Improved cyclotomic conditions leading to new 2-designs: the use of strong difference families1 Simone Costaa , Tao Fengb , Xiaomiao Wangc a
arXiv:1702.07500v1 [math.CO] 24 Feb 2017
b
Dipartimento DICATAM, Via Valotti 9, 1 - 00146 Italy
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China c Department of Mathematics, Ningbo University, Ningbo 315211, P. R. China
[email protected],
[email protected],
[email protected]
Abstract: Strong difference families are an interesting class of discrete structures which can be used to derive relative difference families. Relative difference families are closely related to 2-designs, and have applications in constructions for many significant codes, such as optical orthogonal codes and optical orthogonal signature pattern codes. In this paper, with a careful use of cyclotomic conditions attached to strong difference families, we improve the lower bound on the asymptotic existence results of (Fq1 × Fq2 , Fq1 × {0}, k, λ)-DFs for k ∈ {q1 , q1 + 1}. We improve Buratti’s constructions for 2-(13q, 13, λ) designs and 2-(17q, 17, λ) designs, and establish the existence of seven new 2-(v, k, λ) designs for (v, k, λ) ∈ {(694, 7, 2), (1576, 8, 1), (2025, 9, 1), (765, 9, 2), (1845, 9, 2), (459, 9, 4), (783, 9, 4)}. Keywords: 2-design; relative difference family; strong difference family; Paley difference multiset; cyclotomic class
1
Introduction
Throughout this paper, sets and multisets will be denoted by curly braces { } and square brackets [ ], respectively. When a set or a multiset is fixed with an ordering, we regard it as a sequence, and denote it by ( ). Every union will be understood as multiset union with multiplicities of elements preserved. A∪A∪ · · · ∪A (h times) will be denoted by hA. If A and B are multisets defined on a multiplicative group, then AB denotes the multiset [ab : a ∈ A, b ∈ B]. For a positive integer v, we abbreviate {0, 1, . . . , v − 1} by Zv or Iv , with the former indicating that a cyclic group of this order is acting. A 2-(v, k, λ) design (also called (v, k, λ)-BIBD or balanced incomplete block design) is a pair (V, A) where V is a set of v points and A is a collection of k-subsets of X (called blocks) such that every 2-subset of X is contained in a unique block of A. The concept of 2-(v, k, λ) design was introduced by R. Fisher (1934) to study the design of experiments. These structures turn out to be very useful in many areas such as statistics, information theory, computer science, biology and engineering. A powerful idea to obtain 2-designs is given by the use of difference families and more generally of relative difference families. This idea was implicitly used in many papers (cf. [6]). The concept of relative difference families was initially put forward by M. 1
Supported by NSFC under Grant 11471032, and Fundamental Research Funds for the Central Universities under Grant 2016JBM071, 2016JBZ012 (T. Feng), Zhejiang Provincial Natural Science Foundation of China under Grant LY17A010008, and Natural Science Foundation of Ningbo under Grant 2016A610079 (X. Wang).
1
Buratti in 1998 [10], and systematically developed and discussed by many other authors hereafter (see, for example, [15, 23, 26]). Relative difference families play an important role in the construction of optical orthogonal codes (see, for example, [7, 21, 30]) and optical orthogonal signature pattern codes [27]. Let (G, +) be an abelian group of order g with a subgroup N of order n. A (G, N, k, λ) relative difference family (DF), or (g, n, k, λ)-DF over G relative to N , is a family B = [B1 , B2 , . . . , Br ] of k-subsets of G such that the list ∆B :=
r [
[x − y : x, y ∈ Bi , x 6= y] = λ(G \ N ),
i=1
i.e., every element of G \ N appears exactly λ times in the multiset ∆B while it has no element of N . The members of B are called base blocks and the number r equals to λ(g − n)/(k(k − 1)). When G is cyclic, we say that the (g, n, k, λ)-DF is cyclic. When N = {0}, a relative difference family is simply called a difference family. An automorphism of a 2-(v, k, λ) design D = (X, A) is a permutation on X leaving B invariant. All automorphisms of D form a group, called the full automorphism group of D and denoted by Aut(D). Any subgroup of Aut(D) is called an automorphism group of D. A 2-(v, k, λ) design admitting Zv as its automorphism group is called cyclic. A 2-(v, k, λ) design is said to be 1-rotational if it admits an automorphism consisting of one fixed point and a cycle of length v − 1. The following propositions reveal the relation between relative difference families and 2-designs (cf. [10, 12]). Proposition 1.1 (1) If there exist a (G, N, k, λ)-DF and a 2-(|N |, k, λ) design, then there exists a 2-(|G|, k, λ) design. (2) If there exist a (G, N, k, λ)-DF and a 2-(|N | + 1, k, λ) design, then there exists a 2-(|G| + 1, k, λ) design. Proposition 1.2 (1) If there exists a cyclic (g, k, k, λ)-DF, then there exists a cyclic 2-(g, k, λ) design. (2) If there exists a cyclic (g, k − 1, k, λ)-DF, then there exists a 1-rotational 2-(g + 1, k, λ) design. The target of this paper is to construct relative difference families via strong difference families and to use them to construct new 2-designs. Let S = [F1 , F2 , . . . , Fs ] be a family of multisets of size k of an abelian group G of order g. We say that S is a (G, k, µ) strong difference family, or a (g, k, µ)-SDF over G, if the list ∆S :=
s [
[x − y : x, y ∈ Fi , x 6= y] = µG,
i=1
i.e., every element of G (0 included) appears exactly µ times in the multiset ∆S. The members of S are also called base blocks and the number s equals to µg/(k(k − 1)). Note that µ is necessarily even since the element 0 ∈ G is expressed in even ways as differences in any multiset. Proposition 1.3 A (G, k, µ)-SDF exists only if µ is even and µ|G| ≡ 0 (mod k(k − 1)). 2
M. Buratti [12] in 1999 introduced the concept of strong difference families to establish systematic constructions for relative difference families. He named his main construction as “the fundamental construction” for relative difference families in Theorem 3.1 in [12]. K. Momihara [26] developed Buratti’s technique to give the following theorem. Throughout this paper we always write m p X m 1 2 2 m−1 m) , where U = (e − 1)h (h − 1) Q(e, m) = (U + U + 4e h 4
(1.1)
h=1
for given positive integers e and m. Theorem 1.4 (Theorems 5.8 and 5.9 in [26]) If there exists a (G, k, µ)-SDF with µ = λd, then there exists a (G × Fq , G × {0}, k, λ)-DF • for any even λ and any prime power q ≡ 1 (mod d) with q > Q(d, k − 1); • for any odd λ and any prime power q ≡ d + 1 (mod 2d) with q > Q(d, k − 1). We remark that M. Buratti and A. Pasotti first presented Theorem 1.4 for the case of λ = 1 in their Theorem 5.1 in [16]. Theorem 1.4 shows that any (G, k, µ)-SDF can lead to an infinite family of (G × Fq , G × {0}, k, λ)-DFs for any admissible sufficiently large prime power q. We shall improve Theorem 1.4 in the sense that if the initial SDF in Theorem 1.4 has some particular patterns, then the lower bound on q can be reduced greatly (see Theorems 3.4, 3.5 and 3.6). This enables us to obtain new 2-designs with block sizes 13 and 17 (see Theorem 3.25). This technique vastly improves M. Buratti’s constructions for 2-(13q, 13, λ) designs and 2-(17q, 17, λ) designs in [8, 11]. On the other hand, we recall that, despite the fact that many authors worked on the existence of 2-(v, k, λ) designs with 6 ≤ k ≤ 9, there are still many open cases. In this paper we show that, with a careful application of cyclotomic conditions attached to a strong difference family, it is possible to establish the existence of a 2-(v, k, λ) design in some of the open cases. We can establish the existence of 2-(v, k, λ) designs for (v, k, λ) ∈ {(694, 7, 2), (1576, 8, 1), (2025, 9, 1), (765, 9, 2), (1845, 9, 2), (459, 9, 4), (783, 9, 4)} (see Theorems 2.5, 2.8, 4.3, 4.6 and 4.9).
2
Use of weakly consistent mappings
In this section, we shall review M. Buratti and K. Momihara’s constructions for relative difference families and then present new 2-designs via weakly consistent mappings. Let q be a prime power. As usual we denote by Fq the finite field of order q and by F∗q its multiplicative group. If q ≡ 1 (mod e), then C0e,q will denote the group of nonzero eth powers of Fq and once a primitive element ω of Fq has been fixed, we set Cie,q = ω i · C0e,q for i = 0, 1, . . . , e − 1. Theorem 2.1 [16, 19] Let q ≡ 1 (mod e) be a prime power, let B = {b0 , b1 , . . . , bm−1 } be an arbitrary m-subset of Fq and let (β0 , β1 , . . . , βm−1 ) be an arbitrary element of Zm e . Set X = {x ∈ Fq : x − bi ∈ Cβe,q for i = 0, 1, . . . , m − 1}. Then X is not empty for any i prime power q ≡ 1 (mod e) and q > Q(e, m). 3
The case of m = 3 in Theorem 2.1 was first shown by Buratti [13]. Then a proof similar to that of m = 3 allows Chang and Ji [19], and Buratti and Pasotti [16] to generalize this result to any m. Theorem 2.1 is derived from Weil’s Theorem (see [24], Theorem 5.41) on multiplicative character sums and plays an essential role in the asymptotic existence problem for difference families (cf. [20]). For a positive integer k and a prime power q ≡ 1 (mod e), a mapping φk,q : Ik → Fq is referred to as an Fq -point-labeling of Ik , and a mapping ψk,e : Ik × Ik → Ze is called a Ze pair-labeling of Ik × Ik . Furthermore, a pair of mappings (φk,q , ψk,e ) is weakly consistent if the following conditions are satisfied: (1) φk,q is injective; (2) φk,q (i) − φk,q (j) ∈ Cψe,q
k,e (i,j)
for all i, j ∈ Ik and i 6= j.
