Further results on optimal optical orthogonal codes with weight 4.
A \((v,k,\lambda_a,\lambda_c)\) optical orthogonal code \({\mathcal C}\) (OOC) is a family of (0,1)-sequences with \(k\) 1's and \(v-k\) 0's satisfying the properties: (1) The auto-correlation property: \(\sum x_tx_{t+i}\leq\lambda_a\) for any \((x_0,\ldots,x_{v-1})\in{\mathcal C}\) and any integer \(i\not\equiv0\mod v\). (2) The cross-correlation property: \(\sum x_ty_{t+i}\leq\lambda_c\) for any two different sequences \((x_0,\ldots,x_{v-1}), (y_0,\ldots,y_{v-1})\in{\mathcal C}\) and any integer \(i\). In this paper, the authors prove the existence of optimal \((v,4,1)\)-OOC for \(v=3^nu\), where \(u\) is a product of primes congruent to 1 modulo 4, or \(v=2^nu\), where \(u\) is a product of primes congruent to 1 modulo 6.
scientific article; zbMATH DE number 2062784