BCH and LCD cyclic codes of length $n=\hat{I}»(q^m+1)$ over finite fields

Abstract

BCH and LCD cyclic codes of length $n=\hat{I}»(q^m+1)$ with $\hat{I}»\mid q-1$ are studied. A complete characterization of $q$-cyclotomic cosets modulo $n$ is given: Theorem \ref{th4} provides a necessary and sufficient condition for any $0\le {\hat{I}}^3<n$ 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 ${\mathcal{C}}_{(q,n,2\hat{I}´+1,n-\hat{I}´+1)}$ is raised to $2(\hat{I}´+1)$ (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 ${\mathcal{C}}_{(q,n,\hat{I}´,0)}$ 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 $\hat{I}»=1$.