R.M. Wilson [28] introduced the concept of a pair of weakly consistent mappings to establish asymptotic existence for difference families, but he used different terminology. M. Buratti and A. Pasotti [16] established the following asymptotic existence for a pair of weakly consistent mappings. Lemma 2.2 (Theorem 3.3 in [16]) Let q ≡ 1 (mod e) be a prime power and −1 ∈ Cξe,q . Given any Ze -pair-labeling ψk,e of Ik ×Ik satisfying that ψk,e (i, j) ≡ ψk,e (j, i)+ξ (mod e) for all (i, j) ∈ Ik × Ik and i 6= j, there exists an Fq -point-labeling φk,q of Ik such that (φk,q , ψk,e ) is weakly consistent for any prime power q ≡ 1 (mod e) and q > Q(e, k − 1).
2.1
Revisit of Buratti’s construction for DFs via SDFs
As a corollary of “the fundamental construction” for relative difference families in Theorem 3.1 in [12], M. Buratti presented the following lemma (see Theorem 3.2 in [12]). We outline the proof for completeness. Lemma 2.3 Let q ≡ 1 (mod d) be a prime power and µ = λd. Assume that there exists a (G, k, µ)-SDF S = [F1 , F2 , . . . , Fs ], where Fj = [fj0 , fj1 , . . . , fj,k−1 ], 1 ≤ j ≤ s. If one j can choose appropriate Zd -pair-labelings ψk,d , 1 ≤ j ≤ s, of Ik × Ik satisfying that for each h ∈ G, s [
j [ψk,d (a, b) : fja − fjb = h, (a, b) ∈ Ik × Ik , a 6= b] = λId ,
(2.2)
j=1 j and if there exist Fq -point-labelings φjk,q of Ik such that (φjk,q , ψk,d ) is weakly consistent for all 1 ≤ j ≤ s, then F = [Bj,α : 1 ≤ j ≤ s, α ∈ C0d,q ]
forms a (G × Fq , G × {0}, k, λ)-DF, where Bj,α = {(fj0 , φjk,q (0)), (fj1 , φjk,q (1)), . . . , (fj,k−1 , φjk,q (k − 1))} · {(1, α)}.
(2.3)
Proof Since φjk,q is injective, Bj,α is a set of size k for all 1 ≤ j ≤ s. The size of the multiset F is q−1 µ|G| q−1 λ|G|(q − 1) s· = · = . d k(k − 1) d k(k − 1) 4
❍ ❍❍ f1b f1a ❍❍ ❍
0 4 15 23 37 58 58
0 (4, 0) (15, 0) (23, 1) (37, 1) (58, 0) (58, 1)
4
15
23
(59, 1)
(48, 1) (52, 0)
(40, 0) (44, 1) (55, 1)
(11, (19, (33, (54, (54,
1) 0) 0) 0) 1)
(8, 0) (22, 0) (43, 1) (43, 0)
(14, 1) (35, 1) (35, 0)
37 (26, (30, (41, (49,
0) 1) 1) 0)
58
58
(5, 1) (9, 1) (20, 0) (28, 0) (42, 1)
(5, 0) (9, 0) (20, 1) (28, 1) (42, 0) (0, 1)
(21, 0) (21, 1)
(0, 0)
1 (a, b)) Table 1: (f1a − f1b , ψ7,2
It is readily checked that ∆F =
=
[
∆Bj,α
α∈C0d,q
s [
j=1
[
s [
[(fja − fjb , (φjk,q (a) − φjk,q (b)) · α) : (a, b) ∈ Ik × Ik , a 6= b]
α∈C0d,q j=1
= λ(G × (Fq \ {0})). Note that φjk,q (a) − φjk,q (b) ∈ C d,q j
ψk,d (a,b)
2.2
for all a, b ∈ Ik and a 6= b.
✷
Three new 2-designs
In the following, when we want to give an Fq -point-labeling φk,q of Ik explicitly, we always regard it as a vector of Fkq . Lemma 2.4 There exists a (Z63 × F11 , Z63 × {0}, 7, 1)-DF. Proof First we here construct a (Z63 , 7, 2)-SDF, S = [F1 , F2 , F3 ], where F1 = [0, 4, 15, 23, 37, 58, 58],
F2 = [0, 1, 3, 7, 13, 25, 39],
F3 = [0, 1, 3, 11, 18, 34, 47].
We want to apply Lemma 2.3 with k = 7, d = 2 and λ = 1 to obtain a (Z63 × F11 , Z63 × j {0}, 7, 1)-DF. To present appropriate Z2 -pair-labelings ψ7,2 , 1 ≤ j ≤ 3, of I7 ×I7 satisfying Lemma 2.3, we give Tables 1, 2 and 3, where the rows and columns of Table j are indexed by elements of Fj = [fj0 , fj1 , . . . , fj6 ], and the cell indexed by (fja , fjb ) contains the value j (fja − fjb , ψ7,2 (a, b)) if a 6= b, null otherwise. Take φ17,11 = (0, 3, 5, 6, 8, 1, 10),
φ27,11 = (0, 2, 4, 6, 1, 10, 8),
φ37,11 = (0, 4, 7, 9, 2, 3, 5),
j which are weakly consistent with ψ7,2 given in Tables 1, 2 and 3, respectively. Let
Bj,α = {(fj0 , φj7,11 (0)), (fj1 , φj7,11 (1)), . . . , (fj6 , φj7,11 (6))} · {(1, α)}, where 1 ≤ j ≤ 3 and α ∈ C02,11 . Then by Lemma 2.3, F = [Bj,α : 1 ≤ j ≤ 3, α ∈ C02,11 ] forms a (Z63 × F11 , Z63 × {0}, 7, 1)-DF.
✷ 5
❍ ❍❍ f2b f2a ❍❍ ❍
0 1 3 7 13 25 39
0 (1, 1) (3, 0) (7, 1) (13, 0) (25, 1) (39, 1)
1
3
7
(62, 0)
(60, 1) (61, 0)
(56, 0) (57, 1) (59, 0)
(2, 1) (6, 0) (12, 1) (24, 1) (38, 1)
(4, 1) (10, 1) (22, 1) (36, 0)
(6, 1) (18, 0) (32, 1)
13 (50, (51, (53, (57,
25 1) 0) 0) 0)
(12, 0) (26, 1)
(38, (39, (41, (45, (51,
39 0) 0) 0) 1) 1)
(24, (25, (27, (31, (37, (49,
0) 0) 1) 0) 0) 1)
(14, 0)
2 (a, b)) Table 2: (f2a − f2b , ψ7,2
❍ ❍❍ f3b f3a ❍❍ ❍
0 1 3 11 18 34 47
0 (1, 0) (3, 1) (11, 0) (18, 1) (34, 0) (47, 0)
1
3
11
(62, 1)
(60, 0) (61, 1)
(52, 1) (53, 1) (55, 0)
(2, 0) (10, 0) (17, 0) (33, 1) (46, 0)
(8, 1) (15, 1) (31, 1) (44, 0)
(7, 0) (23, 0) (36, 1)
18 (45, (46, (48, (56,
34 0) 1) 0) 1)
(16, 0) (29, 0)
(29, (30, (32, (40, (47,
47 1) 0) 0) 1) 1)
(16, (17, (19, (27, (34, (50,
1) 1) 1) 0) 1) 0)
(13, 1)
3 (a, b)) Table 3: (f3a − f3b , ψ7,2
Theorem 2.5 There exists a 2-(694, 7, 2) design. Proof By Lemma 2.4, there exists a (693, 63, 7, 1)-DF, which implies the existence of a (693, 63, 7, 2)-DF. Applying Proposition 1.1(2) with a 2-(64, 7, 2) design, which exists by Theorem 2.5 in [1], we get a 2-(694, 7, 2) design. ✷ Combining the known result on the existence of 2-(v, 7, 2) designs from Theorem 2.5 in [1], we have the following corollary. Corollary 2.6 There exists a 2-(v, 7, 2) design if and only if v ≡ 1, 7 (mod 21) with the definite exception of v = 22 and possible exceptions of v ∈ {274, 358, 574, 988, 994}. Lemma 2.7 There exists a (Z27 × Fq , Z27 × {0}, 9, 4)-DF for q ∈ {17, 29}. Proof First we here construct a (Z27 , 9, 8)-SDF, S = [F1 , F2 , F3 ], where F1 = [0, 3, 3, 8, 8, 17, 17, 23, 23], F3 = [0, 1, 2, 3, 19, 6, 11, 13, 17].
F2 = [0, 1, 2, 3, 19, 4, 5, 8, 12],
Take the SDF as the first components of base blocks of the required (Z27 × Fq , Z27 × {0}, 9, 4)-DFs. For q = 17 and α ∈ C02,17 , let B1,α = {(0, 0), (3, 1), (3, 16), (8, 2), (8, 15), (17, 3), (17, 14), (23, 5), (23, 12)} · {(1, α)}, B2,α = {(0, 0), (1, 1), (2, 2), (3, 7), (19, 11), (4, 10), (5, 5), (8, 14), (12, 16)} · {(1, α)}, B3,α = {(0, 0), (1, 16), (2, 15), (3, 10), (19, 6), (6, 3), (11, 2), (13, 12), (17, 13)} · {(1, α)}. 6
For q = 29 and α ∈ C02,29 , let B1,α = {(0, 0), (3, 1), (3, 28), (8, 2), (8, 27), (17, 3), (17, 26), (23, 4), (23, 25)} · {(1, α)}, B2,α = {(0, 0), (1, 1), (2, 2), (3, 4), (19, 11), (4, 15), (5, 5), (8, 13), (12, 21)} · {(1, α)}, B3,α = {(0, 0), (1, 28), (2, 27), (3, 25), (19, 18), (6, 11), (11, 19), (13, 10), (17, 22)} · {(1, α)}. Then by Lemma 2.3, it is readily checked that F = [Bj,α : 1 ≤ j ≤ 3, α ∈ C02,q ] forms a (Z27 × Fq , Z27 × {0}, 9, 4)-DF for q ∈ {17, 29}. Here to save space, we no longer j list the required Z2 -pair-labelings ψ9,2 , but the reader can get them straightforwardly by counting differences from the second components of these base blocks. Note that the coefficients of α are just the required Fq -point-labelings φj9,q . ✷ Theorem 2.8 There exist a 2-(459, 9, 4) design and a 2-(783, 9, 4) design. Proof By Lemma 2.7, there exist a (459, 27, 9, 4)-DF and a (783, 27, 9, 4)-DF. Applying Proposition 1.1(1) with a 2-(27, 9, 4) design, which exists by Theorem 9.1 in [2], we get the required 2-designs. ✷ Combining the known result on the existence of 2-(v, 9, 4) designs from Theorem 9.1 in [2], we have the following corollary. Corollary 2.9 There exists a 2-(v, 9, 4) design if and only if v ≡ 1, 9 (mod 18) with the possible exception of v = 315.
