Asymptotically good quasi-cyclic codes of fractional index

Generalizing the quasi-cyclic codes of index $1\frac{1}{3}$ introduced by Fan et al., we study a more general class of quasi-cyclic codes of fractional index generated by pairs of polynomials. The parity check polynomial and encoder of these codes are obtained. The asymptotic behaviours of the rates and relative distances of this class of codes are studied by using a probabilistic method. We prove that, for any positive real number $\delta$ such that the asymptotic GV-bound at $\frac{k+l}{2}\delta$ is greater than $\frac{1}{2}$, the relative distance of the code is convergent to $\delta$, while the rate is convergent to $\frac{1}{k+l}$. As a result, quasi-cyclic codes of fractional index are asymptotically good.