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 ${\rho }_t(M(m,q))$ of Melas codes $M(m,q)$ over any finite field ${\mathbb{F}}_q$. We determine ${\rho }_2(M(m,q))$ for all $q$, and for a general $t\ge 3$, we prove that ${\rho }_t(M(m,q))\in \left\{2t,2t+1\right\}$ for $q\in \{2,3\}$ and ${\rho }_t(M(m,q))=2t$ for $q\ge 4$ whenever $m$ is sufficiently large. These results extend recent work on the covering radius of Melas codes.

We have updated the references to reflect recent literature on this topic. In addition, we have clarified the relationship between our work and a related in-press article by another group, which addressed the binary Melas codes