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

Official URL: https://dx.doi.org/10.1109/TIT.2025.3544025

For an integer s ≥ 1, let Cs(q0) be the generalized Zetterberg code of length qs0 + 1 over the finite field Fq0 of odd characteristic. Recently, Shi, Helleseth, and Özbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of Cs(q0) for qs0 ≢ 7 (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 qs0 ≡ 7 (mod 8), which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length qs0+1/2, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.