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Paley partial difference sets in groups of order n4 and 9n4 for any odd n>1
Journal of Combinatorial Theory, Series A 117 (2010) 1027–1036
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Journal of Combinatorial Theory, Series A www.elsevier.com/locate/jcta
Paley partial difference sets in groups of order n4 and 9n4 for any odd n > 1 John Polhill Department of Mathematics, Computer Science, and Statistics, Bloomsburg University, Bloomsburg, PA 17815, United States
a r t i c l e
i n f o
a b s t r a c t 1 v −5 v −1 A partial difference set with parameters ( v , v − , 4 , 4 ) is said 2 to be of Paley type. In this paper, we give a recursive theorem that for all odd n > 1 constructs Paley partial difference sets in certain groups of order n4 and 9n4 . We are also able to construct Paley– Hadamard difference sets of the Stanton–Sprott family in groups of order n4 (n4 ± 2) when n4 ± 2 is a prime power and 9n4 (9n4 ± 2) when 9n4 ± 2 is a prime power. Many of these are new parameters for such difference sets, and also give new Hadamard designs and matrices. © 2010 Elsevier Inc. All rights reserved.
Article history: Received 17 February 2009 Available online 4 March 2010 Keywords: Paley partial difference set Paley Hadamard difference set Hadamard difference set Difference set Partial difference set Strongly regular graph
1. Introduction Let G be a finite group of order v, and let D be a subset of G with cardinality k. Then D is a ( v , k, λ)-difference set (DS) provided that the list of differences d1 d2−1 , d1 , d2 ∈ D represents every nonidentity element in G exactly λ times. A difference set, D, is called reversible if d ∈ D implies d−1 ∈ D. The text of Beth, Jungnickel, and Lenz [1] and the survey of Jungnickel [7] are excellent references for DSs. Difference sets are of widespread interest largely due to the fact that a difference set is equivalent to a symmetric design with a regular automorphism group. Hadamard (Menon) difference sets, having parameters (4u 2 , 2u 2 − u , u 2 − u ), are of particular interest due to the fact that their ±1 incidence matrices form Hadamard matrices. Another important family of difference sets is the Hadamard or 1 v −3 Paley–Hadamard difference sets having parameters ( v , v − , 4 ). It is somewhat unfortunate that two 2 distinct families of difference sets are given the name Hadamard, so for this paper we will distinguish them by calling the first Hadamard and the second Paley–Hadamard. When a Paley–Hadamard difference set also has the property that G is the disjoint union of D, the inverses of the elements of D, and 0, it is called a skew Hadamard difference set.
E-mail address: [email protected]. 0097-3165/$ – see front matter doi:10.1016/j.jcta.2010.02.008
© 2010
Elsevier Inc. All rights reserved.
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Now suppose G is 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 = 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) / D and D (−1) = D we call the PDS D regular. The survey article of Ma in G. When the identity e ∈ is an excellent survey on PDSs [9]. Closely related to Paley–Hadamard difference sets are Paley type 1 v −5 v −1 , 4 , 4 ). Paley originally discovered these sets [10] partial difference sets, having parameters ( v , v − 2 along with the Paley–Hadamard difference sets in the context of Hadamard matrices, and in fact both Paley PDSs and Paley–Hadamard difference sets can be used to construct both Hadamard designs and matrices. While Paley PDSs were known to exist in elementary abelian groups since Paley’s work in the 1930’s, the first construction of these sets in nonelementary abelian groups was by Davis in 1994 [3]. This provided an answer to one of the following questions in the survey of Ma [9]: Questions 13.4. Suppose G is an abelian group of order v ≡ 1 (mod 4). If v is not a prime power, does there exist a Paley PDS in G? If v is a prime power, does G need to be elementary abelian? In a previous paper we gave the first construction of Paley PDSs in groups having an order which is not a prime power [11]. However, the Paley PDSs were produced only in those groups of the form Z3 2 × Z p 4t for an odd prime p. In this paper we provide a much more general construction and show that Paley PDSs exist for all group orders n4 and 9n4 where n is odd and n > 1. As a corollary, we will also obtain new parameters for Paley–Hadamard difference sets. The constructions in this paper depend heavily on the existence of building sets, first introduced by Davis and Jedwab [4]. The particular variety of building set that we require is also crucial for the construction of certain types of Hadamard difference sets. For this reason, we will give a brief summary of some of the known results on abelian Hadamard difference sets in Section 2. DSs and PDSs can be studied within the context of the group ring Z[G]. For a subset D in G we write D = d∈ D d and denote D (−1) = d∈ D d−1 . Character theory is a powerful tool in the study of difference sets and partial difference sets in abelian groups. Turyn [13] and Yamamoto [19] first applied character theory to the study of abelian difference sets. A character of an abelian group G is a homomorphism from the group to the complex numbers with modulus 1. The principal character is the homomorphism that maps every element of G to 1; any other character is called nonprincipal. One can naturally extend a character on G to a homomorphism of the group ring Z[G ] as follows: if χ is a character on G, then for an element A = g ∈G a g g let χ ( A ) = g ∈G a g χ ( g ) so that if S is a subset of G, then χ ( S ) = s∈ S χ (s). See [13] for a proof of similar results to the following. Theorem 1.1. (a) Let G be an abelian group of order v with a subset D of cardinality k and let λ be a positive integer satisfying λ( v − 1) = k(k − 1√ ). Then D is a ( v , k, λ)-difference set in G if and only if for every nonprincipal character χ on G, |χ ( D )| = k − λ. / P with k2 = k + λk + (b) Let G be an abelian group of order v with a subset P of cardinality k such that 0 ∈ μ( v − k − 1) for positive integers λ and μ. Then P is a ( v ,√ k, λ, μ)-partial difference set in G if and only if for every nonprincipal character χ on G, χ ( P ) =
(λ−μ)±
(λ−μ)2 +4(k−μ) 2
.
2. Known constructions of Hadamard difference sets Hadamard difference sets are quite likely the most widely studied of all the families of difference sets. A theorem by Turyn [14] shows that if there exist Hadamard difference sets in the abelian groups K × G 1 and K × G 2 where | K | = 4 and |G i | = mi 2 for mi odd, then there is a Hadamard difference set in K × G 1 × G 2 . Therefore, it makes sense to consider groups of the form K × G where |G | = p r , for p a prime and r even. The answer of existence of Hadamard difference sets is completely determined in the case when the group is an abelian 2-group. An abelian group G of order 22d+2 contains a Hadamard difference set if and only if G satisfies exp(G ) 2d+2 (exp(G ) means the order of the element of G with highest additive order) ([6] or [8]).
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For the case where p = 3, a Hadamard difference set exists in every group of the form K × Z3r × Z3r = K × (Z3r )2 for all positive integers r where K is either group of order 4 [1]. A Hadamard difference set does not exist in G × Z3r , where |G | is relatively prime to 3 and r is any positive integer. For p > 3, Davis and Jedwab [5] showed that no Hadamard difference set exists in groups of the form K × (Z pr )2 provided that none exists in K × (Z p )2 , where again K is a group of order 4. Such difference sets do not exist for p > 3. Of particular importance to this paper is the Hadamard difference set construction for K × Z p 4t where | K | = 4 and p is an odd prime. Xia made the first major breakthrough on this problem, constructing difference sets for p ≡ 3 mod 4 in [17]. His approach was simplified by Xiang and Chen [18] using the additive characters of finite fields. Then Wilson and Xiang [16] related the existence of this type of difference set to certain projective sets, and Chen impressively used their work to construct Hadamard difference sets in all groups K × Z p 4t for | K | = 4 and p an odd prime [2]. We will now consider this construction. Let P G (k − 1, q) denote the projective space of dimension k − 1 over F q for q a power of a prime; the corresponding vector space V (k, q) for PG(k − 1, q) will have dimension k over F q . The elements of PG(k − 1, q) are the subspaces of V (k, q), and, in particular, a projective point is a 1-dimensional subspace, a projective line is a 2-dimensional space, and a hyperplane is a (k − 1)-dimensional subspace. A projective (n, k, h1 , h2 ) set O is a proper subset of the projective space PG(k − 1, q) with n points (n = 0) so that O intersects every hyperplane in h1 or h2 points. If we have that O = { y 1 , y 2 , . . . , yn } then let Ω = {x ∈ V (k, q) | x ∈ O}. The following lemma shows that O is a projective set if and only if Ω is a PDS in the additive group of the corresponding vector space under certain conditions. Wilson and Xiang have this result in [16]. Lemma 2.1. O is a projective (n, k, h1 , h2 ) set if and only if χ (Ω) = qh1 − n or qh2 − n for every nonprincipal additive character. Therefore O is a projective (n, k, h1 , h2 ) set if and only if Ω is a PDS in the additive group of V (k, q) and provided that K 2 = K + λ K + μ( v − K − 1) (K is the cardinality of Ω , while λ and μ are the usual PDS parameters). Now consider the projective space Σ3 = PG(3, q) for an odd prime power q = pt . The additive group of the corresponding vector space is the elementary abelian group Z p 4t . Define a spread of Σ3 to be a set of q2 + 1 projective lines which are pairwise disjoint and partition the points of Σ3 . Finally, q4 −1
a subset of Type Q is a projective ( 4(q−1) , 4,
(q−1)2 4
2
, (q+41) ) set in Σ3 .
Theorem 2.2. (See Xiang and Wilson [16].) Suppose that S = { L 1 , L 2 , . . . , L q2 +1 } is a spread of Σ3 . If there q2 +1 2 × Z p 4t
exist two subsets of Type Q , C 0 and C 1 , in Σ3 with the property that |C 0 ∩ L i | = (q + 1)/2 for 1 i q +1 2 2
and |C 1 ∩ L j | = (q + 1)/2 for for | K | = 4.
+ 1 j q2 + 1, then there exists a Hadamard difference set in K
We make some observations regarding this theorem. If two such subsets of Type Q exist, then the sets
C2 =
Li \ C0
and
q 2 +1 1 i 2
C3 =
L j \ C1
q 2 +1 2 2 +1 j q +1
are also subsets of Type Q . Now let W = V (4, q) be the underlying vector space for Σ3 . Then let Ci = { w ∈ W | w ∈ C i } and Li = { w ∈ W | w ∈ L i } .