2.3
Revisit of Momihara’s proof for Theorem 1.4
As we point out in Section 1, Momihara [26] developed Buratti’s technique to give Theorem 1.4, and one of main tasks of this paper is to improve Theorem 1.4 in some circumstances. In order to facilitate the readers to understand the idea of Momihara, we outline the proof of Theorem 1.4 for completeness. This will help readers understand our ideas in the next sections. Proof of Theorem 1.4 By assumption there is a (G, k, µ)-SDF S = [F1 , F2 , . . . , Fs ], where Fj = [fj0 , fj1 , . . . , fj,k−1 ], 1 ≤ j ≤ s. To apply Lemma 2.3, we need to choose j appropriate Zd -pair-labelings ψk,d , 1 ≤ j ≤ s, of Ik × Ik which satisfy the equation (2.2) in Lemma 2.3 and require j j ψk,d (a, b) ≡ ψk,d (b, a) + ξ
(mod d),
(2.4)
for all (a, b) ∈ Ik × Ik and a 6= b, where −1 ∈ Cξd,q . This procedure can always be done.
j Let G2 denote the subgroup of {h ∈ G : 2h = 0}. When λ is even, we can specify ψk,d , 1 ≤ j ≤ s, satisfying (2.4) and S j s j=1 [ψk,d (a, b) : fja − fjb = h, (a, b) ∈ Ik × Ik , a 6= b] = λId , h ∈ G \ G2 , j j=1 [ψk,d (a, b)
Ss
: fja − fjb = h, (a, b) ∈ Ik × Ik , a < b] = λ2 Id , h ∈ G2 . 7
j When λ is odd, we can specify ψk,d , 1 ≤ j ≤ s, satisfying (2.4) and Ss j j=1 [ψk,d (a, b) : fja − fjb = h, (a, b) ∈ Ik × Ik , a 6= b] = λId , j j=1 [ψk,d (a, b)
Ss
h ∈ G \ G2 ,
: fja − fjb = h, (a, b) ∈ Ik × Ik , a < b] = λI d , h ∈ G2 . 2
Note that when λ is odd, the existence of the given (G, k, µ)-SDF implies d is even by Proposition 1.3; when q ≡ d + 1 (mod 2d), −1 ∈ C d,q d . Finally apply Lemma 2.2 to 2
complete the proof.
✷
By Theorem 1.4, any (G, k, µ)-SDF can yield an infinite family of (G × Fq , G × {0}, k, λ)-DFs for any admissible sufficiently large prime power q. We give an example here. Theorem 2.10 There exists a (Z63 ×Fq , Z63 ×{0}, 7, 1)-DF for any prime q ≡ 3 (mod 4) and q ≥ 11. Proof Start from a (Z63 , 7, 2)-SDF, which exists by the proof of Lemma 2.4. Then apply Theorem 1.4 with λ = 1 and d = 2 to obtain a (Z63 × Fq , Z63 × {0}, 7, 1)-DF for any prime power q ≡ 3 (mod 4) and q > Q(2, 6) = 17022.8. When q = 11, the conclusion follows from Lemma 2.4. When q ≡ 3 (mod 4) is a prime and 11 < q ≤ Q(2, 6), it suffices to use Lemma 2.3 to find three pairs of weakly consistent j mappings (φj7,q , ψ7,2 ), 1 ≤ j ≤ 3. A computer search shows that these mappings all exist. The interested reader may get a copy of these data from the authors. ✷
3
DFs from Paley difference multisets
In Section 2, in order to get a (G × Fq , G × {0}, k, λ)-DF, we have to provide multisets [φjk,q (0), φjk,q (1), . . . , φjk,q (k − 1)], 1 ≤ j ≤ s, of second components on Fq that satisfy the assumptions of Lemma 2.3. This is the less trivial part of the procedure and in order to do it we need to use the asymptotical result of Lemma 2.2 or to do a computer research. However, as we have seen in Theorem 1.4, the use of Lemma 2.2 will generally result in a huge lower bound on q. To reduce the lower bound, we request the initial SDF in Theorem 1.4 has some particular patterns, which enable the mappings φjk,q ’s to be related to one another. It will make us have a possibility of taking smaller m (from the notation Q(e, m)) when we use Theorem 2.1. For convenience, we no longer use the terminology of weakly consistent mappings in the remaining part of this paper. This makes our notation simpler. We begin with Paley difference multisets. If a strong difference family only contains one base block, then it is referred to as a difference multiset (cf. [12]) or a regular difference cover (cf. [5]). Lemma 3.1 [12] (1) Let q be an odd prime power. Then {0} ∪ 2F✷ q is an (Fq , q, q − 1)-SDF (called Paley difference multiset of the first type). (2) Let q ≡ 3 (mod 4) be a prime power. Then 2({0} ∪ F✷ q ) is an (Fq , q + 1, q + 1)-SDF (called Paley difference multiset of the second type). 8
Corollary 3.2 Let q1 be an odd prime power. Let λ be any divisor of q1 − 1 and d = (q1 − 1)/λ. Then there exists an (Fq1 × Fq2 , Fq1 × {0}, q1 , λ)-DF • for any even λ and any prime power q2 ≡ 1 (mod d) with q2 > Q(d, q1 − 1); • for any odd λ and any prime power q2 ≡ d + 1 (mod 2d) with q2 > Q(d, q1 − 1). Proof Apply Theorem 1.4 with the first type Paley (Fq1 , q1 , q1 − 1)-SDF to complete the proof. ✷ Corollary 3.3 Let q1 ≡ 3 (mod 4) be a prime power. Let λ be any divisor of q1 + 1 and d = (q1 + 1)/λ. Then there exists an (Fq1 × Fq2 , Fq1 × {0}, q1 + 1, λ)-DF • for any even λ and any prime power q2 ≡ 1 (mod d) with q2 > Q(d, q1 − 1); • for any odd λ and any prime power q2 ≡ d + 1 (mod 2d) with q2 > Q(d, q1 − 1). Proof Apply Theorem 1.4 with the second type Paley (Fq1 , q1 + 1, q1 + 1)-SDF to complete the proof. ✷ The lower bound on q2 in Corollaries 3.2 and 3.3 is very huge even if q2 is small. For example by Corollary 3.2, there is an (Fq1 × Fq2 , Fq1 × {0}, q1 , 1)-DF for any odd prime powers q1 and q2 with q2 ≡ q1 (mod 2(q1 − 1)) and q2 > Q(q1 − 1, q1 − 1). Take q1 = 13. Then Q(12, 12) = 7.94968 × 1027 ! Thus it would be meaningful to develop a new technique to reduce the bound. As main results of this section, we are to prove Theorem 3.4 Let λ be any divisor of (q1 − 1)/4 and d = (q1 − 1)/4λ. (1) There exists an (Fq1 × Fq2 , Fq1 × {0}, q1 , λ)-DF for any prime powers q1 and q2 with q1 ≡ 1, 5 (mod 12), λ(q2 − 1) ≡ 0 (mod q1 − 1) and q2 > Q(d, q1 − 4). (2) Let q1 = 32r be a prime power, and ξ be a primitive 4th root of unity in Fq2 . If λ > 1 or 1 − ξ 6∈ C0d,q2 , then there exists an (Fq1 × Fq2 , Fq1 × {0}, q1 , λ)-DF for any prime powers q1 and q2 with λ(q2 − 1) ≡ 0 (mod q1 − 1) and q2 > Q(d, q1 − 4). Theorem 3.5 Let λ be any divisor of (q1 − 1)/2 and d = (q1 − 1)/2λ. There exists an (Fq1 × Fq2 , Fq1 × {0}, q1 , λ)-DF for any prime powers q1 and q2 with q1 ≡ 1 (mod 2), λ(q2 − 1) ≡ 0 (mod q1 − 1) and q2 > Q(d, q1 − 2). Theorem 3.6 Let λ be any divisor of (q1 + 1)/2 and d = (q1 + 1)/2λ. There exists an (Fq1 × Fq2 , Fq1 × {0}, q1 + 1, λ)-DF for any prime powers q1 and q2 with q1 ≡ 3 (mod 4), λ(q2 − 1) ≡ 0 (mod q1 + 1) and q2 > Q(d, q1 ). Compared with Corollaries 3.2 and 3.3, Theorems 3.4-3.6 not only reduce the lower bound on q2 but also relax the congruence condition on q2 in some circumstances. For example by Theorem 3.4(1), there exists a (Fq1 × Fq2 , Fq1 × {0}, q1 , 1)-DF for any prime powers q1 and q2 with q1 ≡ 1, 5 (mod 12), q2 ≡ 1 (mod q1 −1) and q2 > Q((q1 −1)/4, q1 − 4). Take q1 = 13. Then Q(3, 9) = 9.68583 × 109 , which is much smaller than Q(12, 12). Actually, it is easy to see that Q(e, m) < Q(e, m′ ) for m < m′ and Q(e, m) < Q(e′ , m) for e < e′ . We remark that when q1 ∈ {3, 5}, by Theorems 3.4 and 3.5 one may refind the known results on (q1 q2 , q1 , q1 , 1)-DFs over Fq1 × Fq2 listed in Theorem 16.71 in [3]. 9
3.1
Proof of Theorem 3.4
Let q ≡ 1 (mod e) be a prime power and A be a multisubset of F∗q . If each cyclotomic coset Cle,q for l ∈ Ie contains exactly λ elements of A, then A is said to be a λ-transversal for these cosets. If A is a 1-transversal, A is often referred to as a representative system for the cosets of C0e,q in F∗q . Lemma 3.7 Let q1 ≡ 1 (mod 4) be a prime power and λ be a divisor of (q1 − 1)/4. Write d = (q1 − 1)/4λ. Let q2 be a prime power satisfying λ(q2 − 1) ≡ 0 (mod q1 − 1). Let δ be a generator of C02,q1 and ξ be a primitive 4th root of unity in Fq2 . Take the first type Paley (Fq1 , q1 , q1 − 1)-SDF whose unique base block (f0 , f1 , . . . , fq1 −1 ) = (0, δ, δ, −δ, −δ, δ2 , δ2 , −δ2 , −δ2 , . . . , δ
q1 −1 4
,δ
q1 −1 4
, −δ
q1 −1 4
, −δ
q1 −1 4
).
