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 $k$.

翻译：线性码的广义覆盖半径（GCR）是经典覆盖半径的高维基本推广。虽然最近确定了二元本原双纠错BCH码$\text{BCH}(2,m)$的第二和第三GCR，但其证明依赖于高度复杂的组合论证，且更大阶数$k$下GCR层级的行为仍基本未知。本文引入广义超码引理，该引理利用适当超码的广义汉明重量给出码的GCR下界。应用此引理，我们显著精简并简化了$\rho_2(\text{BCH}(2,m))$和$\rho_3(\text{BCH}(2,m))$已知下界的证明，并建立了$\rho_4(\text{BCH}(2,m))$的新下界。此外，通过将组合论证与Weil型指数和估计相结合，我们研究了一般$k$下的GCR层级，证明当$m$相对于$k$足够大时，有$2k \le \rho_k(\text{BCH}(2,m)) \le 2k+1$成立。