On Euclidean and Hermitian Self-Dual Cyclic Codes over \(\mathbb{F}_{2^r}\)

• Published in: arXiv.org 2016-03
• Main Authors: Consorte, Odessa D, Valdez, Lilibeth D
• Format: Article
• Language: English
• Subjects: Binary system; Codes; Polynomials; Splitting; Unions
• ISSN: 2331-8422

Cyclic and self-dual codes are important classes of codes in coding theory. Jia, Ling and Xing \cite{Jia} as well as Kai and Zhu \cite{Kai} proved that Euclidean self-dual cyclic codes of length \(n\) over \(\mathbb{F}_q\) exist if and only if \(n\) is even and \(q=2^r\), where \(r\) is any positive integer. For \(n\) and \(q\) even, there always exists an \([n, \frac{n}{2}]\) self-dual cyclic code with generator polynomial \(x^{\frac{n}{2}}+1\) called the \textit{trivial self-dual cyclic code}. In this paper we prove the existence of nontrivial self-dual cyclic codes of length \(n=2^\nu \cdot \bar{n}\), where \(\bar{n}\) is odd, over \(\mathbb{F}_{2^r}\) in terms of the existence of a nontrivial splitting \((Z, X_0, X_1)\) of \(\mathbb{Z}_{\bar{n}}\) by \(\mu_{-1}\), where \(Z, X_0,X_1\) are unions of \(2^r\)-cyclotomic cosets mod \(\bar{n}.\) We also express the formula for the number of cyclic self-dual codes over \(\mathbb{F}_{2^r}\) for each \(n\) and \(r\) in terms of the number of \(2^r\)-cyclotomic cosets in \(X_0\) (or in \(X_1\)). We also look at Hermitian self-dual cyclic codes and show properties which are analogous to those of Euclidean self-dual cyclic codes. That is, the existence of nontrivial Hermitian self-dual codes over \(\mathbb{F}_{2^{2 \ell}}\) based on the existence of a nontrivial splitting \((Z, X_0, X_1)\) of \(\mathbb{Z}_{\bar{n}}\) by \(\mu_{-2^\ell}\), where \(Z, X_0,X_1\) are unions of \(2^{2 \ell}\)-cyclotomic cosets mod \(\bar{n}.\) We also determine the lengths at which nontrivial Hermitian self-dual cyclic codes exist and the formula for the number of Hermitian self-dual cyclic codes for each \(n\).