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Each of the four subsets C i in Σ3 gives rise to a corresponding PDS Ci in V (4, q) such that the character values on each is q2 −1 4
q2 −1 4 2
or
q2 −1 4
− q2 for nonprincipal additive characters on V (4, q). The
character sum will be − q for exactly one of the sets C0 , C1 , C2 , C3 for any nonprincipal character. We summarize with the following lemma for convenience. Lemma 2.3. Let C0 , C1 , C2 , C3 be the PDSs in the additive group of W = V (4, q) associated to the four subsets of Type Q C 0 , C 1 , C 2 , C 3 that partition Σ3 . Then for any nonprincipal character on the additive group of W , the character sum will be
q2 −1 4
− q2 for exactly one of the sets C0 , C1 , C2 , C3 and
To each of C0 and C2 we will add call this collection of
q −1 4 2
q2 −1 4
q2 −1 4
for all others.
of the sets L j which are disjoint from C0 and C2 ; we
projective lines B 1 with corresponding set B1 = { w ∈ W | w ∈ B 1 }. The q2 −1
q2 −1
character sum on B1 will be either − 4 or q2 − 4 , and when combined with the PDSs from subsets of Type Q , the character sums are either 0 or ±q2 . q2 −1
We similarly add 4 of the sets Li (disjoint from C1 and C3 ) to C1 and C3 , and call this collection B 0 . When we take the four sets
D0 = C0 ∪ B1 ,
D1 = C1 ∪ B0 ,
D2 = C2 ∪ B1 ,
D3 = C3 ∪ B0 ,
the character sums will be such that for any nonprincipal character χ on G = Z p 4t , χ (Di ) = ±q2 for exactly one i and χ (Dk ) = 0 for all other k. Then the four sets G \ D0 , D1 , D2 , D3 form a
( p −2 p , p 2 , 4, +) covering extended building set using the terminology from [4], so that if K = {a0 , a1 , a2 , a3 } then 4
2
H = a0 (G \ D0 ) ∪ a1 D1 ∪ a2 D2 ∪ a3 D3 is a Hadamard difference set in K × Z p 4t . 3. Extended building sets with amicable Paley PDSs It is known that the group Z p 4t not only has the four sets G \ D0 , D1 , D2 , D3 that form a covering extended building set, but also possess Paley partial difference sets. In this section, we will show that they contain building sets and Paley partial difference sets which relate in a certain way in terms of character sum properties. In the subsequent section, we will employ these related sets in a recursive theorem for generating Paley PDSs in a product group. The material in this section prior to the definition of extended building set with amicable Paley partial difference set was introduced in [11]. We begin by forming two more sets B 2 and B 3 as follows: B2 = (C0 ∪ C2 ) \ B0 and B3 =
(C1 ∪ C3 ) \ B1 so that each of B 2 and B 3 are unions of sums on the Bi are given in the following lemmas. Lemma 3.1. Let
χ (B2 ), χ (B3 ) ∈
q2 +3 4
projective lines. The properties of character
χ be a nonprincipal character on V (4, q). Then χ (B0 ), χ (B1 ) ∈ {− q 4−1 , q2 − 2
2 {− q 4+3 , q2
−
q2 +3 }. 4
q2 −1 } 4
and
Furthermore, exactly one Bi will have a positive character sum.
Proof. The character sums immediately follow from Theorem 1.1. The fact that there will be exactly one positive character sum follows from the fact that Σ3 = B 0 ∪ B 1 ∪ B 2 ∪ B 3 . 2 Lemma 3.2. Let χ be a nonprincipal character on V (4, q). Then either one of χ (B0 ) and χ (B2 ) will take a positive value or one of χ (C0 ) and χ (C2 ) will take a negative value, and it is not possible that both occur. That is to say, exactly one element of the following set will be positive: {χ (B0 ), χ (B2 ), −χ (C0 ), −χ (C2 )}. The same is true replacing {0, 2} with {1, 3}.
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Proof. B0 ∪ B2 = C0 ∪ C2 = S, where S is a Paley PDS. If χ ( S ) > 0, then one of χ (B0 ) and χ (B2 ) will be positive and both χ (C0 ) and χ (C2 ) will be positive. If χ ( S ) < 0, then one of χ (C0 ) and χ (C2 ) will be negative and both χ (B0 ) and χ (B2 ) will be negative. 2 Now we form a Paley partial difference set L in the additive group of V (4, q) by taking L = B1 ∪ B2 . L has special properties with respect to the sets Di as given in the following key lemma. Lemma 3.3. Let χ be a nonprincipal character on V (4, q). For i = 0 or 2, if χ (Di ) = ±q2 then χ ( L ) = For i = 1 or 3, if χ (Di ) = ±q then χ ( L ) = 2
∓q2 −1 2
±q2 −1 2
.
.
χ is principal. Suppose χ (D0 ) = q2 . Then q2 −1 it must be the case that χ (B1 ) is positive. It follows that χ ( L ) = 2 . If instead χ (D0 ) = −q2 then it 2 q −1 −q2 −1 follows that χ (B1 ) is negative. Moreover, χ (C0 ) = 4 − q2 so that χ (C0 ∪ C2 ) = χ (B0 ∪ B2 ) = 2 . This ensures that χ (B0 ) and χ (B2 ) must both be negative. Then it must be the case that χ (B3 ) is −q2 −1 positive, so that χ ( L ) = 2 . The cases i = 0 are similar. 2 Proof. There will be exactly one projective line L i on which
This arrangement of building sets with Paley PDS will prove quite useful, so we will give a careful definition. For ease, we rearrange the building sets so that the first two in the list have character sums of opposite sign as the Paley PDS and the third and fourth have same sign. Specifically, we will call such a combination of sets ( D 0 , D 1 , D 2 , D 3 , P ) in a group G of order n2 an extended building set with amicable Paley partial difference set when the following properties are valid: 2 1. The four sets G \ D 0 , D 1 , D 2 , D 3 form a ( n 2−n , n, 4, +) covering extended building set using the
terminology from [4]. This means that | D i | = n 2−n for all i, and that for any nonprincipal character on G, χ ( D j ) = ±n for a particular j and χ ( D i ) = 0 for i = j. 2. P is a Paley PDS in G. 3. For any nonprincipal character χ on G, if χ ( D i ) = ±n for i = 0, 1 then χ ( P ) = ∓n2−1 . That is, when χ ( D 0 ) (or χ ( D 1 )) is nonzero, χ ( D 0 ) (or χ ( D 1 )) will have the opposite sign as χ ( P ). 4. For any nonprincipal character χ on G, if χ ( D i ) = ±n for i = 2, 3 then χ ( P ) = ±n2−1 . That is, when χ ( D 2 ) (or χ ( D 3 )) is nonzero, χ ( D 2 ) (or χ ( D 3 )) will have the same sign as χ ( P ). 2
If ( D 0 , D 1 , D 2 , D 3 , P ) has the following additional properties then we say it satisfies the composition criteria: 1. P is the union of the disjoint sets ( D 2 ∩ D 3 ) and ((G \ D 0 ) ∩ (G \ D 1 )) − {0}; 2. G − P is the union of the disjoint sets ( D 0 ∩ D 1 ) and ((G \ D 2 ) ∩ (G \ D 3 )). In fact, we have already shown that Z p 4t has such a collection of sets. It is straightforward to show that Z3 2 also has such a collection. Proposition 3.4. For any odd prime p and positive integer t, Z p 4t contains an extended building set with amicable Paley partial difference set that satisfies the composition criteria. Proof. Lemma 3.3 shows that (D1 , D3 , D0 , D2 , L ) is an extended building set with amicable Paley partial difference set satisfying the composition criteria since: L = B 1 ∪ B2 = (D0 ∩ D2 ) ∪ ((Z p 4t \ D1 ) ∩ (Z p 4t \ D3 )) − {0} and similarly for Z p 4t \ L. 2 Proposition 3.5. Z3 2 contains an extended building set with amicable Paley partial difference set that satisfies the composition criteria.
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Proof. Let H 0 , H 1 , H 2 , and H 3 be the subgroups of order 3 in Z3 2 , and H i ∗ = H i \ {(0, 0)}. Then ( H 0 , H 1 , H 2 , H 3 , H 2 ∗ ∪ H 3 ∗ ) can be shown to be an extended building set with amicable Paley partial difference set that satisfies the composition criteria. 2 4. Paley PDSs in non- p-groups Theorem 4.1 (Paley partial difference sets in non-p-groups). Let G 1 be a group of order n2 having a covering extended building set with amicable Paley partial difference set ( D 0 , D 1 , D 2 , D 3 , P ), and let G 2 be a group of order m2 having a covering extended building set with amicable Paley partial difference set ( E 0 , E 1 , E 2 , E 3 , Q ) that satisfies the composition criteria. Then the group G 1 × G 2 has a covering extended building set with amicable Paley partial difference set ( F 0 , F 1 , F 2 , F 3 , R ). Proof. First we define the set R, which we will prove is a Paley PDS. Q c = (G 2 \ Q ) \ {0}.
∪ (G 1 \ D 0 ) × (G 2 \ E 3 ) ∩ Q ∪ D 1 × E 2 ∩ Q c ∪ (G 1 \ D 1 ) × (G 2 \ E 2 ) ∩ Q ∪ (G 1 \ D 2 ) × ( E 0 ∩ Q ) ∪ D 2 × (G 2 \ E 0 ) ∩ Q c ∪ ( G 1 \ D 3 ) × ( E 1 ∩ Q ) ∪ D 3 × ( G 2 \ E 1 ) ∩ Q c ∪ P × { 0} .
