An explicit expression for Euclidean self-dual cyclic codes of length $2^k$ over Galois ring ${\rm GR}(4,m)$

TL;DR

This paper provides a simple, explicit formula and construction method for all Euclidean self-dual cyclic codes of length 2^k over Galois rings, extending prior work that only counted these codes.

Contribution

It introduces a new explicit expression and construction method for all such codes, improving upon previous enumeration results.

Findings

Explicit formula for all codes using combination numbers

Efficient construction method based on matrix properties

Complete listing of codes for lengths 16, 32, 64

Abstract

For any positive integers $m$ and $k$, existing literature only determines the number of all Euclidean self-dual cyclic codes of length $2^k$ over the Galois ring ${\rm GR}(4,m)$, such as in [Des. Codes Cryptogr. (2012) 63:105--112]. Using properties for Kronecker products of matrices of a specific type and column vectors of these matrices, we give a simple and efficient method to construct all these self-dual cyclic codes precisely. On this basis, we provide an explicit expression to accurately represent all distinct Euclidean self-dual cyclic codes of length $2^k$ over ${\rm GR}(4,m)$, using combination numbers. As an application, we list all distinct Euclidean self-dual cyclic codes over ${\rm GR}(4,m)$ of length $2^k$ explicitly, for $k=4,5,6$.