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 × m, 3, 2, 1)-optical orthogonal codes with m ≡ 0 (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 × m, 3, 2, 1)optical orthogonal code is determined for n = 1, 2, m ≡ 0 (mod 4), and n ≡ 0 (mod 3), m ≡ 8 (mod 16) or m ≡ 32 (mod 64) or m ≡ 4, 20 (mod 48).