R = D0 × E3 ∩ Q c
The fact that ( E 0 , E 1 , E 2 , E 3 , Q ) satisfies the composition criteria in G 2 will ensure that R is not a multiset, since the sets from G 2 used in the definition of R will be pairwise disjoint. For example, suppose g ∈ ( E 3 ∩ Q c ) ∩ ((G 2 \ E 0 ) ∩ Q c ). g ∈ E 3 ∩ Q c implies that g ∈ E 0 ∩ E 1 , so g cannot be in (G 2 \ E 0 ) ∩ Q c . We leave the rest of the cases to the reader. The building sets are as follows:
∪ (G 1 \ D 1 ) × ( E 0 ∩ E 1 ) ∪ D 3 × (G 2 \ E 0 ) ∩ E 1 ∪ (G 1 \ D 3 ) × E 0 ∩ (G 2 \ E 1 ) , F 1 = D 0 × (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) ∪ (G 1 \ D 0 ) × ( E 0 ∩ E 1 ) ∪ D 2 × E 0 ∩ (G 2 \ E 1 ) ∪ (G 1 \ D 2 ) × (G 2 \ E 0 ) ∩ E 1 , F 2 = D 1 × E 2 ∩ (G \ E 3 ) ∪ (G 1 \ D 1 ) × E 3 ∩ (G 2 \ E 2 ) ∪ D 3 × (G 2 \ E 2 ) ∩ (G 2 \ E 3 ) ∪ (G 1 \ D 3 ) × ( E 2 ∩ E 3 ) , F 3 = D 0 × (G 2 \ E 2 ) ∩ E 3 ∪ (G 1 \ D 0 ) × E 2 ∩ (G 2 \ E 3 ) ∪ D 2 × (G 2 \ E 2 ) ∩ (G 2 \ E 3 ) ∪ (G 1 \ D 2 ) × ( E 2 ∩ E 3 ) .
F 0 = D 1 × (G 2 \ E 0 ) ∩ (G 2 \ E 1 )
First we check to see that the sets R and F 0 have the correct cardinality. The sets F i for i = 0 will have the same cardinality as F 0 . We will use that ( E 0 , E 1 , E 2 , E 3 , Q ) satisfies the composition criteria in G 2 implies that Q c is a union of the disjoint ( E 3 ∩ Q c ), ( E 2 ∩ Q c ), ((G 2 \ E 0 ) ∩ Q c ), and ((G 2 \ E 1 ) ∩ Q c ), and similarly for Q .
| R | = | D 0 | E 3 ∩ Q c + E 2 ∩ Q c + (G 2 \ E 0 ) ∩ Q c + (G 2 \ E 1 ) ∩ Q c + |G 1 \ D 0 | (G 2 \ E 3 ) ∩ Q + (G 2 \ E 2 ) ∩ Q + | E 0 ∩ Q | + | E 1 ∩ Q | + P × {0} = =
2 2 Q c + n + n |Q | + n − 1
n2 − n
2 2 2 n2 − n m2 − 1 n2 + n m2 − 1 2
2
+
2
2
+
n2 − 1 2
=
(nm)2 − 1 2
.
| F 0 | = | D 1 | (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) + (G 2 \ E 0 ) ∩ E 1 + |G 1 \ D 1 | | E 0 ∩ E 1 | + E 0 ∩ (G 2 \ E 1 ) 2 n − n (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) + (G 2 \ E 0 ) ∩ E 1 = 2
J. Polhill / Journal of Combinatorial Theory, Series A 117 (2010) 1027–1036
+ = =
n
n2 + n 2
2
2
1033
| E 0 ∩ E 1 | + E 0 ∩ (G 2 \ E 1 )
n |G 2 | + | E 0 ∩ E 1 | + E 0 ∩ (G 2 \ E 1 ) − (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) − (G 2 \ E 0 ) ∩ E 1
(nm)2 2
2 (nm)2 − nm n + | E 0 | − |G 2 \ E 0 | = . 2 2
Now suppose that φ is a nonprincipal character on G 1 × G 2 . Then φ = χ ⊗ ψ , where χ is a character on G 1 and ψ is a character on G 2 . To use Theorem 1.1, we need to show that either: −1 and φ( F i ) = ±nm for i = 2, 3 or 1. φ( R ) = ±nm 2
−1 and φ( F i ) = ∓nm for i = 0, 1. 2. φ( R ) = ±nm 2
Case 1: have:
χ is principal on G 1 while ψ is nonprincipal on G 2 . We have χ ( Q ) =
±m−1 2
. Then we
φ( R ) = | D 0 | ψ E 3 ∩ Q c + ψ E 2 ∩ Q c + ψ (G 2 \ E 0 ) ∩ Q c + ψ (G 2 \ E 1 ) ∩ Q c + |G 1 \ D 0 | ψ (G 2 \ E 3 ) ∩ Q + ψ (G 2 \ E 2 ) ∩ Q + ψ( E 0 ∩ Q ) + ψ( E 1 ∩ Q ) + | P | = | D 0 | ψ Q c + |G 1 \ D 0 | ψ( Q ) + | P | n2 − n ∓m − 1 n2 + n ±m − 1 n2 − 1 ±nm − 1 = + + = , 2 2 2 2 2 2 φ( F 0 ) = | D 1 | ψ (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) + ψ (G 2 \ E 0 ) ∩ E 1 + |G 1 \ D 1 | ψ( E 0 ∩ E 1 ) + ψ E 0 ∩ (G 2 \ E 1 ) =
n2 − n 2
n2 + n ψ(G 2 \ E 0 ) + ψ( E 0 ) . 2
Similarly, we get:
φ( F 1 ) = φ( F 2 ) = φ( F 3 ) =
n2 − n 2 n2 − n 2 n2 − n 2
n2 + n ψ(G 2 \ E 1 ) + ψ( E 1 ) , 2
n2 + n ψ(G 2 \ E 3 ) + ψ( E 3 ) , 2
n2 + n ψ(G 2 \ E 2 ) + ψ( E 2 ) . 2
If we are in the case that ψ( E 0 ) = 0, then since ψ( Q ) = ±m2−1 it must be that ψ( E 0 ) = ∓m. It follows that ψ( E 1 ) = ψ( E 2 ) = ψ( E 3 ) = 0. So φ( F 1 ) = φ( F 2 ) = φ( F 3 ) = 0, while φ( F 0 ) = n +n (∓m) 2 2
n2 −n (±m) 2
+
= ∓nm. Therefore, F 0 and R have character sums of opposite sign. The cases when ψ( E i ) =
0 for i = 0 are similar. Case 2: χ is nonprincipal on G 1 while ψ is principal on G 2 .
φ( R ) = χ ( D 0 ) E 3 ∩ Q c − (G 2 \ E 3 ) ∩ Q + χ ( D 1 ) E 2 ∩ Q c − (G 2 \ E 2 ) ∩ Q + χ ( D 2 ) (G 2 \ E 0 ) ∩ Q c − | E 0 ∩ Q | + χ ( D 3 ) (G 2 \ E 1 ) ∩ Q c − | E 1 ∩ Q | + χ ( P ) = χ (D0) |E3| − | Q | + χ (D1) |E2| − | Q | + χ (D2) Q c − |E0| + χ (D1) Q c − |E1| + χ ( P )
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−m + 1 m−1 + χ (D2) + χ (D3) + χ ( P ), = χ (D0) + χ (D1) 2 2 φ( F 0 ) = χ ( D 1 ) (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) − | E 0 ∩ E 1 | + χ ( D 3 ) (G 2 \ E 0 ) ∩ E 1 − E 0 ∩ (G 2 \ E 1 ) = χ ( D 1 )(m) + χ ( D 3 )(0) = mχ ( D 1 ). Similarly we have:
φ( F 1 ) = mχ ( D 0 ),
φ( F 2 ) = mχ ( D 3 ),
φ( F 3 ) = mχ ( D 2 ).
−1 If χ ( D 0 ) = ±n then χ ( P ) = ∓n2−1 , so φ( R ) = ±(n)( −m2+1 ) + ∓n2−1 = ∓nm and φ( F 1 ) = ±nm so 2 the character sums on R and F 1 are of opposite sign. The remaining cases are similar and left to the reader. Case 3: Both χ and ψ are nonprincipal. Before we examine cases, we will simplify the character sums for R and the F i .
φ( R ) = χ ( D 0 ) ψ E 3 ∩ Q c − ψ (G 2 \ E 3 ) ∩ Q + χ ( D 1 ) ψ E 2 ∩ Q c − ψ (G 2 \ E 2 ) ∩ Q + χ ( D 2 ) ψ (G 2 \ E 0 ) ∩ Q c − ψ( E 0 ∩ Q ) + χ D 3 ψ (G 2 \ E 1 ) ∩ Q c − ψ( E 1 ∩ Q ) + χ ( P ) = χ ( D 0 ) ψ( E 3 ) − ψ( Q ) + χ ( D 1 ) ψ( E 2 ) − ψ( Q ) − χ ( D 2 ) ψ( E 0 ) + ψ( Q ) − χ ( D 3 ) ψ( E 1 ) + ψ( Q ) + χ ( P ), φ( F 0 ) = χ ( D 1 ) ψ (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) − ψ( E 0 ∩ E 1 ) + χ ( D 3 ) ψ (G 2 \ E 0 ) ∩ E 1 − ψ E 0 ∩ (G 2 \ E 1 ) = −χ ( D 1 ) ψ( E 0 ) + ψ( E 1 ) + χ ( D 3 ) ψ( E 1 ) − ψ( E 0 ) . Similarly:
φ( F 1 ) = −χ ( D 0 ) ψ( E 0 ) + ψ( E 1 ) + χ ( D 2 ) ψ( E 0 ) − ψ( E 1 ) . φ( F 2 ) = χ ( D 1 ) ψ( E 2 ) − ψ( E 3 ) − χ ( D 3 ) ψ( E 2 ) + ψ( E 3 ) . φ( F 3 ) = χ ( D 0 ) ψ( E 3 ) − ψ( E 2 ) − χ ( D 2 ) ψ( E 2 ) + ψ( E 3 ) . Suppose that ∓m−1
χ ( D 0 ) = ±n so χ ( P ) =
∓n−1 2
. As a subcase suppose also that ψ( E 0 ) = ±m so
χ ( Q ) = 2 . Clearly φ( F 0 ) = φ( F 2 ) = φ( F 3 ) = 0. φ( R ) can be simplified to φ( R ) = χ ( D 0 )(−ψ( Q ))+ χ ( P ) = ∓n( ∓m2−1 ) + ∓n2−1 = (∓n)(∓2 m)−1 . φ( F 1 ) = −χ ( D 0 )(ψ( E 0 )) = −(±n)(±m), which is the oppo-
site sign of φ( R ). The subcase when ψ( E 1 ) is extremely similar. So suppose instead that ψ( E 2 ) = ±m so that χ ( Q ) = ±m2−1 . Now we have clearly that φ( F i ) = 0 except for i = 3. φ( R ) = χ ( D 0 )(−ψ( Q )) +
χ ( P ) = ∓n( ±m2−1 ) +
∓n−1 2
=
(∓n)(±m)−1 2
. φ( F 3 ) = (±n)(∓m), which is of the same sign as φ( R ).
