Quasicyclic Codes of Index $\ell$ over $F_q$ Viewed as $F_q[x]$-Submodules of $F_{q^{\ell}}/\langle x^{m}-1\rangle$

Abstract

Quasicyclic codes of length $n = m \ell$ and index $\ell$ over the finite field $F_q$ are linear codes invariant under cyclic shifts by $\ell$ places. They are shown to be isomorphic to the $F_q[x]/\langle x^{m}-1 \rangle$-submodules of $F_{q^{\ell}}[x]/\langle x^{m}?1\rangle$ where the defining property in this setting is closure under multiplication by $x$ with reduction modulo $x^{m}?1$. Using this representation, the dimension of a $1$-generator code can be determined straightforwardly from the chosen generator, and improved lower bounds on minimum distance are developed. A special case of multi-generator codes, for which the dimension can be algebraically recovered from the generating set is described. Every possible dimension of a quasicyclic code can be obtained in some such special form. Lower bounds on minimum distance are also given for all multi-generator quasicyclic codes.