Complete classification for simple root cyclic codes over local rings $\mathbb{Z}_{p^s}[v]/\langle v^2-pv\rangle$

TL;DR

This paper provides a complete classification of cyclic codes over a specific local ring, including their structure, duals, and self-dual codes, with applications to optimal and extremal codes over finite fields.

Contribution

It offers a full classification of cyclic codes over the ring _{p^s}[v]/\u2208v^2-pv, including duals and self-dual codes, and constructs optimal and extremal codes over finite fields.

Findings

Complete classification of cyclic codes over the ring.

Explicit formulas for code cardinalities and duals.

Construction of optimal and extremal self-dual codes.

Abstract

Let $p$ be a prime integer, $n,s\geq 2$ be integers satisfying ${\rm gcd}(p,n)=1$, and denote $R=\mathbb{Z}_{p^s}[v]/\langle v^2-pv\rangle$. Then $R$ is a local non-principal ideal ring of $p^{2s}$ elements. First, the structure of any cyclic code over $R$ of length $n$ and a complete classification of all these codes are presented. Then the cardinality of each code and dual codes of these codes are given. Moreover, self-dual cyclic codes over $R$ of length $n$ are investigated. Finally, we list some optimal $2$-quasi-cyclic self-dual linear codes over $\mathbb{Z}_4$ of length $30$ and extremal $4$-quasi-cyclic self-dual binary linear $[60,30,12]$ codes derived from cyclic codes over $\mathbb{Z}_{4}[v]/\langle v^2+2v\rangle$ of length $15$.