New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4

Abstract

Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access 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 a construction of an (m, n, 4, 2)-OOSPC from a one-factor of Kohler graph of $$\mathbb {Z}_m\times \mathbb {Z}_n$$ which contains a unique element of order 2. In this paper, we study the existence of one-factor of Kohler graph of $$\mathbb {Z}_m\times \mathbb {Z}_n$$ having three elements of order 2. It is proved that there is a one-factor in the Kohler graph of $$\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}$$ relative to the Sylow 2-subgroup if there is an S-cyclic Steiner quadruple system of order 2p, where $$p\equiv 5\pmod {12}$$ is a prime and $$1\le \epsilon ,\epsilon '\le 2$$ . Using this one-factor, we construct a strictly $$\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}$$ -invariant regular $$G^*(p,2^{\epsilon +\epsilon '},4,3)$$ relative to the Sylow 2-subgroup. By using the known S-cyclic SQS(2p) and a recursive construction for strictly $$\mathbb {Z}_{m}\times \mathbb {Z}_{n}$$ -invariant regular G-designs, we construct more strictly $$\mathbb {Z}_{m}\times \mathbb {Z}_{n}$$ -invariant 3-(mn, 4, 1) packing designs. Consequently, there is an optimal $$(2^{\epsilon }m,2^{\epsilon '}n,4,2)$$ -OOSPC for any $$\epsilon ,\epsilon '\in \{0,1,2\}$$ with $$\epsilon +\epsilon '>0$$ and an optimal (6m, 6n, 4, 2)-OOSPC where m, n are odd integers whose all prime divisors from the set $$\{p\equiv 5\pmod {12}:p$$ is a prime, $$p<$$ 1,500,000}.