Enumeration of Self-Dual Cyclic Codes of some Specific Lengths over Finite Fields

TL;DR

This paper develops efficient algorithms and formulas for enumerating self-dual cyclic codes of specific lengths over finite fields, improving computational methods for these important linear codes.

Contribution

It introduces new formulas and algorithms for counting self-dual cyclic codes of lengths involving powers of two and primes, enhancing computational efficiency.

Findings

Derived alternative formulas for enumeration

Established number theoretical tools for code analysis

Provided efficient algorithms for specific code lengths

Abstract

Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length $n$ over a finite field if and only if $n$ and the field characteristic are even. The enumeration of such codes has been given under both the Euclidean and Hermitian products. However, in each case, the formula for self-dual cyclic codes of length $n$ over a finite field contains a characteristic function which is not easily computed. In this paper, we focus on more efficient ways to enumerate self-dual cyclic codes of lengths $2^\nu p^r$ and $2^\nu p^rq^s$, where $\nu$, $r$, and $s$ are positive integers. Some number theoretical tools are established. Based on these results, alternative formulas and efficient algorithms to determine the number of self-dual cyclic codes of such lengths are provided.