(3.5)
Suppose that one can choose an appropriate multiset (s0 , s1 , . . . , sq1 −1 ) = (0, y1 , −y1 , ξy1 , −ξy1 , y2 , −y2 , ξy2 , −ξy2 , . . . , y q1 −1 , −y q1 −1 , ξy q1 −1 , −ξy q1 −1 ) 4
4
4
(3.6)
4
such that {y1 , y2 , . . . , y(q1 −1)/4 } ⊆ F∗q2 and for each h ∈ Fq1 , [sa − sb : fa − fb = h, (a, b) ∈ Iq1 × Iq1 , a 6= b] = {1, −1, ξ, −ξ} · Dh , where Dh is a λ-transversal for the cosets of C0d,q2 in F∗q2 . Let S be a representative system for the cosets of {1, −1, ξ, −ξ} in C0d,q2 . Let B = {(f0 , s0 ), (f1 , s1 ), . . . , (fq1 −1 , sq1 −1 )}. Then F = [B · {(1, α)} : α ∈ S] forms an (Fq1 × Fq2 , Fq1 × {0}, q1 , λ)-DF. Proof For q1 ≡ 1 (mod 4), δ(q1 −1)/4 = −1 in Fq1 , so the first type Paley (Fq1 , q1 , q1 − 1)SDF can be written as (3.5). Since q2 is odd, yi 6= −yi and ξyi 6= −ξyi for any 1 ≤ i ≤ (q1 − 1)/4, B · {(1, α)} is a set of size q1 for any α ∈ S. It is readily checked that [ [ ∆F = [(h, rα) : r ∈ {1, −1, ξ, −ξ} · Dh ] = λ(Fq1 × F∗q2 ). α∈S h∈Fq1
Note that by the definition of d, d and 4 are both divisors of (q1 − 1)/λ, so combining the assumption λ(q2 − 1) ≡ 0 (mod q1 − 1), we have d and 4 are both divisors of q2 − 1. This makes C0d,q2 and ξ meaningful. ✷ To apply Lemma 3.7, we need to analyze Dh for each h ∈ Fq1 . Remark 3.8 The appearance of Dh is not unique. There are 4(q1 −1)/4 ways to give Dh for any h. For example, it is easy to check that D0 can be taken as {2x1 y1 , 2x2 y2 , . . . , 2x(q1 −1)/4 y(q1 −1)/4 } := 2 · {x1 y1 , x2 y2 , . . . , x(q1 −1)/4 y(q1 −1)/4 } for any xl ∈ {1, −1, ξ, −ξ}, 1 ≤ l ≤ (q1 − 1)/4. In what follows, when we discuss types of Dh , the 4(q1 −1)/4 ways are considered to be the same and correspond to a same type. Note that yi 6= yj for any i 6= j (otherwise, if yi = yj , then 0 ∈ Dh for h = δi − δj , a contradiction). 10
Lemmas 3.9 and 3.10 will follow the notation in Lemma 3.7. Lemma 3.9 For each h ∈ F∗q1 , let Th = [sa − sb : fa − fb = h, (a, b) ∈ Iq1 × Iq1 , a 6= b]. Then Th = {1, −1, ξ, −ξ} · Dh for some Dh ⊂ Fq2 and the size of Dh is (q1 − 1)/4. Furthermore, Dh = D−h and w.l.o.g., Dh consists of elements having the following types: (I) yi − yj , yi + yj ;
(II) yi − yj ξ, yi + yj ξ;
(III) yi (1 − ξ);
(IV ) yi ,
for some yi and yj from (3.6) in Lemma 3.7. Proof The unique base block of the first type Paley (Fq1 , q1 , q1 − 1)-SDF is of the form (0, . . . , δi , δi , −δi , −δi , . . . , δj , δj , −δj , −δj , . . .).
(3.7)
As differences of this SDF, each h ∈ F∗q1 must be of the form ±(δi − δj ), ±(δi + δj ), ±2δi or ±δi for some 1 ≤ i, j ≤ (q1 − 1)/4. Examine the corresponding multiset of this SDF (0, . . . , yi , −yi , ξyi , −ξyi , . . . , yj , −yj , ξyj , −ξyj , . . .).
(3.8)
If h = ±(δi − δj ), then Th ⊃ {1, −1, ξ, −ξ} · {yi − yj , yi + yj }. If h = ±(δi + δj ), then Th ⊃ {1, −1, ξ, −ξ} · {yi − yj ξ, yi + yj ξ}. If h = ±2δi , then Th ⊃ {1, −1, ξ, −ξ} · {yi (1 − ξ)}. If h = ±δi , then Th ⊃ {1, −1, ξ, −ξ} · {yi }. Thus Th = {1, −1, ξ, −ξ} · Dh for some Dh ⊂ Fq2 , Dh = D−h and Dh consists of elements having Types (I)-(IV). Since each h ∈ F∗q1 occurs q1 − 1 times as differences of the Paley SDF, the size of Dh is (q1 − 1)/4. ✷ Lemma 3.10
(1) D0 = 2 · {y1 , y2 , . . . , y(q1 −1)/4 }.
(2) Let q1 ≡ 5 (mod 8). – If h ∈ C02,q1 , then Dh consists of exactly one element with Type (IV ) and (q1 − 5)/4 elements with Types (I) and (II). – If h ∈ C12,q1 , then Dh consists of exactly one element with Type (III) and (q1 − 5)/4 elements with Types (I) and (II). (3) Let q1 ≡ 1 (mod 8). – If h ∈ C02,q1 , then Dh consists of exactly one element with Type (III), exactly one element with Type (IV ) and (q1 − 9)/4 elements with Types (I) and (II). – If h ∈ C12,q1 , then Dh consists of (q1 − 1)/4 elements with Types (I) and (II). (4) Let q1 ≡ 1 (mod 8) and 3 ∤ q1 . If h ∈ C02,q1 , then Dh does not contain elements of the form yi and yi (1 − ξ) at the same time. (5) Let q1 = 9. For each h ∈ C02,9 , Dh must contain elements of the form yi and yi (1 − ξ) at the same time. 11
(6) Let q1 ≡ 1 (mod 4) and T be a representative system for the cosets of {1, −1} in F∗q1 . Any element with Types (I) and (II) must be contained in a unique Dh for some h ∈ T (note that the term “element” here is a symbolic expression, so y1 − y2 and y3 + y4 are different element but they may have the same value). Proof (1) See Remark 3.8. (2) For q1 ≡ 5 (mod 8), (q1 − 1)/4 is odd. Since the size of Dh is (q1 − 1)/4, each Dh , h ∈ F∗q1 , must contain at least one element with Type (III) or (IV). Examining the sequences (3.7) and (3.8), for any h = ±δi ∈ C02,q1 , 1 ≤ i ≤ (q1 − 1)/4, we have yi ∈ Dh . Thus each element with Type (IV) is in Dh for some h ∈ C02,q1 , and any Dh for h ∈ C02,q1 contains only one element with Type (IV). On the other hand, for any h = ±2δi ∈ C12,q1 (note that 2 ∈ C12,q1 for q1 ≡ 5 (mod 8)), 1 ≤ i ≤ (q1 − 1)/4, we have yi (1 − ξ) ∈ Dh . Thus each element with Type (III) is in Dh for some h ∈ C12,q1 , and any Dh for h ∈ C12,q1 contains only one element with Type (III). (3) A similar argument to that in (2) to complete the proof (note that 2 ∈ C02,q1 for q1 ≡ 1 (mod 8)). (4) Any element of the form yi is in Dh for some h = ±δi , and any element of the form yi (1− ξ) is in Dh′ for some h′ = ±2δi . If h = h′ , then δi = ±2δi , which is impossible since q1 is not a power of 3. (5) For q1 = 9, it is readily checked that D±δ = {y1 , y1 (1−ξ)} and D±δ2 = {y2 , y2 (1− ξ)}. (6) Analysing (3.7) and (3.8) carefully, the conclusion is straightforward. Note that by Lemma 3.9, Dh = D−h . ✷ Lemma 3.11 Let r ≥ 4 and Y = {y1 , y2 , . . . , yr } ⊆ F∗q2 . Then for any a ∈ Zr and any r ordered pairs (yi1 , yi2 ) from Y satisfying (1) {yi1 : 1 ≤ i ≤ r} = {yi2 : 1 ≤ i ≤ r} = Y , (2) yi1 6= yi2 , 1 ≤ i ≤ r, there exists a bijection π : Y → Zr such that π(yi1 ) 6≡ π(yi2 ) − a (mod r) for any 1 ≤ i ≤ r. Proof We first give an equivalent description of this lemma couched in the language of graph. Define a directed graph H with vertex set Y . For any two vertices yi and yj , (yi , yj ) is a directed edge of H if and only if (yi , yj ) is a given ordered pair in the assumption. Due to Condition (1), the indegree and outdegree of each vertex of H are both 1, so H can be decomposed into disjoint cycles. Due to Condition (2), the graph contains no loop. Now regard the mapping π as a vertex coloring of H via the color class Zr . The conclusion is that there exists a vertex coloring π such that any two vertices have different colors and π(yi ) 6≡ π(yj ) − a (mod r) for any edge (yi , yj ). When a = 0, any bijection π : Y → Zr can lead to the desired conclusion. When a = r/2, if H consists of one cycle, w.l.o.g., labeled by (y1 , y2 , . . . , yr ) with edges (yi , yi+1 ), 1 ≤ i ≤ r − 1, and (yr , y1 ), then take π(yj ) = j − 1 for 1 ≤ j ≤ r. Since r ≥ 4, π(yj ) 6≡ π(yj+1 ) − a (mod r) for any 1 ≤ j ≤ r − 1 and π(yr ) 6≡ π(y1 ) − a (mod r). If H consists of u disjoint cycles and u ≥ 2, we label these cycles by (y11 , y12 , . . . , y1l1 ) (y21 , y22 , . . . , y2l2 ) · · · (yu1 , yu2 , . . . , yulu ). If none of these cycles has a length of r/2 + 1, 12
