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On covering radius of generalized Zetterberg codes

Haode Yan, Maosheng Xiong

TL;DR

This work develops a unified analytic framework based on character sums and Weil-type bounds to study the covering radius of generalized Zetterberg codes $C_s(q_0)$ over all finite fields. It proves a universal upper bound $\rho(C_s(q_0)) \le 3$ and provides necessary and sufficient conditions for when the radius equals 3, along with explicit exact values for wide ranges of parameters, thus yielding infinite families of quasi-perfect and maximal codes. The analysis treats odd and even $q_0$ with distinct but structurally uniform arguments, improving prior results in the odd case and substantially advancing the even-case understanding. The results have implications for code optimality in the Hamming space and connect number-theoretic techniques to coding theory through precise radius computations.

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.

On covering radius of generalized Zetterberg codes

TL;DR

This work develops a unified analytic framework based on character sums and Weil-type bounds to study the covering radius of generalized Zetterberg codes over all finite fields. It proves a universal upper bound and provides necessary and sufficient conditions for when the radius equals 3, along with explicit exact values for wide ranges of parameters, thus yielding infinite families of quasi-perfect and maximal codes. The analysis treats odd and even with distinct but structurally uniform arguments, improving prior results in the odd case and substantially advancing the even-case understanding. The results have implications for code optimality in the Hamming space and connect number-theoretic techniques to coding theory through precise radius computations.

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.

Paper Structure

This paper contains 18 sections, 31 theorems, 217 equations, 5 tables.

Table of Contents

  1. Introduction
  2. The covering radius
  3. Generalized Zetterberg Codes
  4. Summary of Main Results
  5. Notation
  6. Preliminaries
  7. Generalized Zetterberg and half-Zetterberg codes
  8. Character sums over finite fields
  9. Minimum distance of $C_{s}(q_0)$ and $\widetilde{C_{s}(q_0)}$
  10. Case: $q_0$ is even
  11. Case: $q_0$ is odd
  12. Covering radius of $C_s(q_0)$ is at most 3
  13. Determining covering radius of $C_{s}(q_0)$: $q_0$ odd
  14. Determining covering radius of $C_{s}(q_0)$: $q_0$ even
  15. Discussion: quasi-perfect and maximal codes
  16. ...and 3 more sections

Key Result

Lemma 1

The covering radius of $C_s(q_0)$ is the least positive integer $\rho$ such that for any $\alpha \in {\mathbb{F}}_{q^2}$, there exist $\left(c_1,\ldots,c_{\rho}\right) \in {\mathbb{F}}_{q_0}^{\rho}$ and $\left(x_1,\ldots,x_{\rho}\right) \in H^{\rho}$ such that

Theorems & Definitions (46)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Theorem 1
  • ...and 36 more