The last subcase is that ψ( E 3 ) = ±m so that
φ( R ) =
(±n)(±m)−1
χ (Q ) =
±m−1 2
. Then a similar calculation shows that
while φ( F 3 ) = (±n)(±m), again of the same sign. 2 The case where χ ( D 1 ) = 0 is a symmetric case to χ ( D 0 ) so we leave it to the reader. The cases where χ ( D 2 ) = 0 and χ ( D 3 ) = 0 are also symmetric, so we only prove the former case. Suppose that χ ( D 2 ) = ±n so χ ( P ) = ±n2−1 . As a subcase suppose also that ψ( E 0 ) = ±m so
χ (Q ) =
∓m−1
. Clearly φ( F 0 ) = φ( F 2 ) = φ( F 3 ) = 0. φ( R ) can be simplified to −χ ( D 2 )(ψ( E 0 ) 2 ψ( Q )) + χ ( P ). = ∓n( ±m2−1 ) + ∓n2−1 = (∓n)(±2 m)−1 . φ( F 1 ) = χ ( D 2 )(ψ( E 0 )) = (±n)(±m), which the opposite sign of φ( R ). If instead ψ( E 1 ) = ±m we have χ ( Q ) = ∓m2−1 and again φ( F 0 ) φ( F 2 ) = φ( F 3 ) = 0. φ( R ) can be simplified to −χ ( D 2 )(ψ( E 0 ) + ψ( Q )) + χ ( P ) = ∓n( ∓m2−1 ) + ∓n2−1
+ is
= =
J. Polhill / Journal of Combinatorial Theory, Series A 117 (2010) 1027–1036
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(∓n)(∓m)−1
. φ( F 1 ) = χ ( D 2 )(−ψ( E 1 )) = (±n)(∓m), which is the opposite sign of φ( R ). The remaining 2 cases when ψ( E 2 ) = 0 and ψ( E 3 ) = 0 are symmetric, so we prove the former. ψ( E 2 ) = ±m implies that χ ( Q ) = ±m2−1 . φ( F i ) = 0 except for i = 3. φ( R ) can be simplified to = −χ ( D 2 )(ψ( E 0 ) + ψ( Q )) +
χ ( P ). = ∓n( ±m2−1 ) + ∓n2−1 = (∓n)(±2 m)−1 . φ( F 3 ) = −χ ( D 2 )(ψ( E 2 )) = (∓n)(±m), which is the same sign as φ( R ). 2
Remark. The construction of the building sets F i in Theorem 4.1 is essentially the product theorem of Turyn mentioned previously regarding Williamson matrices and applicable to Hadamard difference sets [14]. Putting the recursive theorem together with the propositions from the previous section gives us the following corollaries. Corollary 4.2. Let n be a positive odd number with n > 1. Then there is a Paley partial difference set in a group of order n4 . Proof. Let the prime factorization of n4 = p 1 4t1 p 2 4t2 p 3 4t3 · · · pk 4tk . Using Theorem 4.1 in conjunction with Proposition 3.4 we obtain a Paley PDS in the group G = Z p 1 4t1 × Z p 2 4t2 × Z p 3 4t3 × · · · × Z pk 4tk . 2 Corollary 4.3. Let n be a positive odd number. Then there is a Paley partial difference set in a group of order 9n4 . Proof. The case n = 1 is known. Otherwise, let the prime factorization of 9n4 = 9p 1 4t1 p 2 4t2 p 3 4t3 · · · pk 4tk . Using Theorem 4.1 together with Propositions 3.4 and 3.5 we obtain a Paley PDS in the group G = Z3 2 × Z p 1 4t1 × Z p 2 4t2 × Z p 3 4t3 × · · · × Z pk 4tk . 2 5. New Paley–Hadamard difference sets We will now show how the Paley PDSs constructed in Theorem 4.1 can be used to construct new Paley–Hadamard difference sets. The following is due to Stanton and Sprott [12]. q(q+2)−1
q(q+2)−3
, )Theorem 5.1. Suppose that q and q + 2 are both prime powers. Then there is a (q(q + 2), 2 2 Hadamard difference set in E A (q) × E A (q + 2), where E A (q) denotes the elementary abelian group of order q and is the additive group of F q . In [15], there are the following generalizations of Stanton and Sprott’s earlier result. G ∗ denotes the nonzero elements of G. 1 v −5 v −1 , 4 , 4 )-Paley partial difference set P with 0 ∈ /P Theorem 5.2. Suppose that the group G has a ( v , v − 2 1 v −1 , ) -skew Hadamard difference set S. Then and that the group G having order v + 2 has a ( v + 2, v + 2 4
∗
D = (G × {0}) ∪ ( P × S ) ∪ ((G ∗ \ P ) × (G \ S )) is a ( v ( v + 2), difference set in the group G × G .
v ( v +2)−1 v ( v +2)−3 , )-Paley–Hadamard 2 4
1 v −5 v −1 Theorem 5.3. Suppose that the group G has a ( v , v − , 4 , 4 )-Paley partial difference set P with 0 ∈ /P 2
3 v −5 , 4 )-skew Hadamard difference set S. Then D = and that the group having order v − 2 has a ( v − 2, v − 2
({0} × G ) ∪ ( P × S ) ∪ ((G ∗ \ P ) × (G ∗ \ S )) is a ( v ( v − 2), set in the group G × G .
v ( v −2)−1 v ( v −2)−3 , )-Paley–Hadamard difference 2 4
Combining Theorem 4.1 with these two results we can get new Stanton–Sprott difference sets. These difference sets will have parameters distinct from previous constructions. Three examples of such Paley–Hadamard DSs are in the following groups: Z3 2 × Z5 4 × Z17 4 × Z469805627 , Z7 4 × Z17 4 × Z200533919 , and Z3 4 × Z5 4 × Z7 4 × Z11 4 × Z13 4 × Z17 4 × Z4245189013920980750627 . In each case, we have a Paley PDS from Theorem 4.1 in the product consisting of all but the last component and a skew
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Hadamard difference set in the latter component since the latter is Z p for a prime p ≡ 3 (mod 4). We summarize with the following corollaries. Corollary 5.4. Suppose that for an odd number n that n4 ± 2 is a prime power. Then there is a Paley–Hadamard difference set in a group of order n4 (n4 ± 2). Corollary 5.5. Suppose that for an odd number n that 9n4 ± 2 is a prime power. Then there is a Paley– Hadamard difference set in a group of order 9n4 (9n4 ± 2). Prior to the Paley–Hadamard difference sets constructed here and in [11], the only known constructions of these DSs were in groups of prime power order, order 2t − 1 in the Singer family, or q(q + 2) for prime powers q and q + 2 in the twin prime power family. Our construction of Paley PDSs provides a generalization of the twin prime power family, where the piece containing the Paley PDS is not required to be of prime power order. Acknowledgments The author would like to thanks the referees for their careful reading of the paper, especially for the suggestion of allowing both G 1 and G 2 to be arbitrary groups with a covering extended building set with amicable Paley partial difference set in Theorem 4.1. References [1] T. Beth, D. Jungnickel, H. Lenz, Design Theory, second ed., Encyclopedia Math. Appl., vol. 78, Cambridge University Press, Cambridge, 1999. [2] Y.Q. Chen, On the existence of abelian Hadamard difference sets and a new family of difference sets, Finite Fields Appl. 3 (1997) 234–256. [3] J.A. Davis, Partial difference sets in p-groups, Arch. Math. 63 (1994) 103–110. [4] J.A. Davis, J. Jedwab, A unifying construction for difference sets, J. Combin. Theory Ser. A 80 (1) (1997) 13–78. [5] J.A. Davis, J. Jedwab, Nested Hadamard difference sets, J. Statist. Plann. Inference 62 (1997) 13–20. [6] J. Jedwab, Generalized perfect arrays and Menon difference sets, Des. Codes Cryptogr. 2 (1992) 19–68. [7] D. Jungnickel, Difference Sets, Contemporary Design Theory, Wiley–Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992, 241–324 pp. [8] R.G. Kraemer, Proof of a conjecture on Hadamard 2-groups, J. Combin. Theory Ser. A 63 (1) (1993) 1–10. [9] S.L. Ma, A survey of partial difference sets, Des. Codes Cryptogr. 4 (1994) 221–261. [10] R.E.A.C. Paley, On orthogonal matrices, J. Math. Phys. 12 (1933) 311–320. [11] J. Polhill, Paley partial difference sets in non-p-groups, Des. Codes Cryptogr. 52 (2009) 163–169. [12] R.G. Stanton, D.A. Sprott, A family of difference sets, Canad. J. Math. 13 (1958) 73–77. [13] R.J. Turyn, Character sums and difference sets, Pacific J. Math. 15 (1965) 319–346. [14] R.J. Turyn, A special class of Williamson matrices and difference sets, J. Combin. Theory Ser. A 36 (1984) 111–115. [15] G. Weng, L. Hu, Some results on skew Hadamard difference sets, Des. Codes Cryptogr. 50 (1) (2009) 93–105. [16] R.M. Wilson, Q. Xiang, Constructions of Hadamard difference sets, J. Combin. Theory Ser. A 77 (1997) 148–160. [17] M.-Y. Xia, Some infinite classes of special Williamson matrices and difference sets, J. Combin. Theory Ser. A 61 (1992) 230–242. [18] Q. Xiang, Y.Q. Chen, On Xia’s construction of Hadamard difference sets, Finite Fields Appl. 2 (1996) 87–95. [19] K. Yamamoto, Decomposition fields of difference sets, Pacific J. Math. 13 (1963) 337–352.
