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.

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