Combinatorial Constructions of Optimal (m,n,4,2)(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 $(m, n, w, \lambda)$-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.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1312.7589 by other author

Optimal 22-D (n×m,3,2,1)(n\times m,3,2,1)-optical orthogonal codes and related equi-difference conflict avoiding codes

This paper focuses on constructions for optimal $2$-D $(n\times m,3,2,1)$-optical orthogonal codes with $m\equiv 0\ ({\rm mod}\ 4)$. An upper bound on the size of such codes is established. It relies heavily on the size of optimal equi-difference $1$-D $(m,3,2,1)$-optical orthogonal codes, which is closely related to optimal equi-difference conflict avoiding codes with weight $3$. The exact number of codewords of an optimal $2$-D $(n\times m,3,2,1)$-optical orthogonal code is determined for $n=1,2$, $m\equiv 0 \pmod{4}$, and $n\equiv 0 \pmod{3}$, $m\equiv 8 \pmod{16}$ or $m\equiv 32 \pmod{64}$ or $m\equiv 4,20 \pmod{48}$