Contents lists available at ScienceDirect
Journal of Combinatorial Theory, Series A www.elsevier.com/locate/jcta
Paley partial difference sets in groups of order n4 and 9n4 for any odd n > 1 John Polhill Department of Mathematics, Computer Science, and Statistics, Bloomsburg University, Bloomsburg, PA 17815, United States
a r t i c l e
i n f o
a b s t r a c t 1 v −5 v −1 A partial difference set with parameters ( v , v − , 4 , 4 ) is said 2 to be of Paley type. In this paper, we give a recursive theorem that for all odd n > 1 constructs Paley partial difference sets in certain groups of order n4 and 9n4 . We are also able to construct Paley– Hadamard difference sets of the Stanton–Sprott family in groups of order n4 (n4 ± 2) when n4 ± 2 is a prime power and 9n4 (9n4 ± 2) when 9n4 ± 2 is a prime power. Many of these are new parameters for such difference sets, and also give new Hadamard designs and matrices. © 2010 Elsevier Inc. All rights reserved.
Article history: Received 17 February 2009 Available online 4 March 2010 Keywords: Paley partial difference set Paley Hadamard difference set Hadamard difference set Difference set Partial difference set Strongly regular graph
1. Introduction Let G be a finite group of order v, and let D be a subset of G with cardinality k. Then D is a ( v , k, λ)-difference set (DS) provided that the list of differences d1 d2−1 , d1 , d2 ∈ D represents every nonidentity element in G exactly λ times. A difference set, D, is called reversible if d ∈ D implies d−1 ∈ D. The text of Beth, Jungnickel, and Lenz [1] and the survey of Jungnickel [7] are excellent references for DSs. Difference sets are of widespread interest largely due to the fact that a difference set is equivalent to a symmetric design with a regular automorphism group. Hadamard (Menon) difference sets, having parameters (4u 2 , 2u 2 − u , u 2 − u ), are of particular interest due to the fact that their ±1 incidence matrices form Hadamard matrices. Another important family of difference sets is the Hadamard or 1 v −3 Paley–Hadamard difference sets having parameters ( v , v − , 4 ). It is somewhat unfortunate that two 2 distinct families of difference sets are given the name Hadamard, so for this paper we will distinguish them by calling the first Hadamard and the second Paley–Hadamard. When a Paley–Hadamard difference set also has the property that G is the disjoint union of D, the inverses of the elements of D, and 0, it is called a skew Hadamard difference set.
E-mail address: [email protected]. 0097-3165/$ – see front matter doi:10.1016/j.jcta.2010.02.008
© 2010
Elsevier Inc. All rights reserved.
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Now suppose G is 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 = 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) / D and D (−1) = D we call the PDS D regular. The survey article of Ma in G. When the identity e ∈ is an excellent survey on PDSs [9]. Closely related to Paley–Hadamard difference sets are Paley type 1 v −5 v −1 , 4 , 4 ). Paley originally discovered these sets [10] partial difference sets, having parameters ( v , v − 2 along with the Paley–Hadamard difference sets in the context of Hadamard matrices, and in fact both Paley PDSs and Paley–Hadamard difference sets can be used to construct both Hadamard designs and matrices. While Paley PDSs were known to exist in elementary abelian groups since Paley’s work in the 1930’s, the first construction of these sets in nonelementary abelian groups was by Davis in 1994 [3]. This provided an answer to one of the following questions in the survey of Ma [9]: Questions 13.4. Suppose G is an abelian group of order v ≡ 1 (mod 4). If v is not a prime power, does there exist a Paley PDS in G? If v is a prime power, does G need to be elementary abelian? In a previous paper we gave the first construction of Paley PDSs in groups having an order which is not a prime power [11]. However, the Paley PDSs were produced only in those groups of the form Z3 2 × Z p 4t for an odd prime p. In this paper we provide a much more general construction and show that Paley PDSs exist for all group orders n4 and 9n4 where n is odd and n > 1. As a corollary, we will also obtain new parameters for Paley–Hadamard difference sets. The constructions in this paper depend heavily on the existence of building sets, first introduced by Davis and Jedwab [4]. The particular variety of building set that we require is also crucial for the construction of certain types of Hadamard difference sets. For this reason, we will give a brief summary of some of the known results on abelian Hadamard difference sets in Section 2. DSs and PDSs can be studied within the context of the group ring Z[G]. For a subset D in G we write D = d∈ D d and denote D (−1) = d∈ D d−1 . Character theory is a powerful tool in the study of difference sets and partial difference sets in abelian groups. Turyn [13] and Yamamoto [19] first applied character theory to the study of abelian difference sets. A character of an abelian group G is a homomorphism from the group to the complex numbers with modulus 1. The principal character is the homomorphism that maps every element of G to 1; any other character is called nonprincipal. One can naturally extend a character on G to a homomorphism of the group ring Z[G ] as follows: if χ is a character on G, then for an element A = g ∈G a g g let χ ( A ) = g ∈G a g χ ( g ) so that if S is a subset of G, then χ ( S ) = s∈ S χ (s). See [13] for a proof of similar results to the following. Theorem 1.1. (a) Let G be an abelian group of order v with a subset D of cardinality k and let λ be a positive integer satisfying λ( v − 1) = k(k − 1√ ). Then D is a ( v , k, λ)-difference set in G if and only if for every nonprincipal character χ on G, |χ ( D )| = k − λ. / P with k2 = k + λk + (b) Let G be an abelian group of order v with a subset P of cardinality k such that 0 ∈ μ( v − k − 1) for positive integers λ and μ. Then P is a ( v ,√ k, λ, μ)-partial difference set in G if and only if for every nonprincipal character χ on G, χ ( P ) =
(λ−μ)±
(λ−μ)2 +4(k−μ) 2
.
2. Known constructions of Hadamard difference sets Hadamard difference sets are quite likely the most widely studied of all the families of difference sets. A theorem by Turyn [14] shows that if there exist Hadamard difference sets in the abelian groups K × G 1 and K × G 2 where | K | = 4 and |G i | = mi 2 for mi odd, then there is a Hadamard difference set in K × G 1 × G 2 . Therefore, it makes sense to consider groups of the form K × G where |G | = p r , for p a prime and r even. The answer of existence of Hadamard difference sets is completely determined in the case when the group is an abelian 2-group. An abelian group G of order 22d+2 contains a Hadamard difference set if and only if G satisfies exp(G ) 2d+2 (exp(G ) means the order of the element of G with highest additive order) ([6] or [8]).
J. Polhill / Journal of Combinatorial Theory, Series A 117 (2010) 1027–1036
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For the case where p = 3, a Hadamard difference set exists in every group of the form K × Z3r × Z3r = K × (Z3r )2 for all positive integers r where K is either group of order 4 [1]. A Hadamard difference set does not exist in G × Z3r , where |G | is relatively prime to 3 and r is any positive integer. For p > 3, Davis and Jedwab [5] showed that no Hadamard difference set exists in groups of the form K × (Z pr )2 provided that none exists in K × (Z p )2 , where again K is a group of order 4. Such difference sets do not exist for p > 3. Of particular importance to this paper is the Hadamard difference set construction for K × Z p 4t where | K | = 4 and p is an odd prime. Xia made the first major breakthrough on this problem, constructing difference sets for p ≡ 3 mod 4 in [17]. His approach was simplified by Xiang and Chen [18] using the additive characters of finite fields. Then Wilson and Xiang [16] related the existence of this type of difference set to certain projective sets, and Chen impressively used their work to construct Hadamard difference sets in all groups K × Z p 4t for | K | = 4 and p an odd prime [2]. We will now consider this construction. Let P G (k − 1, q) denote the projective space of dimension k − 1 over F q for q a power of a prime; the corresponding vector space V (k, q) for PG(k − 1, q) will have dimension k over F q . The elements of PG(k − 1, q) are the subspaces of V (k, q), and, in particular, a projective point is a 1-dimensional subspace, a projective line is a 2-dimensional space, and a hyperplane is a (k − 1)-dimensional subspace. A projective (n, k, h1 , h2 ) set O is a proper subset of the projective space PG(k − 1, q) with n points (n = 0) so that O intersects every hyperplane in h1 or h2 points. If we have that O = { y 1 , y 2 , . . . , yn } then let Ω = {x ∈ V (k, q) | x ∈ O}. The following lemma shows that O is a projective set if and only if Ω is a PDS in the additive group of the corresponding vector space under certain conditions. Wilson and Xiang have this result in [16]. Lemma 2.1. O is a projective (n, k, h1 , h2 ) set if and only if χ (Ω) = qh1 − n or qh2 − n for every nonprincipal additive character. Therefore O is a projective (n, k, h1 , h2 ) set if and only if Ω is a PDS in the additive group of V (k, q) and provided that K 2 = K + λ K + μ( v − K − 1) (K is the cardinality of Ω , while λ and μ are the usual PDS parameters). Now consider the projective space Σ3 = PG(3, q) for an odd prime power q = pt . The additive group of the corresponding vector space is the elementary abelian group Z p 4t . Define a spread of Σ3 to be a set of q2 + 1 projective lines which are pairwise disjoint and partition the points of Σ3 . Finally, q4 −1
a subset of Type Q is a projective ( 4(q−1) , 4,
(q−1)2 4
2
, (q+41) ) set in Σ3 .
Theorem 2.2. (See Xiang and Wilson [16].) Suppose that S = { L 1 , L 2 , . . . , L q2 +1 } is a spread of Σ3 . If there q2 +1 2 × Z p 4t
exist two subsets of Type Q , C 0 and C 1 , in Σ3 with the property that |C 0 ∩ L i | = (q + 1)/2 for 1 i q +1 2 2
and |C 1 ∩ L j | = (q + 1)/2 for for | K | = 4.
+ 1 j q2 + 1, then there exists a Hadamard difference set in K
We make some observations regarding this theorem. If two such subsets of Type Q exist, then the sets
C2 =
Li \ C0
and
q 2 +1 1 i 2
C3 =
L j \ C1
q 2 +1 2 2 +1 j q +1
are also subsets of Type Q . Now let W = V (4, q) be the underlying vector space for Σ3 . Then let Ci = { w ∈ W | w ∈ C i } and Li = { w ∈ W | w ∈ L i } .
