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

• DOI: 10.1007/s10623-018-0549-3
• Publication Date: 2018-09-09T23:35:54Z
• Authors: Tao Feng; Lidong Wang; Xiaomiao Wang
• Keywords: Code (set theory)

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 $$n=1,2$$ , $$m\equiv 0 \pmod {4}$$ , and $$n\equiv 0 \pmod {3}$$ , $$m\equiv 8 \pmod {16}$$ or $$m\equiv 32 \pmod {64}$$ or $$m\equiv 4,20 \pmod {48}$$ .