The generalized covering radii of Melas codes

Abstract

The generalized covering radii have recently emerged as fundamental parameters of linear codes with applications to database linear querying. In this paper, we study the generalized covering radii $ρ_t(M(m,q))$ of Melas codes $M(m,q)$ over any finite field $\mathbb{F}_q$. We determine $ρ_2(M(m,q))$ for all $q$, and for a general $t \ge 3$, we prove that $ρ_t(M(m,q)) \in \left\{2t,2t+1\right\}$ for $q \in \{2,3\}$ and $ρ_t(M(m,q))=2t$ for $q \ge 4$ whenever $m$ is sufficiently large. These results extend recent work on the covering radius of Melas codes.