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The dimension and Bose distance of some BCH codes of length $\frac{q^{m}-1}λ$

Run Zheng, Nung-Sing Sze, Zejun Huang

TL;DR

This work advances the parameter understanding of BCH codes of length $n=\frac{q^m-1}{\lambda}$ with $\lambda\mid (q-1)$ by deriving explicit dimension formulas and Bose-distance characterizations for a substantially broader range of designed distances $\delta$ than previously known, specifically for narrow-sense codes when $m\ge4$ and $2\le\delta\le\frac{q^{\lfloor(2m-1)/3\rfloor+1}-1}{\lambda}+1$. The authors leverage a key coset-leader correspondence between modulo $n$ and modulo $\lambda n$, along with detailed coset-structure results and auxiliary counting lemmas, to reduce to the primitive BCH setting and to obtain closed-form expressions involving functions $N(\cdot)$, $f(\cdot)$, $\tilde f(\cdot)$ and $g(\cdot)$. They further extend these results to certain non-narrow-sense BCH codes and provide explicit parameter formulas for cases where $\lambda\delta=a q^{h+k}+b$, including concrete corollaries demonstrating good or optimal parameters in several examples. Overall, the paper broadens the regime in which BCH code parameters can be exactly determined, facilitating the construction of high-performance codes of length $n=\frac{q^m-1}{\lambda}$ for practical applications.

Abstract

BCH codes are important error correction codes, widely utilized due to their robust algebraic structure, multi-error correcting capability, and efficient decoding algorithms. Despite their practical importance and extensive study, their parameters, including dimension, minimum distance and Bose distance, remain largely unknown in general. This paper addresses this challenge by investigating the dimension and Bose distance of BCH codes of length $(q^m - 1)/λ$ over the finite field $\mathbb{F}_q$, where $λ$ is a positive divisor of $q - 1$. Specifically, for narrow-sense BCH codes of this length with $m \geq 4$, we derive explicit formulas for their dimension for designed distance $2 \leq δ\leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/λ + 1$. We also provide explicit formulas for their Bose distance in the range $2 \leq δ\leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/λ$. These ranges for $δ$ are notably larger than the previously known results for this class of BCH codes. Furthermore, we extend these findings to determine the dimension and Bose distance for certain non-narrow-sense BCH codes of the same length. Applying our results, we identify several BCH codes with good parameters.

The dimension and Bose distance of some BCH codes of length $\frac{q^{m}-1}λ$

TL;DR

This work advances the parameter understanding of BCH codes of length with by deriving explicit dimension formulas and Bose-distance characterizations for a substantially broader range of designed distances than previously known, specifically for narrow-sense codes when and . The authors leverage a key coset-leader correspondence between modulo and modulo , along with detailed coset-structure results and auxiliary counting lemmas, to reduce to the primitive BCH setting and to obtain closed-form expressions involving functions , , and . They further extend these results to certain non-narrow-sense BCH codes and provide explicit parameter formulas for cases where , including concrete corollaries demonstrating good or optimal parameters in several examples. Overall, the paper broadens the regime in which BCH code parameters can be exactly determined, facilitating the construction of high-performance codes of length for practical applications.

Abstract

BCH codes are important error correction codes, widely utilized due to their robust algebraic structure, multi-error correcting capability, and efficient decoding algorithms. Despite their practical importance and extensive study, their parameters, including dimension, minimum distance and Bose distance, remain largely unknown in general. This paper addresses this challenge by investigating the dimension and Bose distance of BCH codes of length over the finite field , where is a positive divisor of . Specifically, for narrow-sense BCH codes of this length with , we derive explicit formulas for their dimension for designed distance . We also provide explicit formulas for their Bose distance in the range . These ranges for are notably larger than the previously known results for this class of BCH codes. Furthermore, we extend these findings to determine the dimension and Bose distance for certain non-narrow-sense BCH codes of the same length. Applying our results, we identify several BCH codes with good parameters.

Paper Structure

This paper contains 22 sections, 22 theorems, 191 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some known results on -cyclotomic cosets modulo
  4. Auxillary Lemmas
  5. The dimension of for
  6. The Bose distance of
  7. BCH codes with
  8. Some non-narrow-sense BCH codes
  9. Conclusion
  10. Proof of Lemma \ref{['le1']}
  11. Proof of Lemma \ref{['lll8']}
  12. Proof of Lemma \ref{['le0']}
  13. Proof of Lemma \ref{['adl']}
  14. Proof of Lemma \ref{['ll9']}
  15. Proof of Lemma \ref{['lema12']}
  16. ...and 7 more sections

Key Result

Lemma 1

Suppose that $n$ and $\lambda$ are two integers such that $\mathrm{gcd}(n,q)=1$ and $\mathrm{gcd}(\lambda,q)=1$. Let $a\in [0,n-1]$ be an integer. Then

Theorems & Definitions (50)

  • Lemma 1
  • proof
  • Corollary 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • Lemma 2
  • ...and 40 more