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Each of the four subsets C i in Σ3 gives rise to a corresponding PDS Ci in V (4, q) such that the character values on each is q2 −1 4
q2 −1 4 2
or
q2 −1 4
− q2 for nonprincipal additive characters on V (4, q). The
character sum will be − q for exactly one of the sets C0 , C1 , C2 , C3 for any nonprincipal character. We summarize with the following lemma for convenience. Lemma 2.3. Let C0 , C1 , C2 , C3 be the PDSs in the additive group of W = V (4, q) associated to the four subsets of Type Q C 0 , C 1 , C 2 , C 3 that partition Σ3 . Then for any nonprincipal character on the additive group of W , the character sum will be
q2 −1 4
− q2 for exactly one of the sets C0 , C1 , C2 , C3 and
To each of C0 and C2 we will add call this collection of
q −1 4 2
q2 −1 4
q2 −1 4
for all others.
of the sets L j which are disjoint from C0 and C2 ; we
projective lines B 1 with corresponding set B1 = { w ∈ W | w ∈ B 1 }. The q2 −1
q2 −1
character sum on B1 will be either − 4 or q2 − 4 , and when combined with the PDSs from subsets of Type Q , the character sums are either 0 or ±q2 . q2 −1
We similarly add 4 of the sets Li (disjoint from C1 and C3 ) to C1 and C3 , and call this collection B 0 . When we take the four sets
D0 = C0 ∪ B1 ,
D1 = C1 ∪ B0 ,
D2 = C2 ∪ B1 ,
D3 = C3 ∪ B0 ,
the character sums will be such that for any nonprincipal character χ on G = Z p 4t , χ (Di ) = ±q2 for exactly one i and χ (Dk ) = 0 for all other k. Then the four sets G \ D0 , D1 , D2 , D3 form a
( p −2 p , p 2 , 4, +) covering extended building set using the terminology from [4], so that if K = {a0 , a1 , a2 , a3 } then 4
2
H = a0 (G \ D0 ) ∪ a1 D1 ∪ a2 D2 ∪ a3 D3 is a Hadamard difference set in K × Z p 4t . 3. Extended building sets with amicable Paley PDSs It is known that the group Z p 4t not only has the four sets G \ D0 , D1 , D2 , D3 that form a covering extended building set, but also possess Paley partial difference sets. In this section, we will show that they contain building sets and Paley partial difference sets which relate in a certain way in terms of character sum properties. In the subsequent section, we will employ these related sets in a recursive theorem for generating Paley PDSs in a product group. The material in this section prior to the definition of extended building set with amicable Paley partial difference set was introduced in [11]. We begin by forming two more sets B 2 and B 3 as follows: B2 = (C0 ∪ C2 ) \ B0 and B3 =
(C1 ∪ C3 ) \ B1 so that each of B 2 and B 3 are unions of sums on the Bi are given in the following lemmas. Lemma 3.1. Let
χ (B2 ), χ (B3 ) ∈
q2 +3 4
projective lines. The properties of character
χ be a nonprincipal character on V (4, q). Then χ (B0 ), χ (B1 ) ∈ {− q 4−1 , q2 − 2
2 {− q 4+3 , q2
−
q2 +3 }. 4
q2 −1 } 4
and
Furthermore, exactly one Bi will have a positive character sum.
Proof. The character sums immediately follow from Theorem 1.1. The fact that there will be exactly one positive character sum follows from the fact that Σ3 = B 0 ∪ B 1 ∪ B 2 ∪ B 3 . 2 Lemma 3.2. Let χ be a nonprincipal character on V (4, q). Then either one of χ (B0 ) and χ (B2 ) will take a positive value or one of χ (C0 ) and χ (C2 ) will take a negative value, and it is not possible that both occur. That is to say, exactly one element of the following set will be positive: {χ (B0 ), χ (B2 ), −χ (C0 ), −χ (C2 )}. The same is true replacing {0, 2} with {1, 3}.
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Proof. B0 ∪ B2 = C0 ∪ C2 = S, where S is a Paley PDS. If χ ( S ) > 0, then one of χ (B0 ) and χ (B2 ) will be positive and both χ (C0 ) and χ (C2 ) will be positive. If χ ( S ) < 0, then one of χ (C0 ) and χ (C2 ) will be negative and both χ (B0 ) and χ (B2 ) will be negative. 2 Now we form a Paley partial difference set L in the additive group of V (4, q) by taking L = B1 ∪ B2 . L has special properties with respect to the sets Di as given in the following key lemma. Lemma 3.3. Let χ be a nonprincipal character on V (4, q). For i = 0 or 2, if χ (Di ) = ±q2 then χ ( L ) = For i = 1 or 3, if χ (Di ) = ±q then χ ( L ) = 2
∓q2 −1 2
±q2 −1 2
.
.
χ is principal. Suppose χ (D0 ) = q2 . Then q2 −1 it must be the case that χ (B1 ) is positive. It follows that χ ( L ) = 2 . If instead χ (D0 ) = −q2 then it 2 q −1 −q2 −1 follows that χ (B1 ) is negative. Moreover, χ (C0 ) = 4 − q2 so that χ (C0 ∪ C2 ) = χ (B0 ∪ B2 ) = 2 . This ensures that χ (B0 ) and χ (B2 ) must both be negative. Then it must be the case that χ (B3 ) is −q2 −1 positive, so that χ ( L ) = 2 . The cases i = 0 are similar. 2 Proof. There will be exactly one projective line L i on which
This arrangement of building sets with Paley PDS will prove quite useful, so we will give a careful definition. For ease, we rearrange the building sets so that the first two in the list have character sums of opposite sign as the Paley PDS and the third and fourth have same sign. Specifically, we will call such a combination of sets ( D 0 , D 1 , D 2 , D 3 , P ) in a group G of order n2 an extended building set with amicable Paley partial difference set when the following properties are valid: 2 1. The four sets G \ D 0 , D 1 , D 2 , D 3 form a ( n 2−n , n, 4, +) covering extended building set using the
terminology from [4]. This means that | D i | = n 2−n for all i, and that for any nonprincipal character on G, χ ( D j ) = ±n for a particular j and χ ( D i ) = 0 for i = j. 2. P is a Paley PDS in G. 3. For any nonprincipal character χ on G, if χ ( D i ) = ±n for i = 0, 1 then χ ( P ) = ∓n2−1 . That is, when χ ( D 0 ) (or χ ( D 1 )) is nonzero, χ ( D 0 ) (or χ ( D 1 )) will have the opposite sign as χ ( P ). 4. For any nonprincipal character χ on G, if χ ( D i ) = ±n for i = 2, 3 then χ ( P ) = ±n2−1 . That is, when χ ( D 2 ) (or χ ( D 3 )) is nonzero, χ ( D 2 ) (or χ ( D 3 )) will have the same sign as χ ( P ). 2
If ( D 0 , D 1 , D 2 , D 3 , P ) has the following additional properties then we say it satisfies the composition criteria: 1. P is the union of the disjoint sets ( D 2 ∩ D 3 ) and ((G \ D 0 ) ∩ (G \ D 1 )) − {0}; 2. G − P is the union of the disjoint sets ( D 0 ∩ D 1 ) and ((G \ D 2 ) ∩ (G \ D 3 )). In fact, we have already shown that Z p 4t has such a collection of sets. It is straightforward to show that Z3 2 also has such a collection. Proposition 3.4. For any odd prime p and positive integer t, Z p 4t contains an extended building set with amicable Paley partial difference set that satisfies the composition criteria. Proof. Lemma 3.3 shows that (D1 , D3 , D0 , D2 , L ) is an extended building set with amicable Paley partial difference set satisfying the composition criteria since: L = B 1 ∪ B2 = (D0 ∩ D2 ) ∪ ((Z p 4t \ D1 ) ∩ (Z p 4t \ D3 )) − {0} and similarly for Z p 4t \ L. 2 Proposition 3.5. Z3 2 contains an extended building set with amicable Paley partial difference set that satisfies the composition criteria.
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Proof. Let H 0 , H 1 , H 2 , and H 3 be the subgroups of order 3 in Z3 2 , and H i ∗ = H i \ {(0, 0)}. Then ( H 0 , H 1 , H 2 , H 3 , H 2 ∗ ∪ H 3 ∗ ) can be shown to be an extended building set with amicable Paley partial difference set that satisfies the composition criteria. 2 4. Paley PDSs in non- p-groups Theorem 4.1 (Paley partial difference sets in non-p-groups). Let G 1 be a group of order n2 having a covering extended building set with amicable Paley partial difference set ( D 0 , D 1 , D 2 , D 3 , P ), and let G 2 be a group of order m2 having a covering extended building set with amicable Paley partial difference set ( E 0 , E 1 , E 2 , E 3 , Q ) that satisfies the composition criteria. Then the group G 1 × G 2 has a covering extended building set with amicable Paley partial difference set ( F 0 , F 1 , F 2 , F 3 , R ). Proof. First we define the set R, which we will prove is a Paley PDS. Q c = (G 2 \ Q ) \ {0}.
∪ (G 1 \ D 0 ) × (G 2 \ E 3 ) ∩ Q ∪ D 1 × E 2 ∩ Q c ∪ (G 1 \ D 1 ) × (G 2 \ E 2 ) ∩ Q ∪ (G 1 \ D 2 ) × ( E 0 ∩ Q ) ∪ D 2 × (G 2 \ E 0 ) ∩ Q c ∪ ( G 1 \ D 3 ) × ( E 1 ∩ Q ) ∪ D 3 × ( G 2 \ E 1 ) ∩ Q c ∪ P × { 0} .
