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On generalized covering radii of binary primitive double-error-correcting BCH codes

Maosheng Xiong, Chi Hoi Yip

Abstract

The generalized covering radii (GCR) of linear codes are a fundamental higher-dimensional extension of the classical covering radius. While the second and third GCR of binary primitive double-error-correcting BCH codes, $\text{BCH}(2,m)$, were recently determined, their proofs relied on highly complex combinatorial arguments, and the behavior of the GCR hierarchy for larger orders $k$ has remained largely unexplored. In this paper, we introduce the Generalized Supercode Lemma, which lower-bounds the GCR of a code using the generalized Hamming weights of an appropriate supercode. Applying this lemma, we significantly streamline and simplify the proofs for the known lower bounds of $ρ_2(\text{BCH}(2,m))$ and $ρ_3(\text{BCH}(2,m))$, and we establish a new lower bound for $ρ_4(\text{BCH}(2,m))$. Furthermore, by combining combinatorial arguments with Weil-type exponential sum estimates, we investigate the GCR hierarchy for general $k$, proving that $2k \le ρ_k(\text{BCH}(2,m)) \le 2k+1$ whenever $m$ is sufficiently large compared to $k$.

On generalized covering radii of binary primitive double-error-correcting BCH codes

Abstract

The generalized covering radii (GCR) of linear codes are a fundamental higher-dimensional extension of the classical covering radius. While the second and third GCR of binary primitive double-error-correcting BCH codes, , were recently determined, their proofs relied on highly complex combinatorial arguments, and the behavior of the GCR hierarchy for larger orders has remained largely unexplored. In this paper, we introduce the Generalized Supercode Lemma, which lower-bounds the GCR of a code using the generalized Hamming weights of an appropriate supercode. Applying this lemma, we significantly streamline and simplify the proofs for the known lower bounds of and , and we establish a new lower bound for . Furthermore, by combining combinatorial arguments with Weil-type exponential sum estimates, we investigate the GCR hierarchy for general , proving that whenever is sufficiently large compared to .
Paper Structure (16 sections, 14 theorems, 92 equations)

This paper contains 16 sections, 14 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Fundamental Parameters of Linear Codes
  5. Generalized Hamming Weights and Generalized Covering Radii
  6. Primitive BCH Codes
  7. Weil-Type Estimates
  8. Lower bounds of $\rho_3$ and $\rho_4$
  9. The Generalized Supercode Lemma
  10. On $\rho_3(\mathrm{BCH}(2,m))$
  11. On $\rho_4(\mathrm{BCH}(2,m))$
  12. General bounds on $\rho_k(\mathrm{BCH}(2,m))$
  13. Lower bound for large $m$
  14. Upper bound for large $m$
  15. Conclusion
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $r \ge 1$. The $r$-th generalized covering radius $\rho_r$ of the code $\mathrm{BCH}(2,m)$ is the least positive integer $t$ such that for any $r$ vectors $\left[\right], \cdots, \left[\right]$ in ${\mathbb{F}}_{2^m}^2$, there exists $t$ values $x_1,\cdots, x_t \in \mathbb{F}_{2^m}^*$ such that

Theorems & Definitions (23)

  • Lemma 2.1: YohananovbOzbudakh
  • Lemma 2.2: Weil's bound WeilLidlfLici
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1: The Generalized Supercode Lemma
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 13 more