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

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, λ)-OOSPC and a (λ+ 1)-(mn,w, 1) packing design admitting an automorphism group isomorphic to Zm × Zn. 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 Zm×Zn-invariant s-fan designs, strictly Zm ×Zn-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.