R = D0 × E3 ∩ Q c
The fact that ( E 0 , E 1 , E 2 , E 3 , Q ) satisfies the composition criteria in G 2 will ensure that R is not a multiset, since the sets from G 2 used in the definition of R will be pairwise disjoint. For example, suppose g ∈ ( E 3 ∩ Q c ) ∩ ((G 2 \ E 0 ) ∩ Q c ). g ∈ E 3 ∩ Q c implies that g ∈ E 0 ∩ E 1 , so g cannot be in (G 2 \ E 0 ) ∩ Q c . We leave the rest of the cases to the reader. The building sets are as follows:
∪ (G 1 \ D 1 ) × ( E 0 ∩ E 1 ) ∪ D 3 × (G 2 \ E 0 ) ∩ E 1 ∪ (G 1 \ D 3 ) × E 0 ∩ (G 2 \ E 1 ) , F 1 = D 0 × (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) ∪ (G 1 \ D 0 ) × ( E 0 ∩ E 1 ) ∪ D 2 × E 0 ∩ (G 2 \ E 1 ) ∪ (G 1 \ D 2 ) × (G 2 \ E 0 ) ∩ E 1 , F 2 = D 1 × E 2 ∩ (G \ E 3 ) ∪ (G 1 \ D 1 ) × E 3 ∩ (G 2 \ E 2 ) ∪ D 3 × (G 2 \ E 2 ) ∩ (G 2 \ E 3 ) ∪ (G 1 \ D 3 ) × ( E 2 ∩ E 3 ) , F 3 = D 0 × (G 2 \ E 2 ) ∩ E 3 ∪ (G 1 \ D 0 ) × E 2 ∩ (G 2 \ E 3 ) ∪ D 2 × (G 2 \ E 2 ) ∩ (G 2 \ E 3 ) ∪ (G 1 \ D 2 ) × ( E 2 ∩ E 3 ) .
F 0 = D 1 × (G 2 \ E 0 ) ∩ (G 2 \ E 1 )
First we check to see that the sets R and F 0 have the correct cardinality. The sets F i for i = 0 will have the same cardinality as F 0 . We will use that ( E 0 , E 1 , E 2 , E 3 , Q ) satisfies the composition criteria in G 2 implies that Q c is a union of the disjoint ( E 3 ∩ Q c ), ( E 2 ∩ Q c ), ((G 2 \ E 0 ) ∩ Q c ), and ((G 2 \ E 1 ) ∩ Q c ), and similarly for Q .
| R | = | D 0 | E 3 ∩ Q c + E 2 ∩ Q c + (G 2 \ E 0 ) ∩ Q c + (G 2 \ E 1 ) ∩ Q c + |G 1 \ D 0 | (G 2 \ E 3 ) ∩ Q + (G 2 \ E 2 ) ∩ Q + | E 0 ∩ Q | + | E 1 ∩ Q | + P × {0} = =
2 2 Q c + n + n |Q | + n − 1
n2 − n
2 2 2 n2 − n m2 − 1 n2 + n m2 − 1 2
2
+
2
2
+
n2 − 1 2
=
(nm)2 − 1 2
.
| F 0 | = | D 1 | (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) + (G 2 \ E 0 ) ∩ E 1 + |G 1 \ D 1 | | E 0 ∩ E 1 | + E 0 ∩ (G 2 \ E 1 ) 2 n − n (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) + (G 2 \ E 0 ) ∩ E 1 = 2
J. Polhill / Journal of Combinatorial Theory, Series A 117 (2010) 1027–1036
+ = =
n
n2 + n 2
2
2
1033
| E 0 ∩ E 1 | + E 0 ∩ (G 2 \ E 1 )
n |G 2 | + | E 0 ∩ E 1 | + E 0 ∩ (G 2 \ E 1 ) − (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) − (G 2 \ E 0 ) ∩ E 1
(nm)2 2
2 (nm)2 − nm n + | E 0 | − |G 2 \ E 0 | = . 2 2
Now suppose that φ is a nonprincipal character on G 1 × G 2 . Then φ = χ ⊗ ψ , where χ is a character on G 1 and ψ is a character on G 2 . To use Theorem 1.1, we need to show that either: −1 and φ( F i ) = ±nm for i = 2, 3 or 1. φ( R ) = ±nm 2
−1 and φ( F i ) = ∓nm for i = 0, 1. 2. φ( R ) = ±nm 2
Case 1: have:
χ is principal on G 1 while ψ is nonprincipal on G 2 . We have χ ( Q ) =
±m−1 2
. Then we
φ( R ) = | D 0 | ψ E 3 ∩ Q c + ψ E 2 ∩ Q c + ψ (G 2 \ E 0 ) ∩ Q c + ψ (G 2 \ E 1 ) ∩ Q c + |G 1 \ D 0 | ψ (G 2 \ E 3 ) ∩ Q + ψ (G 2 \ E 2 ) ∩ Q + ψ( E 0 ∩ Q ) + ψ( E 1 ∩ Q ) + | P | = | D 0 | ψ Q c + |G 1 \ D 0 | ψ( Q ) + | P | n2 − n ∓m − 1 n2 + n ±m − 1 n2 − 1 ±nm − 1 = + + = , 2 2 2 2 2 2 φ( F 0 ) = | D 1 | ψ (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) + ψ (G 2 \ E 0 ) ∩ E 1 + |G 1 \ D 1 | ψ( E 0 ∩ E 1 ) + ψ E 0 ∩ (G 2 \ E 1 ) =
n2 − n 2
n2 + n ψ(G 2 \ E 0 ) + ψ( E 0 ) . 2
Similarly, we get:
φ( F 1 ) = φ( F 2 ) = φ( F 3 ) =
n2 − n 2 n2 − n 2 n2 − n 2
n2 + n ψ(G 2 \ E 1 ) + ψ( E 1 ) , 2
n2 + n ψ(G 2 \ E 3 ) + ψ( E 3 ) , 2
n2 + n ψ(G 2 \ E 2 ) + ψ( E 2 ) . 2
If we are in the case that ψ( E 0 ) = 0, then since ψ( Q ) = ±m2−1 it must be that ψ( E 0 ) = ∓m. It follows that ψ( E 1 ) = ψ( E 2 ) = ψ( E 3 ) = 0. So φ( F 1 ) = φ( F 2 ) = φ( F 3 ) = 0, while φ( F 0 ) = n +n (∓m) 2 2
n2 −n (±m) 2
+
= ∓nm. Therefore, F 0 and R have character sums of opposite sign. The cases when ψ( E i ) =
0 for i = 0 are similar. Case 2: χ is nonprincipal on G 1 while ψ is principal on G 2 .
φ( R ) = χ ( D 0 ) E 3 ∩ Q c − (G 2 \ E 3 ) ∩ Q + χ ( D 1 ) E 2 ∩ Q c − (G 2 \ E 2 ) ∩ Q + χ ( D 2 ) (G 2 \ E 0 ) ∩ Q c − | E 0 ∩ Q | + χ ( D 3 ) (G 2 \ E 1 ) ∩ Q c − | E 1 ∩ Q | + χ ( P ) = χ (D0) |E3| − | Q | + χ (D1) |E2| − | Q | + χ (D2) Q c − |E0| + χ (D1) Q c − |E1| + χ ( P )
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−m + 1 m−1 + χ (D2) + χ (D3) + χ ( P ), = χ (D0) + χ (D1) 2 2 φ( F 0 ) = χ ( D 1 ) (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) − | E 0 ∩ E 1 | + χ ( D 3 ) (G 2 \ E 0 ) ∩ E 1 − E 0 ∩ (G 2 \ E 1 ) = χ ( D 1 )(m) + χ ( D 3 )(0) = mχ ( D 1 ). Similarly we have:
φ( F 1 ) = mχ ( D 0 ),
φ( F 2 ) = mχ ( D 3 ),
φ( F 3 ) = mχ ( D 2 ).
−1 If χ ( D 0 ) = ±n then χ ( P ) = ∓n2−1 , so φ( R ) = ±(n)( −m2+1 ) + ∓n2−1 = ∓nm and φ( F 1 ) = ±nm so 2 the character sums on R and F 1 are of opposite sign. The remaining cases are similar and left to the reader. Case 3: Both χ and ψ are nonprincipal. Before we examine cases, we will simplify the character sums for R and the F i .
φ( R ) = χ ( D 0 ) ψ E 3 ∩ Q c − ψ (G 2 \ E 3 ) ∩ Q + χ ( D 1 ) ψ E 2 ∩ Q c − ψ (G 2 \ E 2 ) ∩ Q + χ ( D 2 ) ψ (G 2 \ E 0 ) ∩ Q c − ψ( E 0 ∩ Q ) + χ D 3 ψ (G 2 \ E 1 ) ∩ Q c − ψ( E 1 ∩ Q ) + χ ( P ) = χ ( D 0 ) ψ( E 3 ) − ψ( Q ) + χ ( D 1 ) ψ( E 2 ) − ψ( Q ) − χ ( D 2 ) ψ( E 0 ) + ψ( Q ) − χ ( D 3 ) ψ( E 1 ) + ψ( Q ) + χ ( P ), φ( F 0 ) = χ ( D 1 ) ψ (G 2 \ E 0 ) ∩ (G 2 \ E 1 ) − ψ( E 0 ∩ E 1 ) + χ ( D 3 ) ψ (G 2 \ E 0 ) ∩ E 1 − ψ E 0 ∩ (G 2 \ E 1 ) = −χ ( D 1 ) ψ( E 0 ) + ψ( E 1 ) + χ ( D 3 ) ψ( E 1 ) − ψ( E 0 ) . Similarly:
φ( F 1 ) = −χ ( D 0 ) ψ( E 0 ) + ψ( E 1 ) + χ ( D 2 ) ψ( E 0 ) − ψ( E 1 ) . φ( F 2 ) = χ ( D 1 ) ψ( E 2 ) − ψ( E 3 ) − χ ( D 3 ) ψ( E 2 ) + ψ( E 3 ) . φ( F 3 ) = χ ( D 0 ) ψ( E 3 ) − ψ( E 2 ) − χ ( D 2 ) ψ( E 2 ) + ψ( E 3 ) . Suppose that ∓m−1
χ ( D 0 ) = ±n so χ ( P ) =
∓n−1 2
. As a subcase suppose also that ψ( E 0 ) = ±m so
χ ( Q ) = 2 . Clearly φ( F 0 ) = φ( F 2 ) = φ( F 3 ) = 0. φ( R ) can be simplified to φ( R ) = χ ( D 0 )(−ψ( Q ))+ χ ( P ) = ∓n( ∓m2−1 ) + ∓n2−1 = (∓n)(∓2 m)−1 . φ( F 1 ) = −χ ( D 0 )(ψ( E 0 )) = −(±n)(±m), which is the oppo-
site sign of φ( R ). The subcase when ψ( E 1 ) is extremely similar. So suppose instead that ψ( E 2 ) = ±m so that χ ( Q ) = ±m2−1 . Now we have clearly that φ( F i ) = 0 except for i = 3. φ( R ) = χ ( D 0 )(−ψ( Q )) +
χ ( P ) = ∓n( ±m2−1 ) +
∓n−1 2
=
(∓n)(±m)−1 2
. φ( F 3 ) = (±n)(∓m), which is of the same sign as φ( R ).
