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

Abstract

This paper focuses on constructions for optimal $2$-D $(n\times m,3,2,1)$-optical orthogonal codes with $m\equiv 0(\mathrm{mod}4)$. 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 $n=1,2$, $m\equiv 0(\mathrm{mod}4)$, and $n\equiv 0(\mathrm{mod}3)$, $m\equiv 8(\mathrm{mod}16)$ or $m\equiv 32(\mathrm{mod}64)$ or $m\equiv 4,20(\mathrm{mod}48)$.