区组大小为3的半循环Holey群可分设计
Semi-cyclic holey group divisible designs with block size three
摘要：
本文讨论了（n，m^t）型半循环Holey群可分设计的存在性问题，其中（n，m^t）型半循环Holey群可分设计表示为（n，m^t）型3-SCHGDD，其区组长度为3.当n=3时，（3，m ^t）型3-SCHGDD等价于（3，mt;m）-循环Holey差分矩阵，记为（3，mt;m）-CHDM。证明了存在（3，mt;m）-CHDM当且仅当（t-1）m \equiv 0（mod 2）和t \geq 3（m \equiv 0（mod 2）和t=3除外）.当n \geq 4时，考虑t奇数的情况。如果t \equiv 1（mod 2）和n \geq 4，则存在类型为（n，m^t）的3-SCHGDD当且仅当t \geq 3和（t-1）n（n-1）m \equiv 0（mod 6），除了n=6和8的一些可能的例外。本文的主要结果最近被作者用来构造权为3的不同自相关和互相关约束的最优二维光正交码。
英文摘要：
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\equiv 0 (mod 2) and t\geq 3 with the exception of m\equiv 0 (mod 2) and t=3. When n\geq 4, the case of t odd is considered. It is established that if t\equiv 1 (mod 2) and n\geq 4, then there exists a 3-SCHGDD of type (n,m^t) if and only if t\geq 3 and (t-1)n(n-1)m\equiv 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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