Multi-twisted codes as free modules over principal ideal domains

We begin by introducing the simple algebraic structure of cyclic, constacyclic, quasi-cyclic (QC), quasi-twisted (QT), generalized quasi-cyclic (GQC), and multi-twisted (MT) codes over finite fields. Then, we establish the correspondence between these codes and submodules of the free \mathbb{F}_q[x]-module \left(\mathbb{F}_q[x]\right)^\ell. We show that an MT code is a linear code over the principal ideal domain (PID) \mathbb{F}_q[x]. Hence, a basis of this code exists and is used to build a generator matrix with polynomial entries, called the generator polynomial matrix (GPM). The Hermite normal form of matrices over PIDs is exploited to achieve the reduced GPMs of MT codes. Some properties of the reduced GPM are introduced, for example, the identical equation. A formula for a GPM of the dual code of an MT code is established. At this point, special attention is paid to QC codes. We characterize GPMs for QC codes that combine reversibility and self-duality/self-orthogonality. We show the existence of binary self-orthogonal reversible QC codes that have the best known parameters as linear codes.

کلیدواژه ها:

Dual code ، Generator polynomial matrix ، Hermite normal form ، Multi-twisted code