Quasi-Cyclic Codes of Index $1\frac {1}{3}$

Abstract

We introduce quasi-cyclic codes of index 1(1/3), and construct a class of such codes generated by pairs of polynomials. By investigating the pair of circulant matrices associated with the generator pair of polynomials, we obtain the generator matrix of any code of the class. Using a probabilistic method, we prove that, for any positive real number $\delta $ such that the asymptotic GV-bound at $2\delta $ is greater than 1/2, the probability that the relative minimal distance of the code in the class is greater than $\delta $ is almost 1; and the probability that the rate of the code equals to 1/4 is also almost 1. An obvious consequence is that the quasi-cyclic codes of index 1(1/3) are asymptotically good.