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In this paper we discuss the existence problem for a semi-cyclic holey group divisible design of type (n, m t ) with block size 3, which is denoted by a 3-SCHGDD of type (n, m t ). When n = 3, a 3-SCHGDD of type (3, m t ) is equivalent to a (3, mt; m)-cyclic holey difference matrix, denoted by a (3, mt; m)-CHDM.It is shown that there is a (3, mt; m)-CHDM if and only if (t − 1)m ≡ 0 (mod 2) and t ≥ 3 with the exception of m ≡ 0 (mod 2) and t = 3. When n ≥ 4, the case of t odd is considered. It is established that if t ≡ 1 (mod 2) and n ≥ 4, then there exists a 3-SCHGDD of type (n, m t ) if and only if t ≥ 3 and (t − 1)n(n − 1)m ≡ 0 (mod 6) with some possible exceptions of n = 6 and 8. The main results in this paper have been used to construct optimal two-dimensional optical orthogonal codes with weight 3 and different auto-and cross-correlation constraints by the authors recently.

Abstract:We consider the existence problem for a semi-cyclic holey group divisible design of type (n, m t ) with block size 3, which is denoted by a 3-SCHGDD of type (n, m t ). When t is odd and n = 8 or t is doubly even and t = 8, the existence problem is completely solved; when t is singly even, many infinite families are obtained. Applications of our results to twodimensional balanced sampling plans and optimal two-dimensional optical orthogonal codes are also discussed.

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)

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