Constructions of Optimal Optical Orthogonal Codes with Weights Set 5,7 via Cyclic Packing

Abstract

Double-weight optical orthogonal codes are variable-weight optical orthogonal codes (OOCs), which have been widely applied in optical networks and systems. Some works have been devoted to optimal $\left(n,W,1,Q\right)$ -OOCs with $\max \left\{w:w\in W\right\}\le 6$ . So far, there is no explicit construction of optimal $\left(n,W,1,Q\right)$ -OOCs with $W=\left\{\mathrm{5,7}\right\}$ . It is known that heavier-weight codewords have better code performance than lighter-weight codewords. So, in this paper, we use cyclic packing to construct two infinite classes of optimal OOCs with weights set $\left\{\mathrm{5,7}\right\}$ explicitly, for any prime $p\equiv 3\left(\mathrm{mod}4\right)$ and $p\ge 7$ . In addition, for $1\le t<p-1/2$ , by breaking $t$ blocks of size 7 into 3 of $\left(31p,\left\{\mathrm{5,7}\right\},1,\left\{1/2,1/2\right\}\right)$ -OOCs and $\left(41p,\left\{\mathrm{5,7}\right\},1,\left\{2/3,1/3\right\}\right)$ -OOCs, we obtain new infinite classes of optimal $\left(31p,\left\{\mathrm{3,5,7}\right\},1,\left\{7t/p-1+6t,p-1/2\left(p-1+6t\right),p-1-2t/2\left(p-1+6t\right)\right\}\right)$ -OOCs and $\left(41p,\left\{\mathrm{3,5,7}\right\},1,\left\{14t/3\left(p-1+4t\right),2\left(p-1\right)/3\left(p-1+4t\right),p-1-2t/3\left(p-1+4t\right)\right\}\right)$ -OOCs, respectively.