Improved Johnson bounds for optical orthogonal codes with /spl lambda/ > 1 and some optimal constructions

Abstract

Optical orthogonal codes (OOC) are used as spreading sequences for optical CDMA networks. An OOC is a family of constant weight binary codes with a pre-specified maximum correlation parameter (MCP). Johnson in his 1962 paper introduced three bounds for constant weight codes, that we call bounds A, B, and hybrid. Subsequently Chung et al. adapted Johnson bound A to generate a bound for OOCs, which has been widely used to prove the optimality of OOCs. Johnson bound B has been used in a prior work of this paper's authors to prove the optimality of some OOCs. In this paper we give an improvement of this bound, and based on that prove the optimality of some other constructions which were not known to be optimal. Using the results from Agrell et al., 2000 paper we also give an improvement of Johnson hybrid bound for constant weight codes, and then use it to generate a bound for OOCs. Finally, we introduce a new family of OOCs, based on flats in an affine geometry. While OOCs based on lines and hyperplanes are optimal, we can't say much about other OOCs resulting from this construction. We show that the hybrid bound gives tighter bound than the other two bounds in some regions for this construction. Recently lot of interest has been shown to find all optimal OOCs with weight 4 and 5 and MCP 1 and 2. Using affine geometry construction, a new family of optimal OOCs with weight 4 and MCP 2 is introduced