Determining the Covering Radius of All Generalized Zetterberg Codes in Odd Characteristic
Minjia Shi, Shitao Li, Tor Helleseth, Ferruh Özbudak · IEEE Transactions on Information Theory · 2025
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.