Further results on optimal optical orthogonal codes with weight 4

Abstract

By a ( v, k,1)-OOC we mean an optical orthogonal code of length v, weight k, and correlation constraints 1. In this paper, we take advantage of the equivalence between such codes and cyclic packings of pairs to make further investigation regarding the existence of a ( v,4,1)-OOC. It is proved that an optimal ( v,4,1)-OOC exists whenever v=3 n u with u a product of primes congruent to 1 modulo 4, or v=2 n u with u a product of primes congruent to 1 modulo 6, where n is an arbitrary positive integer and n≠2 in the case v=2 n u. A strong indication about the existence of an optimal (2 2 u,4,1)-OOC with u a product of primes congruent to 1 modulo 6 has been given in (M. Buratti, Des. Codes Cryptogr. 26 (2002) 111–125). The results in this paper are obtained mainly by means of a great deal of direct constructions, including using Weil's theorem with more than one independent variations.