# An efficient method to construct self-dual cyclic codes of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$

## TL;DR

This paper presents an explicit method for constructing all self-dual cyclic codes of length p^s over a specific finite chain ring, utilizing combinatorial identities and matrix properties.

## Contribution

It introduces a novel explicit representation, enumeration, and an efficient construction method for self-dual cyclic codes over the given ring.

## Key findings

- Explicit enumeration of self-dual cyclic codes of length p^s
- Development of an efficient construction method
- Application of combinatorial identities to code characterization

## Abstract

Let $p$ be an odd prime number, $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $s$ a positive integer. Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over $\mathbb{F}_p$ with a specific type. On that basis, we give an explicit representation and enumeration for all distinct self-dual cyclic codes of length $p^s$ over the finite chain ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2=0)$. Moreover, We provide an efficient method to construct every self-dual cyclic code of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ precisely.