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

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 $k$.