More results on optimal optical orthogonal codes with weight four

Abstract

Optical orthogonal codes (OOCs) were introduced by Salehi, as signature sequences to facilitate multiple access in optical fibre networks. The existence of optimal (v, 3,1)-OOCs had been solved completely. For k ≥ 4, although there are some partial results, the existence of optimal (v, k, 1)-OOCs is far from settled. In this paper it is proved that if there exists a (g, 4, 1)-PDF, then (1) there exists an optimal ((g+2)v, 4, 1)-OOC for each integer v whose prime factors are all congruent to 1 modulo 4; (2) there exists an optimal ((g+1)v, 4, 1)-OOC for each integer v whose prime factors are all congruent to 1 modulo 6; (3) if g ≡ 1 (mod 48), then there exists an optimal ((g+7)v, 4, 1)-OOC for each integer v whose prime factors are all congruent to 1 modulo 6. By using the known results on (g, 4,1)-PDFs, many new optimal optical orthogonal codes with weight four are obtained.