The last subcase is that ψ( E 3 ) = ±m so that
φ( R ) =
(±n)(±m)−1
χ (Q ) =
±m−1 2
. Then a similar calculation shows that
while φ( F 3 ) = (±n)(±m), again of the same sign. 2 The case where χ ( D 1 ) = 0 is a symmetric case to χ ( D 0 ) so we leave it to the reader. The cases where χ ( D 2 ) = 0 and χ ( D 3 ) = 0 are also symmetric, so we only prove the former case. Suppose that χ ( D 2 ) = ±n so χ ( P ) = ±n2−1 . As a subcase suppose also that ψ( E 0 ) = ±m so
χ (Q ) =
∓m−1
. Clearly φ( F 0 ) = φ( F 2 ) = φ( F 3 ) = 0. φ( R ) can be simplified to −χ ( D 2 )(ψ( E 0 ) 2 ψ( Q )) + χ ( P ). = ∓n( ±m2−1 ) + ∓n2−1 = (∓n)(±2 m)−1 . φ( F 1 ) = χ ( D 2 )(ψ( E 0 )) = (±n)(±m), which the opposite sign of φ( R ). If instead ψ( E 1 ) = ±m we have χ ( Q ) = ∓m2−1 and again φ( F 0 ) φ( F 2 ) = φ( F 3 ) = 0. φ( R ) can be simplified to −χ ( D 2 )(ψ( E 0 ) + ψ( Q )) + χ ( P ) = ∓n( ∓m2−1 ) + ∓n2−1
+ is
= =
J. Polhill / Journal of Combinatorial Theory, Series A 117 (2010) 1027–1036
1035
(∓n)(∓m)−1
. φ( F 1 ) = χ ( D 2 )(−ψ( E 1 )) = (±n)(∓m), which is the opposite sign of φ( R ). The remaining 2 cases when ψ( E 2 ) = 0 and ψ( E 3 ) = 0 are symmetric, so we prove the former. ψ( E 2 ) = ±m implies that χ ( Q ) = ±m2−1 . φ( F i ) = 0 except for i = 3. φ( R ) can be simplified to = −χ ( D 2 )(ψ( E 0 ) + ψ( Q )) +
χ ( P ). = ∓n( ±m2−1 ) + ∓n2−1 = (∓n)(±2 m)−1 . φ( F 3 ) = −χ ( D 2 )(ψ( E 2 )) = (∓n)(±m), which is the same sign as φ( R ). 2
Remark. The construction of the building sets F i in Theorem 4.1 is essentially the product theorem of Turyn mentioned previously regarding Williamson matrices and applicable to Hadamard difference sets [14]. Putting the recursive theorem together with the propositions from the previous section gives us the following corollaries. Corollary 4.2. Let n be a positive odd number with n > 1. Then there is a Paley partial difference set in a group of order n4 . Proof. Let the prime factorization of n4 = p 1 4t1 p 2 4t2 p 3 4t3 · · · pk 4tk . Using Theorem 4.1 in conjunction with Proposition 3.4 we obtain a Paley PDS in the group G = Z p 1 4t1 × Z p 2 4t2 × Z p 3 4t3 × · · · × Z pk 4tk . 2 Corollary 4.3. Let n be a positive odd number. Then there is a Paley partial difference set in a group of order 9n4 . Proof. The case n = 1 is known. Otherwise, let the prime factorization of 9n4 = 9p 1 4t1 p 2 4t2 p 3 4t3 · · · pk 4tk . Using Theorem 4.1 together with Propositions 3.4 and 3.5 we obtain a Paley PDS in the group G = Z3 2 × Z p 1 4t1 × Z p 2 4t2 × Z p 3 4t3 × · · · × Z pk 4tk . 2 5. New Paley–Hadamard difference sets We will now show how the Paley PDSs constructed in Theorem 4.1 can be used to construct new Paley–Hadamard difference sets. The following is due to Stanton and Sprott [12]. q(q+2)−1
q(q+2)−3
, )Theorem 5.1. Suppose that q and q + 2 are both prime powers. Then there is a (q(q + 2), 2 2 Hadamard difference set in E A (q) × E A (q + 2), where E A (q) denotes the elementary abelian group of order q and is the additive group of F q . In [15], there are the following generalizations of Stanton and Sprott’s earlier result. G ∗ denotes the nonzero elements of G. 1 v −5 v −1 , 4 , 4 )-Paley partial difference set P with 0 ∈ /P Theorem 5.2. Suppose that the group G has a ( v , v − 2 1 v −1 , ) -skew Hadamard difference set S. Then and that the group G having order v + 2 has a ( v + 2, v + 2 4
∗
D = (G × {0}) ∪ ( P × S ) ∪ ((G ∗ \ P ) × (G \ S )) is a ( v ( v + 2), difference set in the group G × G .
v ( v +2)−1 v ( v +2)−3 , )-Paley–Hadamard 2 4
1 v −5 v −1 Theorem 5.3. Suppose that the group G has a ( v , v − , 4 , 4 )-Paley partial difference set P with 0 ∈ /P 2
3 v −5 , 4 )-skew Hadamard difference set S. Then D = and that the group having order v − 2 has a ( v − 2, v − 2
({0} × G ) ∪ ( P × S ) ∪ ((G ∗ \ P ) × (G ∗ \ S )) is a ( v ( v − 2), set in the group G × G .
v ( v −2)−1 v ( v −2)−3 , )-Paley–Hadamard difference 2 4
Combining Theorem 4.1 with these two results we can get new Stanton–Sprott difference sets. These difference sets will have parameters distinct from previous constructions. Three examples of such Paley–Hadamard DSs are in the following groups: Z3 2 × Z5 4 × Z17 4 × Z469805627 , Z7 4 × Z17 4 × Z200533919 , and Z3 4 × Z5 4 × Z7 4 × Z11 4 × Z13 4 × Z17 4 × Z4245189013920980750627 . In each case, we have a Paley PDS from Theorem 4.1 in the product consisting of all but the last component and a skew
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Hadamard difference set in the latter component since the latter is Z p for a prime p ≡ 3 (mod 4). We summarize with the following corollaries. Corollary 5.4. Suppose that for an odd number n that n4 ± 2 is a prime power. Then there is a Paley–Hadamard difference set in a group of order n4 (n4 ± 2). Corollary 5.5. Suppose that for an odd number n that 9n4 ± 2 is a prime power. Then there is a Paley– Hadamard difference set in a group of order 9n4 (9n4 ± 2). Prior to the Paley–Hadamard difference sets constructed here and in [11], the only known constructions of these DSs were in groups of prime power order, order 2t − 1 in the Singer family, or q(q + 2) for prime powers q and q + 2 in the twin prime power family. Our construction of Paley PDSs provides a generalization of the twin prime power family, where the piece containing the Paley PDS is not required to be of prime power order. Acknowledgments The author would like to thanks the referees for their careful reading of the paper, especially for the suggestion of allowing both G 1 and G 2 to be arbitrary groups with a covering extended building set with amicable Paley partial difference set in Theorem 4.1. References [1] T. Beth, D. Jungnickel, H. Lenz, Design Theory, second ed., Encyclopedia Math. Appl., vol. 78, Cambridge University Press, Cambridge, 1999. [2] Y.Q. Chen, On the existence of abelian Hadamard difference sets and a new family of difference sets, Finite Fields Appl. 3 (1997) 234–256. [3] J.A. Davis, Partial difference sets in p-groups, Arch. Math. 63 (1994) 103–110. [4] J.A. Davis, J. Jedwab, A unifying construction for difference sets, J. Combin. Theory Ser. A 80 (1) (1997) 13–78. [5] J.A. Davis, J. Jedwab, Nested Hadamard difference sets, J. Statist. Plann. Inference 62 (1997) 13–20. [6] J. Jedwab, Generalized perfect arrays and Menon difference sets, Des. Codes Cryptogr. 2 (1992) 19–68. [7] D. Jungnickel, Difference Sets, Contemporary Design Theory, Wiley–Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992, 241–324 pp. [8] R.G. Kraemer, Proof of a conjecture on Hadamard 2-groups, J. Combin. Theory Ser. A 63 (1) (1993) 1–10. [9] S.L. Ma, A survey of partial difference sets, Des. Codes Cryptogr. 4 (1994) 221–261. [10] R.E.A.C. Paley, On orthogonal matrices, J. Math. Phys. 12 (1933) 311–320. [11] J. Polhill, Paley partial difference sets in non-p-groups, Des. Codes Cryptogr. 52 (2009) 163–169. [12] R.G. Stanton, D.A. Sprott, A family of difference sets, Canad. J. Math. 13 (1958) 73–77. [13] R.J. Turyn, Character sums and difference sets, Pacific J. Math. 15 (1965) 319–346. [14] R.J. Turyn, A special class of Williamson matrices and difference sets, J. Combin. Theory Ser. A 36 (1984) 111–115. [15] G. Weng, L. Hu, Some results on skew Hadamard difference sets, Des. Codes Cryptogr. 50 (1) (2009) 93–105. [16] R.M. Wilson, Q. Xiang, Constructions of Hadamard difference sets, J. Combin. Theory Ser. A 77 (1997) 148–160. [17] M.-Y. Xia, Some infinite classes of special Williamson matrices and difference sets, J. Combin. Theory Ser. A 61 (1992) 230–242. [18] Q. Xiang, Y.Q. Chen, On Xia’s construction of Hadamard difference sets, Finite Fields Appl. 2 (1996) 87–95. [19] K. Yamamoto, Decomposition fields of difference sets, Pacific J. Math. 13 (1963) 337–352.