Semi-cyclic holey group divisible designs with block size three

2013/04/11 by Tao Feng, Xiaomiao Wang, Yanxun Chang · 1 citation
Mathematics · #math.CO #msc:05B30 #msc:94B25

paper · pdf · doi:10.1007/s10623-013-9859-7

arxiv created 2013/04/11 · arxiv updated 2014/04/24

Abstract

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.

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• Semi-cyclic holey group divisible designs with block size three and applications to sampling designs and optical orthogonal codes