Optimal $2$-D $(n\times m,3,2,1)$-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 $m\equiv 4,20 \pmod{48}$.