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

2026-03-22

AI Summary

This study investigates the hierarchical structure of higher-order generalized covering radii (GCR) of binary primitive double-error-correcting BCH codes, denoted BCH(2,m), a problem that remains largely unresolved for larger orders \(k\). By introducing a generalized supercode lemma and leveraging tools from generalized Hamming weights, combinatorial analysis, and Weil-type exponential sum estimates, the authors streamline existing proofs for the lower bounds of \(\rho_2\) and \(\rho_3\), and establish a new lower bound for \(\rho_4\) for the first time. Moreover, they derive tight bounds for the general \(k\)-th order GCR: for sufficiently large \(m\), it holds that \(2k \leq \rho_k(\text{BCH}(2,m)) \leq 2k+1\), revealing an essentially linear growth behavior of the generalized covering radii with respect to the order \(k\).

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $k$.

Problem

• generalized covering radii
• BCH codes
• binary primitive codes
• double-error-correcting
• higher-dimensional extension

Innovation

• Generalized covering radius
• BCH codes
• Generalized Supercode Lemma
• Generalized Hamming weights
• Weil exponential sums