Generator Polynomial Matrices of Reversed and Reversible Quasi-Cyclic Codes

Quasi-cyclic (QC) codes over the finite field F
_q
correspond to certain F
_q
[x]-modules. We describe a QC code C by a generator polynomial matrix. The code obtained by reversing the codewords of C is called the reversed code of C and denoted by R. If R = C, then C is a reversible code. In this work, we prove a formula for a generator polynomial matrix of the reversed code R of a prescribed QC code C. Then, we present a necessary and sufficient condition in terms of the proven formula to ensure the reversibility of any QC code. Moreover, we characterize the reduced generator polynomial matrices of reversible QC codes. As an application, we apply our theoretical results to QC codes with the best known parameters. Computer search is used to show the existence of reversible QC codes that achieve the upper bounds on the minimum distance of linear codes. Several binary reversible QC codes with the best known parameters are provided.