Determining the covering radius of all generalized Zetterberg codes in odd characteristic

Abstract

For an integer $s\ge 1$, let ${\mathcal{C}}_s(q_0)$ be the generalized Zetterberg code of length $q_0^s+1$ over the finite field $\F_{q_0}$ of odd characteristic. Recently, Shi, Helleseth, and Özbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of ${\mathcal{C}}_s(q_0)$ for $q_0^s\not\equiv 7(\mathrm{mod}8)$, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for $q_0^s\equiv 7(\mathrm{mod}8)$, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length $\frac{q_0^s+1}{2}$, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.