On the locality of quasi-cyclic codes over finite fields

A code is said to have locality r if any coordinate value in a codeword of that code can be recovered by at most r other coordinates. In this paper, we have studied the locality of quasi-cyclic codes over finite fields. The generator matrix of a quasi-cyclic code can be represented in the form of circulant matrices. We have obtained a bound on the locality of the code in terms of the weights of the associated polynomials to these circulant matrices. We have further analyzed the bounds on the locality, particularly in the case of 1-generator quasi-cyclic codes. An algorithm to find the locality of a quasi-cyclic code is also presented. We have given a construction of 1-generator quasi-cyclic codes with locality at most r using the zeros of its generator polynomial. Some examples have been given to illustrate the results presented in the paper.