Optimal \((9v,4,1)\) optical orthogonal codes (Q2719168)

A \((v,k,1)\) optical orthogonal code is a family of \((0,1)\)-sequences such that the weight of each sequence is \(k\) and the auto- and crosscorrelation is at most \(1\). The code is optimal if the number of sequences is \(\lfloor\frac{v-1}{k(k-1)}\rfloor\) (which is the theoretically achievable maximum). Using cyclotomy and Weil's bound on character sums, the authors construct optimal \((9v,4,1)\)-codes for all \(v\) which are products of primes congruent to \(1\) modulo \(4\). The paper provides fairly direct constructions for primes \(p\) using cyclotomy. However, the constructions require the existence of suitable elements in the finite field \({\mathbb F}_p\). To show the existence of such elements, Weil's bound is used.