and 3.3 (for the second example below) to construct new negative Latin square type PDSs. Example 4.2 Let G = (Z2 )6 and G0 = (Z8 × Z8 ). By Theorem 2.2 with f = r = 3 and n = 8 we get a (212 , 21(26 + 1), −26 + 212 + 3(21), 212 + 21)-negative Latin square type PDS in each of G × G, G × G0 , and G0 × G0 .
5
Example 4.3 Let G = (Z2 )8 and G0 = (Z16 × Z16 ). By Theorem 2.1 with f = 3, r = 5, and n = 16 we get a (216 , 85(28 + 1), −26 + 852 + 3(85), 852 + 85)-negative Latin square type PDS in each of G × G and G × G0 . In [15], less general product theorems were used to construct various PDSs in groups of order 256 with exponent 2 or 4. Many of these could be proved by using Theorem 3.5. We have shown that Theorems 2.1 and 2.2 are applicable to p-groups where p is any prime. Theoretically, one could apply them to groups whose order is not a prime power. The difficulty is that in this case the required PDSs in the input groups are difficult to construct, if it is even possible at all. Perhaps if more general theorems could be proved where the input groups don’t need to be abelian, one might have greater success finding the input PDSs. When we partition the groups into PDSs as we do with our product theorems we can think of this as partitioning the complete graph with vertex set the group elements into strongly regular graphs. The examples in this paper have graphs which are of negative Latin square type. A theorem of Van Dam [21] will allow us to put the results in the context of association schemes. Theorem 4.4 (Van Dam [21]) Let {G1 , G2 , · · · Gd } be an edge-decomposition of the complete graph on a set X, where each Gi is strongly regular. If the Gi are all of Latin square type or all of negative Latin square type, then the decomposition is a d-class amorphic association scheme on X. In [16] and [4], product theorems are given so that the groups G and G0 are partitioned into f PDSs and the number of PDSs that can be formed in G × G0 is also f . The resulting PDSs can then be used recursively with the product theorems. In this paper, if we use f input PDSs in both G and G0 , we get b f2 c PDSs in the product group and can not get a recursion.
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