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 (OCDMA) 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 a point-regular 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 see that in some cases an optimal (m, n, 4, 2)-OOSPC can not achieve the Johnson bound. We also use Witt’s inversive planes to obtain optimal \((p, p, p+1, 2)\)-OOSPCs for all primes \(p\ge 3\).