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

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 Köhler 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 Köhler 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 Köhler 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}.