BCH and LCD cyclic codes of length $n=\lambda(q^m+1)$ over finite fields

TL;DR

This paper characterizes BCH and LCD cyclic codes of length n=lambda(q^m+1), determines their dimensions, bounds their minimum distances, and classifies LCD codes, extending previous results for lambda=1.

Contribution

It provides a complete characterization of q-cyclotomic cosets, explicit coset leaders, and conditions for duality and LCD properties for these codes.

Findings

Explicit formulas for coset leaders and cyclotomic cosets.

Determination of code dimensions and improved minimum distance bounds.

Complete enumeration of LCD cyclic codes of the given length.

Abstract

BCH and LCD cyclic codes of length $n=\lambda(q^m+1)$ with $\lambda\mid q-1$ are studied. A complete characterization of $q$-cyclotomic cosets modulo $n$ is given: Theorem \ref{th4} provides a necessary and sufficient condition for any $0\le \gamma<n$ to be a coset leader, and for odd $m$, the two largest coset leaders are explicitly determined (Theorem \ref{th9} and Theorem \ref{th14}). Based on these results, the dimensions of several families of BCH codes are determined, and the lower bound on the minimal distance of $\mathcal{C}_{(q,n,2\delta+1,n-\delta+1)}$ is raised to $2(\delta+1)$ (Theorem \ref{th15}--\ref{th5}). Notably, several of these codes are optimal. When $m$ is odd, the necessary and sufficient condition for the BCH code $\mathcal{C}_{(q,n,\delta,0)}$ to be dually-BCH is proved (Theorem \ref{th11}). Finally, an exact enumeration of all LCD cyclic codes of this length is derived (Theorem \ref{th3}). All of the above results extend previous results that were limited to $\lambda=1$.