A class of 1-generator repeated root quasi-cyclic codes

Abstract

Let $$q$$ be a power of a prime integer $$p, m=p^em_0$$ and $$|q|_{m_{0}}$$ the order of $$q$$ modulo $$m_0$$ . By use of finite commutative chain ring theory, an algorithm to construct all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial over a finite field $$F_q$$ of length $$mn$$ and index $$n$$ , under the condition that $$\mathrm {gcd}(|q|_{m_0},n)=1$$ , are given.