Optimal (9<i>v</i>, 4, 1) Optical Orthogonal Codes

Abstract

Optimal (9p,4,1) optical orthogonal codes (OOCs) are constructed for all primes p congruent to 1 modulo 4. Direct constructions with explicit codewords are presented for the case $p \equiv 13 \ \mbox{mod} \ 24$, and Weil's theorem on character sums is used to settle the cases $p \equiv 1,5,17 \ \mbox{mod} \ 24$. By applying a known recursive construction, optimal (9v,4,1)-OOCs are obtained for all v, a product of primes congruent to 1 modulo 4.