Linking Reversed and Dual Codes of Quasi-Cyclic Codes

Abstract

It is known that quasi-cyclic (QC) codes over the finite field 𝔽_q correspond to certain 𝔽_q[x]-modules. A QC code C is specified by a generator polynomial matrix G whose rows generate C as an 𝔽_q[x]-module. The reversed code of C, denoted by R, is the code obtained by reversing all codewords of C while the dual code of C is denoted by C^⊥. We call C reversible, self-orthogonal, and self-dual if R = C, C^⊥ ⊇ C, and C^⊥ = C, respectively. In this study, for a given C, we find an explicit formula for a generator polynomial matrix of R. A necessary and sufficient condition for C to be reversible is derived from this formula. In addition, we reveal the relations among C, R, and C^⊥. Specifically, we give conditions on G corresponding to C^⊥ ⊇ R, C^⊥ ⊆ R, and C = R = C^⊥. As an application, we employ these theoretical results to the construction of QC codes with best parameters. Computer search is used to show that there exist various binary reversible self-orthogonal QC codes that achieve the upper bounds on the minimum distance of linear codes.