Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes

The existence problem for a semicyclic group divisible design (SCGDD) of type m n with block size 4 and index unity, denoted by 4-SCGDD, has been studied for any odd integer m to construct a kind of two-dimensional optical orthogonal codes (2-D OOCs) with the AM-OPPW (at most one-pulse per wavelength) restriction. In this paper, the existence of a 4-SCGDD of type m n is determined for any even integer m , with some possible exceptions. A complete asymptotic existence result for k -SCGDDs of type m n is also obtained for all larger k and odd integer m . All these SCGDDs are used to derive new 2-D OOCs with the AM-OPPW restriction, which are optimal in the sense of their sizes.