Pj1 −1 then take π(yj1 ,j2 ) = i=1 li + j2 − 1 for 1 ≤ j1 ≤ u and 1 ≤ j2 ≤ li . If there is a cycle of length r/2 + 1, then r > 4 (otherwise when r = 4, H contains an isolated vertex, a contradiction). W.l.o.g., l1 = r/2 + 1. Take π(y1,j2 ) = j2 − 1 for 1 ≤ j2 ≤ l1 − 1, Pj1 −1 π(y1,l1 ) = r/2 + 1, π(y2,1 ) = r/2, and π(yj1 ,j2 ) = i=1 li + j2 − 1 for 2 ≤ j1 ≤ u, 1 ≤ j2 ≤ li , (j1 , j2 ) 6= (2, 1). When a 6= 0 and a 6= r/2, we will color the vertices of H one by one. • If H consists of one cycle, the length of the cycle is at least 4. W.l.o.g, the cycle is labeled by (y1 , y2 , . . . , yr ) with edges (yi , yi+1 ), 1 ≤ i ≤ r − 1, and (yr , y1 ). We shall start from y1 and end at yr . Step 1. Set π(y1 ) = 0 and π(y2 ) = r −a. Since a ∈ Zr and a 6∈ {0, r/2}, π(y2 ) 6= π(y1 ) and π(y1 ) 6≡ π(y2 ) − a (mod r). It follows that the values for π(y1 ) and π(y2 ) are feasible so far. Step 2. Consider π(y3 ). We hope π(y3 ) 6∈ [π(y1 ), π(y2 ), π(y2 ) + a] := A3 . Since r ≥ 4, Zr \ A3 6= ∅, which allows us to pick up a π(y3 ) such that the values for π(y1 ), π(y2 ) and π(y3 ) are feasible so far. Step 3. Use a similar process for other yi , i > 3, step by step. It will always be successful until π(yr−1 ). Now consider π(yr−1 ). We hope π(yr−1 ) 6∈ {π(yi ) : 1 ≤ i ≤ r − 2} ∪ {π(yr−2 ) + a} := Ar−1 . Since Zr \ Ar−1 6= ∅, we can pick up a π(yr−1 ) such that the values for π(yi ), 1 ≤ i ≤ r − 1, are feasible so far. Step 4. The key step is to take π(yr ). Actually, for the mapping π, only one value (denoted by η) of Zr is left now, so π(yr ) has to be taken as η. If η 6∈ {π(yi ) : 1 ≤ i ≤ r − 1} ∪ [π(yr−1 ) + a, r − a], where the number r − a is from the edge (yr , y1 ) (note that π(y1 ) = 0), then the process ends successfully. Otherwise, since η 6∈ {π(yi ) : 1 ≤ i ≤ r − 1}, η must be π(yr−1 ) + a or r − a. However, η 6= r − a because of π(y2 ) = r − a, so η = π(yr−1 ) + a. Write β = π(yr−1 ), so η = β + a. Now we swap the values of π(yr−1 ) and π(yr ), i.e., set π(yr−1 ) = β + a and π(yr ) = β. Then the process ends successfully. This is because if π(yr−1 ) = β + a = π(yr−2 ) + a, then β = π(yr−2 ), a contradiction; if π(yr ) = β = π(yr−1 ) + a = β + 2a, then 2a = 0, a contradiction. • If H consists of u disjoint cycles and u ≥ 2, we label these cycles by (y11 , y12 , . . . , y1l1 ) (y21 , y22 , . . . , y2l2 ) · · · (yu1 , yu2 , . . . , yulu ). If there is at least one cycle of length 3, then w.l.o.g., assume that lu ≥ 3. We will color the vertices of H one by one. First set π(y11 ) = r −a and π(yu1 ) = 0. Then by similar argument to that for H being a one cycle, one can complete the proof. Note that lu ≥ 3 is crucial since when the process reaches to the step to assign a value for π(yu−1,lu−1 ), we hope π(yu−1,lu−1 ) 6∈ {π(y11 ), π(y12 ), . . . , π(yu−1,lu−1 −1 )} ∪ [π(yu1 ), π(yu−1,lu−1 −1 ) + a, π(yu−1,1 ) − a] := A; since the length of the last cycle in H is at least 3, the size of A is at most r − 1; it follows that we can always pick up a feasible π(yu−1,lu−1 ). If each cycle in H is of length 2, then l1 = l2 = · · · = lu = 2 and u = r/2. Take π(yj1 ,j2 ) = j1 − 1 + 2r (j2 − 1) for 1 ≤ j1 ≤ r/2 and 1 ≤ j2 ≤ 2. Since a 6= r/2, it is easy to see that π(yj1 ,1 ) 6≡ π(yj1 ,2 ) ± a (mod r). ✷ Proof of Theorem 3.4 By Lemma 3.7, it suffices to find appropriate multiset (s0 , s1 , . . . , sq1 −1 ) such that for each h ∈ Fq1 , Dh is a λ-transversal for the cosets of C0d,q2 in F∗q2 . Let T be a representative system for the cosets of {1, −1} in F∗q1 . By Lemma 3.9, Dh = D−h for any h. Thus it suffices to analyze Dh for h ∈ {0} ∪ T . When q1 = 5, which implies λ = 1 and d = 1, C01,q2 is just F∗q2 . In this case each Dh contains only one element, so the conclusion is straightforward. 13
When q1 ≡ 5 (mod 8) is a prime power and q1 > 5, by Lemma 3.10(1) and (2), D0 = 2 · {y1 , y2 , . . . , y(q1 −1)/4 }; if h ∈ C02,q1 , then Dh consists of exactly one element with Type (IV) and (q1 − 5)/4 elements with Types (I) and (II); if h ∈ C12,q1 , then Dh consists of exactly one element with Type (III) and (q1 − 5)/4 elements with Types (I) and (II). Since by Lemma 3.10(6), any element with Types (I) and (II) is contained in a unique Dh for some h ∈ T , applying Theorem 2.1, one can always pick up appropriate y1 , y2 , . . . , y(q1 −1)/4 such that each Dh , h ∈ {0} ∪ T , is a λ-transversal for the cosets of C0d,q2 in F∗q2 for any divisor λ of (q1 − 1)/4 and any prime power q2 with λ(q2 − 1) ≡ 0 (mod q1 − 1) and q2 > Q(d, q1 − 4), where the number q1 − 4 is from the fact that given any 1 ≤ i ≤ (q1 − 1)/4, count the number of cyclotomic conditions on yi , which is 4 × ((q1 − 5)/4) + 1 = q1 − 4 since every pair of yi and yj for any j 6= i can contribute 4 cyclotomic conditions and yi itself contributes one cyclotomic condition. When q1 ≡ 1 (mod 8) is a prime power, by Lemma 3.10(3) if h ∈ C02,q1 , then Dh consists of exactly one element with Type (III), exactly one element with Type (IV) and (q1 − 9)/4 elements with Types (I) and (II); if h ∈ C12,q1 , then Dh consists of (q1 − 1)/4 elements with Types (I) and (II). Since by Lemma 3.10(6), any element with Types (I) and (II) is contained in a unique Dh for some h ∈ T , to apply Theorem 2.1, one thing to be careful of is to examine Dh ’s, h ∈ C02,q1 , of the form [yih , yjh (1 − ξ), · · · ] such that each of these Dh ’s is a λ-transversal for the cosets of C0d,q2 in F∗q2 . When λ > 1, it is easy to do it. It suffices to consider the case of λ = 1. In such a case, d = (q1 − 1)/4. If ih 6= jh for any h ∈ C02,q1 , then yih 6= yjh for any h ∈ C02,q1 by Remark 3.8. By d,q2 Lemma 3.10(5), q1 6= 9. So assume that q1 ≥ 17. Let 1 − ξ ∈ Cd−a for some 0 ≤ a < d. To apply Theorem 2.1, we need to find a bijection π : {y1 , y2 , . . . , yd } → Zd such that π(yih ) 6≡ π(yjh ) + d − a (mod d) for any Dh = [yih , yjh (1 − ξ), · · · ], h ∈ {δ, δ2 , . . . , δd } (note that examining (3.5) and (3.6), we have {yih : h ∈ {δ, δ2 , . . . , δd }} = {yjh : h ∈ {δ, δ2 , . . . , δd }} = {y1 , y2 , . . . , yd }). By Lemma 3.11, when q1 ≥ 17, i.e., d ≥ 4, such a bijection exists. Thus we can apply Theorem 2.1 to complete the proof. If ih′ = jh′ for some h′ ∈ C02,q1 , then by Lemma 3.10(4), q1 must be a power of 3, which implies q1 must be a power of 9 because of q1 ≡ 1 (mod 8). In this case, since λ = 1, if 1 − ξ 6∈ C0d,q2 , then yih′ and yih′ (1 − ξ) are not in the same coset. Thus we can apply Theorem 2.1 to complete the proof. ✷ In order to facilitate the reader to understand the use and the proof of Theorem 3.4 and Lemma 3.7, we present several examples below. Example 3.12 There exists an (F13 × Fq , F13 × {0}, 13, 1)-DF for all primes q ≡ 1 (mod 12) with the possible exceptions of q ∈ E13 = {37, 61, 73, 97, 109, 181, 313, 337, 349, 373, 409, 421, 541, 577, 829, 853, 1129, 1741, 2473}. Proof Applying Theorem 3.4 with λ = 1, q1 = 13 and q2 = q, we get an (F13 × Fq , F13 × {0}, 13, 1)-DF for all primes q ≡ 1 (mod 12) and q > Q(3, 9) = 9.68583 × 109 . Take the first type Paley (F13 , 13, 12)-SDF: (0, 4, 4, −4, −4, 3, 3, −3, −3, −1, −1, 1, 1). Let ξ be a primitive 4th root of unity in Fq and B = {(0, 0), (4, y1 ), (4, −y1 ), (−4, y1 ξ), (−4, −y1 ξ), (3, y2 ), (3, −y2 ), (−3, y2 ξ), (−3, −y2 ξ), (−1, y3 ), (−1, −y3 ), (1, y3 ξ), (1, −y3 ξ)}. We follow the notation in Lemmas 3.7 and 3.9. Set Th = {1, −1, ξ, −ξ} · Dh where Dh = D13−h , h ∈ F13 . It is readily checked that 14
D0 D2 D4 D6
= 2 · [y1 , y2 , y3 ], = [y3 (1 − ξ), y3 − y2 ξ, y3 + y2 ξ], = [y1 , y3 − y2 , y3 + y2 ], = [y2 (1 − ξ), y2 − y1 ξ, y2 + y1 ξ].
