Constructions of Optimal Optical Orthogonal Codes with Weights Set $\left\{\mathrm{5,7}\right\}$ 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 n , W , 1 , Q -OOCs with max w : w ∈ W ≤ 6 . So far, there is no explicit construction of optimal n , W , 1 , Q -OOCs with W = 5,7 . 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 5,7 explicitly, for any prime p ≡ 3 mod 4 and p ≥ 7 . In addition, for 1 ≤ t < p − 1 / 2 , by breaking t blocks of size 7 into 3 of 31 p , 5,7 , 1 , 1 / 2 , 1 / 2 -OOCs and 41 p , 5,7 , 1 , 2 / 3 , 1 / 3 -OOCs, we obtain new infinite classes of optimal 31 p , 3,5,7 , 1 , 7 t / p − 1 + 6 t , p − 1 / 2 p − 1 + 6 t , p − 1 − 2 t / 2 p − 1 + 6 t -OOCs and 41 p , 3,5,7 , 1 , 14 t / 3 p − 1 + 4 t , 2 p − 1 / 3 p − 1 + 4 t , p − 1 − 2 t / 3 p − 1 + 4 t -OOCs, respectively.