An efficient algorithm for constructing reversible quasi-cyclic codes via Chinese remainder theorem

Regarding quasi-cyclic codes as certain polynomial matrices, we show that all reversible quasi-cyclic codes are decomposed into reversible linear codes of shorter lengths corresponding to the coprime divisors of the polynomials with the form of one minus x to the power of m. This decomposition brings us an efficient method to construct reversible quasi-cyclic codes. We also investigate the reversibility and the self-duality of the linear codes corresponding to the coprime divisors of the polynomials. Specializing to the cases where the number of cyclic sections is not more than two, we give necessary and sufficient conditions for the divisors of the polynomials for which the self-dual codes are reversible and the reversible codes of half-length-dimension are self-dual. Our theorems are utilized to search reversible self-dual quasi-cyclic codes with two cyclic sections over binary and quaternary fields of lengths up to seventy and thirty-six, respectively, together with the maximums of their minimum weights.