D1 = [y2 − y1 , y2 + y1 , y3 ], D3 = [y2 , y3 − y1 ξ, y3 + y1 ξ], D5 = [y1 (1 − ξ), y3 − y1 , y3 + y1 ],
With the aid of computer, we can pick up appropriate y1 , y2 , y3 in F∗q for all primes q ≡ 1 (mod 12), q ≤ Q(3, 9) and q 6∈ E13 such that each Dh , h ∈ F13 , is a representative system for the cosets of C03,q in F∗q . The interested reader may get a copy of these data from the authors. Let S be a representative system for the coset of {1, −1, ξ, −ξ} in C03,q . Then by Lemma 3.7, the base blocks B · {(1, α)}, α ∈ S, form an (F13 × Fq , F13 × {0}, 13, 1)-DF. For instance, one choice of y1 , y2 , y3 such that each Dh , h ∈ F13 , is a representative system for the cosets of C03,q in F∗q is listed below, according to the class of 1 − ξ in F∗q . if 1 − ξ y1 y2 y2 − y1 y2 + y1 y2 − y1 ξ
∈ ∈ ∈ ∈ ∈ ∈
0 0 1 0 1 0
1 0 1 0 1 0
2 0 1 0 1 1
if 1 − ξ y2 + y1 ξ y3 y3 − y1 y3 + y1 y3 − y1 ξ
∈ ∈ ∈ ∈ ∈ ∈
0 2 2 1 2 0
1 1 2 0 2 0
2 2 2 0 1 0
if 1 − ξ y3 + y1 ξ y3 − y2 y3 + y2 y3 − y2 ξ y3 + y2 ξ
∈ ∈ ∈ ∈ ∈ ∈
0 2 1 2 0 1
1 2 1 2 1 2
2 2 1 2 0 2
To save space, we write i instead of Ci3,q in the above table.
✷
Example 3.13 There exists an (F13 × Fq , F13 × {0}, 13, 3)-DF for all primes q ≡ 1 (mod 4) and q > 9. Proof Applying Theorem 3.4 with λ = 3, q1 = 13 and q2 = q, we get an (F13 × Fq , F13 × {0}, 13, 3)-DF for all primes q ≡ 1 (mod 4) and q > Q(1, 9) = 9. ✷ Example 3.14 There exists an (F17 × Fq , F17 × {0}, 17, 1)-DF for all primes q ≡ 1 (mod 16) and q > Q(4, 13) = 3.44807 × 1017 , or q ∈ S17 ∪ {p : p is a prime, p ≡ 1 (mod 16), 6673 ≤ p ≤ 9857}, where S17 = {17, 881, 1297, 1601, 1873, 2017, 2129, 2657, 2753, 2801, 2897, 3089, 3121, 3217, 3313, 3361, 3617, 3697, 3761, 3793, 3889, 4001, 4049, 4129, 4241, 4273, 4289, 4481, 4561, 4657, 4721, 4801, 4817, 4993, 5009, 5233, 5281, 5297, 5393, 5441, 5521, 5569, 5857, 5953, 6113, 6257, 6337, 6449, 6529}. Proof Applying Theorem 3.4 with λ = 1, q1 = 17 and q2 = q, we get an (F17 × Fq , F17 × {0}, 17, 1)-DF for all primes q ≡ 1 (mod 16) and q > Q(4, 13). Take the first type Paley (F17 , 17, 16)-SDF: (0, −8, −8, 8, 8, −4, −4, 4, 4, −2, −2, 2, 2, −1, −1, 1, 1). Let ξ be a primitive 4th root of unity in Fq and B = {(0, 0), (−8, y1 ), (−8, −y1 ), (8, y1 ξ), (8, −y1 ξ), (−4, y2 ), (−4, −y2 ), (4, y2 ξ), (4, −y2 ξ), (−2, y3 ), (−2, −y3 ), (2, y3 ξ), (2, −y3 ξ), (−1, y4 ), (−1, −y4 ), (1, y4 ξ), (1, −y4 ξ)}. We follow the notation in Lemmas 3.7 and 3.9. Set Th = {1, −1, ξ, −ξ} · Dh where Dh = D17−h , h ∈ F17 . It is readily checked that
15
D0 D2 D4 D6 D8
= 2 · [y1 , y2 , y3 , y4 ], = [y3 , y3 − y2 , y3 + y2 , y4 (1 − ξ)], = [y2 , y2 − y1 , y2 + y1 , y3 (1 − ξ)], = [y3 − y1 , y3 + y1 , y3 − y2 ξ, y3 + y2 ξ], = [y1 , y2 (1 − ξ), y4 − y1 ξ, y4 + y1 ξ].
D1 D3 D5 D7
= [y1 (1 − ξ), y4 , y4 − y3 , y4 + y3 ], = [y4 − y2 , y4 + y2 , y4 − y3 ξ, y4 + y3 ξ], = [y2 − y1 ξ, y2 + y1 ξ, y4 − y2 ξ, y4 + y2 ξ], = [y3 − y1 ξ, y3 + y1 ξ, y4 − y1 , y4 + y1 ],
With the aid of computer, we can pick up appropriate y1 , y2 , y3 , y4 in F∗q for q ∈ S17 ∪ {p : p is a prime, p ≡ 1 (mod 16), 6673 ≤ p ≤ 9857} such that each Dh , h ∈ F17 , is a representative system for the cosets of C04,q in F∗q . The interested reader may get a copy of these data from the authors. Let S be a representative system for the coset of {1, −1, ξ, −ξ} in C04,q . Then by Lemma 3.7, the base blocks B · {(1, α)}, α ∈ S, form an (F17 × Fq , F17 × {0}, 17, 1)-DF. ✷ Example 3.15 There exists an (F17 × Fq , F17 × {0}, 17, 2)-DF for all primes q ≡ 1 (mod 8). Proof Applying Theorem 3.4 with λ = 2, q1 = 17 and q2 = q, we get an (F17 × Fq , F17 × {0}, 17, 2)-DF for all primes q ≡ 1 (mod 8) and q > Q(2, 13) = 2.03024 × 109 . For primes q ≡ 1 (mod 8) and q ≤ Q(2, 13), we examine the block B in the proof of Example 3.14. By computer search, we can pick up appropriate y1 , y2 , y3 , y4 in F∗q such that each Dh , h ∈ F17 , is a 2-transversal for the cosets of C02,q in F∗q . The interested reader may get a copy of these data from the authors. Let S be a representative system for the coset of {1, −1, ξ, −ξ} in C02,q . Then by Lemma 3.7, the base blocks B · {(1, α)}, α ∈ S form an (F17 × Fq , F17 × {0}, 17, 2)-DF. ✷ Example 3.16 There exists an (F17 × Fq , F17 × {0}, 17, 4)-DF for all primes q ≡ 1 (mod 4) and q > 13. Proof Applying Theorem 3.4 with λ = 4, q1 = 17 and q2 = q, we get an (F17 × Fq , F17 × {0}, 17, 4)-DF for all primes q ≡ 1 (mod 4) and q > Q(1, 13) = 13. ✷
3.2
Proof of Theorem 3.5
Lemma 3.17 Let q1 ≡ 1 (mod 2) be a prime power and λ be a divisor of (q1 − 1)/2. Write d = (q1 − 1)/2λ. Let q2 be a prime power satisfying λ(q2 − 1) ≡ 0 (mod q1 − 1). Let δ be a generator of C02,q1 . Take the first type Paley (Fq1 , q1 , q1 − 1)-SDF whose unique base block (f0 , f1 , . . . , fq1 −1 ) = (0, δ, δ, δ2 , δ2 , . . . , δ
q1 −1 2
,δ
q1 −1 2
).
