Semi-cyclic holey group divisible designs with block size three

TL;DR

This paper investigates the existence conditions for semi-cyclic holey group divisible designs with block size three, providing new theoretical results and applications to optical orthogonal codes.

Contribution

It establishes necessary and sufficient conditions for the existence of 3-SCHGDDs and related difference matrices, advancing combinatorial design theory.

Findings

Existence of (3,mt;m)-CHDM characterized by modular conditions.

Conditions for 3-SCHGDD existence when n≥4 and t is odd.

Application of results to construct optimal optical orthogonal codes.

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\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.