Combinatorial Constructions of Optimal $(m, n,4,2)$ Optical Orthogonal Signature Pattern Codes

TL;DR

This paper introduces new combinatorial design-based constructions for optimal optical orthogonal signature pattern codes, expanding the known families and highlighting cases where the Johnson bound is not attainable.

Contribution

It presents novel construction methods for optimal (m, n, 4, 2)-OOSPCs using various combinatorial designs, increasing the number of known infinite families.

Findings

New infinite families of optimal (m, n, 4, 2)-OOSPCs

Construction methods using $bZ_m imes bZ_n$-invariant designs

Cases where optimal codes do not reach the Johnson bound

Abstract

Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access (CDMA) network for 2-dimensional image transmission. There is a one-to-one correspondence between an $(m, n, w, \lambda)$-OOSPC and a $(\lambda+1)$-$(mn,w,1)$ packing design admitting an automorphism group isomorphic to $\mathbb{Z}_m\times \mathbb{Z}_n$. In 2010, Sawa gave the first infinite class of $(m, n, 4, 2)$-OOSPCs by using $S$-cyclic Steiner quadruple systems. In this paper, we use various combinatorial designs such as strictly $\mathbb{Z}_m\times \mathbb{Z}_n$-invariant $s$-fan designs, strictly $\mathbb{Z}_m\times \mathbb{Z}_n$-invariant $G$-designs and rotational Steiner quadruple systems to present some constructions for $(m, n, 4, 2)$-OOSPCs. As a consequence, our new constructions yield more infinite families of optimal $(m, n, 4, 2)$-OOSPCs. Especially, we shall see that in some cases an optimal $(m, n, 4, 2)$-OOSPC can not achieve the Johnson bound.