(3.9)
Suppose that one can choose an appropriate multiset (s0 , s1 , . . . , sq1 −1 ) = (0, y1 , −y1 , y2 , −y2 , . . . , y q1 −1 , −y q1 −1 ) 2
2
such that {y1 , y2 , . . . , y(q1 −1)/2 } ⊆ F∗q2 and for each h ∈ Fq1 , [sa − sb : fa − fb = h, (a, b) ∈ Iq1 × Iq1 , a 6= b] = {1, −1} · Dh , 16
(3.10)
where Dh is a λ-transversal for the cosets of C0d,q2 in F∗q2 . Let S be a representative system for the cosets of {1, −1} in C0d,q2 . Let B = {(f0 , s0 ), (f1 , s1 ), . . . , (fq1 −1 , sq1 −1 )}. Then F = [B · {(1, α)} : α ∈ S] forms an (Fq1 × Fq2 , Fq1 × {0}, q1 , λ)-DF. Proof Since q2 is odd, yi 6= −yi for any 1 ≤ i ≤ (q1 − 1)/2, B · {(1, α)} is a set of size q1 for any α ∈ S. It is readily checked that [ [ ∆F = [(h, rα) : r ∈ {1, −1} · Dh ] = λ(Fq1 × F∗q2 ). α∈S h∈Fq1
Note that by the definition of d, d is a divisor of (q1 − 1)/λ, so combining the assumption λ(q2 − 1) ≡ 0 (mod q1 − 1), we have d is also a divisor of q2 − 1. This makes C0d,q2 meaningful. ✷ Similar arguments to those in Lemmas 3.9 and 3.10, we have Lemma 3.18 Follow the notation in Lemma 3.17. (1) D0 = 2 · {y1 , y2 , . . . , y(q1 −1)/2 }. (2) For each h ∈ F∗q1 , let Th = [sa − sb : fa − fb = h, (a, b) ∈ Iq1 × Iq1 , a 6= b]. Then Th = {1, −1} · Dh for some Dh ⊂ Fq2 and the size of Dh is (q1 − 1)/2. Furthermore, Dh = D−h and w.l.o.g., Dh consists of elements having types (I) yi − yj , yi + yj , and (II) yi . Proof of Theorem 3.5 Combine the results of Lemmas 3.17 and 3.18. Then apply Theorem 2.1 to complete the proof. Note that the number q1 − 2 in Q(d, q1 − 2) is from the fact that given any 1 ≤ i ≤ (q1 − 1)/2, count the number of cyclotomic conditions on yi , which is 2 × ((q1 − 3)/2) + 1 = q1 − 2 since every pair of yi and yj for any j 6= i can contribute 2 cyclotomic conditions and yi itself contributes one cyclotomic condition. ✷
3.3
Proof of Theorem 3.6
Lemma 3.19 Let q1 ≡ 3 (mod 4) be a prime power and λ be a divisor of (q1 + 1)/2. Write d = (q1 + 1)/2λ. Let q2 be a prime power satisfying λ(q2 − 1) ≡ 0 (mod q1 + 1). Let δ be a generator of C02,q1 . Take the second type Paley (Fq1 , q1 + 1, q1 + 1)-SDF whose unique base block (f0 , f1 , . . . , fq1 ) = (0, 0, δ, δ, δ2 , δ2 , . . . , δ
q1 −1 2
,δ
q1 −1 2
).
(3.11)
Suppose that one can choose an appropriate multiset (s0 , s1 , . . . , sq1 ) = (y, −y, y1 , −y1 , y2 , −y2 , . . . , y q1 −1 , −y q1 −1 ) 2
17
2
(3.12)
such that {y, y1 , y2 , . . . , y(q1 −1)/2 } ⊆ F∗q2 and for each h ∈ Fq1 , [sa − sb : fa − fb = h, (a, b) ∈ Iq1 +1 × Iq1 +1 , a 6= b] = {1, −1} · Dh , where Dh is a λ-transversal for the cosets of C0d,q2 in F∗q2 . Let S be a representative system for the cosets of {1, −1} in C0d,q2 . Let B = {(f0 , s0 ), (f1 , s1 ), . . . , (fq1 , sq1 )}. Then F = [B · {(1, α)} : α ∈ S] forms an (Fq1 × Fq2 , Fq1 × {0}, q1 + 1, λ)-DF. Proof Since q2 is odd, y 6= −y and yi 6= −yi for any 1 ≤ i ≤ (q1 − 1)/2, B · {(1, α)} is a set of size q1 + 1 for any α ∈ S. It is readily checked that [ [ ∆F = [(h, rα) : r ∈ {1, −1} · Dh ] = λ(Fq1 × F∗q2 ). α∈S h∈Fq1
Note that by the definition of d, d is a divisor of (q1 + 1)/λ, so combining the assumption λ(q2 − 1) ≡ 0 (mod q1 + 1), we have d is also a divisor of q2 − 1. This makes C0d,q2 meaningful. ✷ Similar arguments to those in Lemmas 3.9 and 3.10, we have Lemma 3.20 Follow the notation in Lemma 3.19. (1) D0 = 2 · {y, y1 , y2 , . . . , y(q1 −1)/2 }. (2) For each h ∈ F∗q1 , let Th = [sa − sb : fa − fb = h, (a, b) ∈ Iq1 +1 × Iq1 +1 , a 6= b]. Then Th = {1, −1} · Dh for some Dh ⊂ Fq2 and the size of Dh is (q1 + 1)/2. Furthermore, Dh = D−h and w.l.o.g., Dh consists of elements having types (I) yi − yj , yi + yj , and (II) yi − y, yi + y. Proof of Theorem 3.6 Combine the results of Lemmas 3.19 and 3.20. Then apply Theorem 2.1 to complete the proof. Note that the number q1 in Q(d, q1 ) is from the fact that given any 1 ≤ i ≤ (q1 − 1)/2, count the number of cyclotomic conditions on yi , which is 2 × ((q1 − 3)/2) + 2 + 1 = q1 since every pair of yi and yj for any j 6= i and every pair of yi and y for any i can both contribute 2 cyclotomic conditions and yi itself contributes one cyclotomic condition. Similarly, the number of cyclotomic conditions on y is also 2 × ((q1 − 1)/2) + 1 = q1 . ✷ The proof of Theorem 3.6 is much easier than that of Theorem 3.4. When q1 ≡ 3 (mod 4), −1 ∈ C12,q1 . The second type Paley (Fq1 , q1 + 1, q1 + 1)-SDF cannot be written as (0, 0, δ, δ, −δ, −δ, δ 2 , δ2 , −δ2 , −δ2 , . . . , δ(q1 −1)/4 , δ(q1 −1)/4 , −δ(q1 −1)/4 , −δ(q1 −1)/4 ). Thus it is not necessary to introduce ξ in the choice of multiset (s0 , s1 , . . . , sq1 ) (in fact the use of ξ will not lead to the reduction of the lower bound on q2 ). It follows that no Dh of the form [yi , yj (1 − ξ), · · · ] exists, which makes the analysis of Theorem 3.6 easier. 18
3.4
New 2-designs
Start from the relative difference families in Theorems 3.4 and 3.5. Then apply Proposition 1.1(1) with a trivial 2-(q1 , q1 , λ) design. We obtain the following two theorems. Theorem 3.21 Let λ be any divisor of (q1 − 1)/4 and d = (q1 − 1)/4λ. (1) There exists a 2-(q1 q2 , q1 , λ) design for any prime powers q1 and q2 with q1 ≡ 1, 5 (mod 12), λ(q2 − 1) ≡ 0 (mod q1 − 1) and q2 > Q(d, q1 − 4). (2) Let q1 = 32r be a prime power, and ξ be a primitive 4th root of unity in Fq2 . If λ > 1 or 1 − ξ 6∈ C0d,q2 , then there exists a 2-(q1 q2 , q1 , λ) design for any prime powers q1 and q2 with λ(q2 − 1) ≡ 0 (mod q1 − 1) and q2 > Q(d, q1 − 4). Theorem 3.22 Let λ be any divisor of (q1 − 1)/2 and d = (q1 − 1)/2λ. There exists a 2-(q1 q2 , q1 , λ) design for any prime powers q1 and q2 with q1 ≡ 1 (mod 2), λ(q2 − 1) ≡ 0 (mod q1 − 1) and q2 > Q(d, q1 − 2). Start from the relative difference families in Theorem 3.6. Then apply Proposition 1.1(2) with a trivial 2-(q1 + 1, q1 + 1, λ) design. We obtain the following theorem. Theorem 3.23 Let λ be any divisor of (q1 + 1)/2 and d = (q1 + 1)/2λ. There exists a 2(q1 q2 , q1 + 1, λ) design for any prime powers q1 and q2 with q1 ≡ 3 (mod 4), λ(q2 − 1) ≡ 0 (mod q1 + 1) and q2 > Q(d, q1 ). Remark 3.24 If q1 and q2 are both prime and q1 6= q2 , then since Fq1 ×Fq2 is isomorphic to Zq1q2 , applying Proposition 1.2, we have that all 2-(q1 q2 , q1 , λ) designs from Theorems 3.21 and 3.22 are cyclic, while all 2-(q1 q2 , q1 + 1, λ) designs from Theorem 3.23 are 1rotational. Combining the results of Examples 3.12-3.16 and applying Proposition 1.1(1), we have Theorem 3.25 (1) There exists a 2-(13q, 13, 1) design for all primes q ≡ 1 (mod 12) with the possible exceptions of q ∈ E13 = {37, 61, 73, 97, 109, 181, 313, 337, 349, 373, 409, 421, 541, 577, 829, 853, 1129, 1741, 2473}. (2) There exists a 2-(13q, 13, 3) design for all primes q ≡ 1 (mod 4) and q > 9. (3) There exists a 2-(17q, 17, 1) design for all primes q ≡ 1 (mod 16) and q > Q(4, 13) = 3.44807 × 1017 , or q ∈ S17 ∪ {p : p is a prime, p ≡ 1 (mod 16), 6673 ≤ p ≤ 9857} (see Example 3.14 for details of S17 ). (4) There exists a 2-(17q, 17, 2) design for all primes q ≡ 1 (mod 8). (5) There exists a 2-(17q, 17, 4) design for all primes q ≡ 1 (mod 4) and q > 13. M. Buratti discussed constructions for 2-(13q, 13, λ) designs and 2-(17q, 17, λ) designs in [8,11]. His results rely heavily on cyclotomic conditions of some specific numbers. For example, his construction for 2-(17q, 17, 2) designs requires 2 is not a 4th power in Fq [11]. Thus Theorem 3.25 improves Buratti’s results greatly. Remark 3.26 If q 6= 13 in Theorem 3.25(1) and (2), or q 6= 17 in Theorem 3.25(3), (4) and (5), then applying Proposition 1.2(1), we have that all 2-designs from Theorem 3.25 are cyclic. 19
4
New 2-designs from SDFs with particular patterns
Section 3 is devoted to providing new 2-designs from Paley difference multisets. As a special SDF, Paley difference multisets contain only one base block. In this section, we will present a new technique to construct 2-designs via SDFs having at least two base blocks. We require these SDFs have certain particular patterns. Lemma 4.1 There exists a (Z63 , 8, 8)-SDF. Proof Take A1 = [20, 20, −20, −20, 29, 29, −29, −29], A2 = A3 = A4 = A5 = [0, 1, 3, 7, 19, 34, 42, 53], A6 = A7 = A8 = A9 = [0, 1, 4, 6, 26, 36, 43, 51]. Then the multiset [Ai : 1 ≤ i ≤ 9] forms a (Z63 , 8, 8)-SDF.
