Cyclic codes over $\mathbb{F}_{2^m}[u]/\langle u^k\rangle$ of oddly even length

TL;DR

This paper characterizes cyclic codes over a specific finite ring of length twice an odd integer, providing explicit representations, enumeration formulas, and analysis of dual and self-dual codes.

Contribution

It offers explicit descriptions and counting formulas for cyclic codes over the ring _{2^m}[u]/_{2^m} + u_{2^m} + _{2^m} u^{k-1} with length 2n, including duality and self-duality properties.

Findings

Explicit representation of cyclic codes over R

Formulas for counting codewords and codes

Analysis of dual and self-dual cyclic codes

Abstract

Let $\mathbb{F}_{2^m}$ be a finite field of characteristic $2$ and $R=\mathbb{F}_{2^m}[u]/\langle u^k\rangle=\mathbb{F}_{2^m} +u\mathbb{F}_{2^m}+\ldots+u^{k-1}\mathbb{F}_{2^m}$ ($u^k=0$) where $k\in \mathbb{Z}^{+}$ satisfies $k\geq 2$. For any odd positive integer $n$, it is known that cyclic codes over $R$ of length $2n$ are identified with ideals of the ring $R[x]/\langle x^{2n}-1\rangle$. In this paper, an explicit representation for each cyclic code over $R$ of length $2n$ is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over $R$ of length $2n$ is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over $R$ of length $2n$ are investigated. (AAECC-1522)