An explicit representation and enumeration for self-dual cyclic codes over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ of length $2^s$

Abstract

Let ${\mathbb{F}}_{2^m}$ be a finite field of cardinality $2^m$ and $s$ a positive integer. Using properties for Kronecker product of matrices and calculation for linear equations over ${\mathbb{F}}_{2^m}$, an efficient method for the construction of all distinct self-dual cyclic codes with length $2^s$ over the finite chain ring ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ $(u^2=0)$ is provided. On that basis, an explicit representation for every self-dual cyclic code of length $2^s$ over ${\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}$ and an exact formula to count the number of all these self-dual cyclic codes are given.