Further results on optimal optical orthogonal codes with weight 4 Original Research Article

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=3nu with u a product of primes congruent to 1 modulo 4, or v=2nu 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=2nu. A strong indication about the existence of an optimal (22u,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 Weilʹs theorem with more than one independent variations.