On generalized covering radii of binary primitive double-error-correcting BCH codes
This work addresses a fundamental problem in coding theory for researchers, providing incremental theoretical insights into the GCR of specific BCH codes.
The paper tackles the problem of determining the generalized covering radii (GCR) hierarchy for binary primitive double-error-correcting BCH codes, introducing the Generalized Supercode Lemma to simplify proofs for known lower bounds and establish a new lower bound for the fourth GCR, while proving that the k-th GCR is between 2k and 2k+1 for large m relative to k.
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 $Ï_2(\text{BCH}(2,m))$ and $Ï_3(\text{BCH}(2,m))$, and we establish a new lower bound for $Ï_4(\text{BCH}(2,m))$. Furthermore, by combining combinatorial arguments with Weil-type exponential sum estimates, we investigate the GCR hierarchy for general $k$, proving that $2k \le Ï_k(\text{BCH}(2,m)) \le 2k+1$ whenever $m$ is sufficiently large compared to $k$.