✷
Lemma 4.2 There exists a (Z63 × F25 , Z63 × {0}, 8, 1)-DF. Proof Take the (Z63 , 8, 8)-SDF from Lemma 4.1 as the first components of base blocks of the required (Z63 × F25 , Z63 × {0}, 8, 1)-DF. Take p(x) = x2 + 2x + 3 to be a primitive polynomial of degree 2 over F5 . Then F25 is the splitting field of p(x) over F5 and, denoted by ω a root of p(x) = 0, ω is a generator of the multiplicative group of F25 . Let ξ = ω 6 and = {(20, 1), (20, −1), (−20, ξ), (−20, −ξ), (29, ω), (29, −ω), (−29, ωξ), (−29, −ωξ)}, = {(0, 0), (1, 1), (3, ω), (7, ω 2 ), (19, ω 3 ), (34, ω 4 ), (42, ω 7 ), (53, ω 10 )}, = {(0, 0), (1, ω), (4, ω 4 ), (6, ω 20 ), (26, ω 14 ), (36, ω 12 ), (43, ω 15 ), (51, ω 17 )}, = B2 · {(1, −1)}, B4 = B2 · {(1, ξ)}, B5 = B2 · {(1, −ξ)}, = B6 · {(1, −1)}, B8 = B6 · {(1, ξ)}, B9 = B6 · {(1, −ξ)}, S S We set 9i=1 ∆Bi = 62 h=0 {h} × Th . It is readily checked that Th = {1, −1, ξ, −ξ} · Dh , Dh = D63−h , h ∈ Z63 , and each Dh is a representative system for the cosets of C02,25 in F∗25 . Let S be a representative system for the coset of {1, −1, ξ, −ξ} in C02,25 . Then the base blocks Bi · {(1, α)}, 1 ≤ i ≤ 9 and α ∈ S, form a (Z63 × F25 , Z63 × {0}, 8, 1)-DF. ✷ B1 B2 B6 B3 B7
Theorem 4.3 There exists a 2-(1576, 8, 1) design. Proof By Lemma 4.2, there exists a (1575, 63, 8, 1)-DF. Applying Proposition 1.1(2) with a 2-(64, 8, 1) design, which exists by Table 3.3 in [4], we get the required design. ✷ Lemma 4.4 There exists a (Zv , 9, 8)-SDF for v ∈ {45, 81}. Proof Take v = 45:
v = 81:
A1 A2 A4 A1 A2 A6
= [0, 2, 2, 15, 15, 23, 23, 33, 33], = A3 = [0, 1, 4, 5, 6, 7, 13, 22, 33], = A5 = [0, 2, 5, 11, 21, 25, 28, 36, 40]; = [0, 4, 4, −4, −4, 37, 37, −37, −37], = A3 = A4 = A5 = [0, 1, 4, 6, 17, 18, 38, 63, 72], = A7 = A8 = A9 = [0, 2, 7, 27, 30, 38, 53, 59, 69].
Then the multiset [Ai : 1 ≤ i ≤ v/9] forms a (Zv , 9, 8)-SDF. 20
✷
Lemma 4.5 There exists a (Z81 × F25 , Z81 × {0}, 9, 1)-DF. Proof Take the (Z81 , 9, 8)-SDF from Lemma 4.4 as the first components of base blocks of the required (Z81 × F25 , Z81 × {0}, 9, 1)-DF. Take p(x) = x2 + 2x + 3 to be a primitive polynomial of degree 2 over F5 . Then F25 is the splitting field of p(x) over F5 and, denoted by ω a root of p(x) = 0, ω is a generator of the multiplicative group of F25 . Let ξ = ω 6 and = {(0, 0), (4, 1), (4, −1), (−4, ξ), (−4, −ξ), (37, ω), (37, −ω), (−37, ωξ), (−37, −ωξ)}, = {(0, 0), (1, 1), (4, ω), (6, ω 2 ), (17, ω 3 ), (18, ω 4 ), (38, ω 5 ), (63, ω 7 ), (72, ω 8 )}, = {(0, 0), (2, 1), (7, ω 4 ), (27, ω 17 ), (30, ω 2 ), (38, ω 18 ), (53, ω 8 ), (59, ω 10 ), (69, ω 14 )}, = B2 · {(1, −1)}, B4 = B2 · {(1, ξ)}, B5 = B2 · {(1, −ξ)}, = B6 · {(1, −1)}, B8 = B6 · {(1, ξ)}, B9 = B6 · {(1, −ξ)}, S S We set 9i=1 ∆Bi = 80 h=0 {h} × Th . It is readily checked that Th = {1, −1, ξ, −ξ} · Dh , Dh = D81−h , h ∈ Z81 , and each Dh is a representative system for the cosets of C02,25 in F∗25 . Let S be a representative system for the coset of {1, −1, ξ, −ξ} in C02,25 . Then the blocks Bi · {(1, α)}, 1 ≤ i ≤ 9 and α ∈ S form a (Z81 × F25 , Z81 × {0}, 9, 1)-DF. ✷ B1 B2 B6 B3 B7
Theorem 4.6 There exists a 2-(2025, 9, 1) design. Proof By Lemma 4.5, there exists a (2025, 81, 9, 1)-DF. Applying Proposition 1.1(1) with a 2-(81, 9, 1) design, which exists by Table 3.3 in [4], we get the required design. ✷ According to Table 3.3 in [4], Theorems 4.3 and 4.6 provide the first examples of a 2-(1576, 8, 1) design and a 2-(2025, 9, 1) design so far. Lemma 4.7 There exists a (Z45 × F17 , Z45 × {0}, 9, 2)-DF. Proof Take the (Z45 , 9, 8)-SDF from Lemma 4.4 as the first components of base blocks of the required (Z45 × F17 , Z45 × {0}, 9, 2)-DF. Let B1 = {(0, 0), (2, 1), (2, −1), (15, 2), (15, −2), (23, 3), (23, −3), (33, 5), (33, −5)}, B2 = {(0, 0), (1, 1), (4, 2), (5, 3), (6, 6), (7, 9), (13, 4), (22, 11), (33, 15)}, B4 = {(0, 0), (2, 3), (5, 8), (11, 6), (21, 12), (25, 7), (28, 9), (36, 2), (40, 13)}, B3 = B2 · {(1, −1)}, B5 = B4 · {(1, −1)}. S5 S44 We set i=1 ∆Bi = h=0 {h} × Th . It is readily checked that Th = {1, −1} · Dh , Dh = D45−h , h ∈ Z45 , and each Dh is a 2-transversal for the cosets of C02,17 in F∗17 . Let S be a representative system for the coset of {1, −1} in C02,17 . Then the base blocks Bi ·{(1, α)}, 1 ≤ i ≤ 5 and α ∈ S, form a (Z45 × F17 , Z45 × {0}, 9, 2)-DF. ✷ Lemma 4.8 There exists a (Z45 × F41 , Z45 × {0}, 9, 1)-DF. Proof Take the (Z45 , 9, 8)-SDF from Lemma 4.4 as the first components of base blocks of the required (Z45 × F41 , Z45 × {0}, 9, 1)-DF. Let B1 B2 B4 B3
= {(0, 0), (2, 1), (2, −1), (15, 2), (15, −2), (23, 3), (23, −3), (33, 6), (33, −6)}, = {(0, 0), (1, 1), (4, 7), (5, 21), (6, 12), (7, 15), (13, 24), (22, 4), (33, 34)}, = {(0, 0), (2, 3), (5, 31), (11, 32), (21, 15), (25, 9), (28, 40), (36, 25), (40, 35)}, = B2 · {(1, −1)}, B5 = B4 · {(1, −1)}. 21
S S We set 5i=1 ∆Bi = 44 h=0 {h} × Th . It is readily checked that Th = {1, −1} · Dh , Dh = D45−h , h ∈ Z45 , and each Dh is a representative system for the cosets of C04,41 in F∗41 . Let S be a representative system for the coset of {1, −1} in C04,41 . Then the base blocks Bi · {(1, α)}, 1 ≤ i ≤ 5 and α ∈ S form a (Z45 × F41 , Z45 × {0}, 9, 1)-DF. ✷ Theorem 4.9 There exist a 2-(765, 9, 2) design and a 2-(1845, 9, 2) design. Proof By Lemmas 4.7 and 4.8, there exist a (765, 45, 9, 2)-DF and a (1845, 45, 9, 2)-DF (note that two (1845, 45, 9, 1)-DFs can produce a (1845, 45, 9, 2)-DF). Applying Proposition 1.1(1) with a 2-(45, 9, 2) design, which exists by Theorem 8.2 in [2], we get the required 2-designs. ✷ Combining the known result on the existence of 2-(v, 9, 2) designs from Theorem 8.2 in [2], we have Corollary 4.10 There exists a 2-(v, 9, 2) design if and only if v ≡ 1, 9 (mod 36) with the possible exceptions of v ∈ {189, 253, 505, 837, 1197, 1837}.
5
Concluding remarks
By a careful application of cyclotomic conditions attached to strong difference families, this paper improves the lower bound on the asymptotic existence results of (Fq1 × Fq2 , Fq1 × {0}, k, λ)-DFs for k ∈ {q1 , q1 + 1}, and presents seven new 2-designs. Future directions are two-fold. One is to systematically analyze possible patterns of SDFs that help to produce new 2-designs from the point of view of asymptotic existence or concrete designs. Section 4 just provides several specific examples. We believe routine extensions are possible. We point out that M. Buratti et al. [14, 17] essentially made use of a Paley SDF of the first type, called a strong difference map from a graph-theoretical perspective, to investigate the constructions for i-perfect cycle decompositions. The other direction is to generalize the techniques used in this paper to construct other kinds of difference families, such as resolvable difference families (cf. [9]) and partitioned difference families (cf. [18]), which can be used to construct frequency hopping sequences (cf. [22]) and constant composition codes (cf. [29]), etc.
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