Determining the Covering Radius of All Generalized Zetterberg Codes in Odd Characteristic

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 ${\mathbb {F}}_{q_{0}}$ of odd characteristic. Recently, Shi et al. determined the covering radius of ${\mathcal {C}}_{s}(q_{0})$ for $q_{0}^{s} \cancel {\equiv }7 \pmod {8}$ , 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 $q_{0}^{s} \equiv 7 \pmod {8}$ , 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.