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

TL;DR

This paper completely determines the covering radius of all generalized Zetterberg codes in odd characteristic, solving an open problem by employing finite field arithmetic and algebraic curves, and introduces related twisted codes with quasi-perfect properties.

Contribution

It develops a new technique to determine the covering radius for all cases of generalized Zetterberg codes, including previously unresolved instances.

Findings

Covering radius determined for all generalized Zetterberg codes.

Introduction of twisted half generalized Zetterberg codes with similar properties.

Identification of some quasi-perfect codes derived from these results.

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 \"{O}zbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of $\mathcal{C}_s(q_0)$ for $q_0^s \not \equiv 7 \pmod{8}$, 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.