On covering radius of generalized Zetterberg codes

TL;DR

This paper uses advanced number theoretic techniques to analyze the covering radius of generalized Zetterberg codes over all finite fields, establishing upper bounds and exact values for many parameters, and identifying many quasi-perfect and maximal codes.

Contribution

It introduces a unified analytic framework that simplifies the study of the covering radius of generalized Zetterberg codes across all finite fields, extending previous results especially in even characteristic.

Findings

Covering radius is at most 3 in all cases.

Exact covering radius determined for a broad range of parameters.

Infinitely many quasi-perfect and maximal codes identified.

Abstract

We employ analytic number theoretic techniques, specifically character sums and Weil type estimates, to study the covering radius of the generalized Zetterberg codes over all finite fields. Although the even and odd field cases require distinct technical treatment, the proofs follow a unified analytic framework that is substantially simpler and more transparent than previous approaches. We prove that the covering radius is at most 3 in all cases, and determine its exact value for a wide range of parameters. In even characteristic, our results fill the gap left by recent studies focused solely on odd characteristic; for odd characteristic, the range of parameters for which the covering radius is exactly determined is considerably broader than previously known. Combined with the corresponding minimum distance results, we obtain infinitely many quasi-perfect and maximal codes within this family.