New 2-designs from strong difference families

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 (F p ×F q , F p ×{0}, k, λ)-DFs for k ∈ {p, p + 1}. We improve Buratti's existence results 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)

Strong difference families of special types

“…The Paley (p, p, p − 1)-SDFs of the 1st type in Lemma 1(1) satisfy the hypothesis of Proposition 3 when p ≡ 1 (mod 4) since −1 ∈ C 2,p 0 . Then by Remark 3, we have that there exists an (F p × F q , F p × {0}, p, 1)-DF for any prime powers p and q with p ≡ 1, 5 (mod 12), p = 17, q ≡ 1 (mod p − 1) and q > Q((p − 1)/4, p − 4) (see also Theorem 3.8(1) of [16]).…”

“…Example 6. By Proposition 3, the following (Z 45 , 5, 4)-SDF is of type 4 with respect to any prime power q ≡ 1 (mod 4): Other SDFs of type 4 which satisfy the hypothesis of Proposition 3 can be found in [16]. They are listed in Table 2, which will be used to construct relative difference families later.…”

“…This paper is devoted to constructing strong difference families of types 2 and 4 (see Propositions 2 and 3) in Sections 2 and 3, respectively, by using cyclotomic conditions that generalize the procedure of [16,17]. Then apply Proposition 1 to obtain new asymptotic existence results (see Theorems 2 and 4) and existence results for DFs (see Theorems 3 and 5).…”

On silver and golden optical orthogonal codes

“…The elder constructions are surveyed in [2]. More recent constructions can be found in [8,9,11,14,15,21,22,25]. Here one often tries to have each ∆ g B a complete system of representatives for the cosets of the subgroup of F * q of index µ, namely the group C µ of non-zero µ-th powers of F q .…”

“…Thus Theorem 4.1 can be applied. The reduction modulo p of the Fibonacci sequence up to its 34-th term is (0, 1, 1, 2, 3,5,8,13,21,34,55,18,2,20,22,42,64,35,28,63,20,12,32,44,5,49,54,32,15,47,62,38,29,67,25).…”