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Multi-twisted Codes as Free Modules Over Principal Ideal Domains by ElDin, Ramy Taki is a scholarly article available to read on EtoBox.

What is Multi-twisted Codes as Free Modules Over Principal Ideal Domains about?

We begin this chapter by introducing the simple algebraic structure of cyclic codes over finite fields. This structure undergoes a series of generalizations to present algebraic descriptions of constacyclic, quasi-cyclic (QC), quasi-twisted (QT), generalized quasi-cyclic (GQC), and multi-twisted (MT) codes. The correspondence between these codes and submodules of the free $\mathbb{F}_q[x]$-module $\left(\mathbb{F}_q[x]\right)^\ell$ is established. Thus, any of these codes corresponds to a free linear code over the principal ideal domain (PID) $\mathbb{F}_q[x]$. 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 $\mathcal{C}^\perp$ of a MT code is given. At this point, special attention is paid to QC codes. For a QC code $\mathcal{C}$, we define its reversed code $\mathcal{R}$. We call $\mathcal{C}$ reversible or self-dual if $\mathcal{R}=\mathcal{C}$ or $\mathcal{C}^\perp=\m

Author

ElDin, Ramy Taki

Published

2022

Language

EN