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

TL;DR

This paper introduces a new lemma to simplify the analysis of generalized covering radii of binary primitive double-error-correcting BCH codes, establishing bounds and exploring the hierarchy for larger orders.

Contribution

The paper presents the Generalized Supercode Lemma, simplifying proofs of GCR bounds and establishing new bounds for higher orders in BCH codes.

Findings

Streamlined proofs for known GCR bounds

New lower bound for the fourth GCR

Bounds for GCR hierarchy when m is large

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 $\rho_2(\text{BCH}(2,m))$ and $\rho_3(\text{BCH}(2,m))$, and we establish a new lower bound for $\rho_4(\text{BCH}(2,m))$. Furthermore, by combining combinatorial arguments with Weil-type exponential sum estimates, we investigate the GCR hierarchy for general $k$, proving that $2k \le \rho_k(\text{BCH}(2,m)) \le 2k+1$ whenever $m$ is sufficiently large compared to $k$.