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

Jinle Liu, Hongfeng Wu, Li Zhu

Abstract

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

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

Abstract

BCH and LCD cyclic codes of length with are studied. A complete characterization of -cyclotomic cosets modulo is given: Theorem \ref{th4} provides a necessary and sufficient condition for any to be a coset leader, and for odd , 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 is raised to (Theorem \ref{th15}--\ref{th5}). Notably, several of these codes are optimal. When is odd, the necessary and sufficient condition for the BCH code 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 .
Paper Structure (10 sections, 32 theorems, 76 equations)

This paper contains 10 sections, 32 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coset leaders and sizes of $q$-cyclotomic cosets modulo $n$
  4. BCH codes and their dual
  5. Parameters of BCH codes of length $\lambda(q^m+1)$
  6. The dual codes of BCH codes of length $\lambda(q^m+1)$
  7. The Enumeration of LCD cyclic codes of length $\lambda(q^{m}+1)$ over $\mathbb{F}_q$
  8. Enumeration of cyclotomic cosets
  9. Enumeration of LCD cyclic codes of length $\lambda(q^m+1)$ over $\mathbb{F}_q$
  10. Concluding remarks

Key Result

Lemma 2.1

lichengju The irreducible polynomial $f_{\gamma}(x)$ is self-reciprocal if and only if $n-\gamma\in C_{\gamma}$.

Theorems & Definitions (52)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • ...and 42 more