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Nodal surfaces in $\mathbb{P}^3$ and coding theory

Sascha Kurz

TL;DR

This work investigates the binary codes associated to nodal hypersurfaces in $\mathbb{P}^3$, focusing on sextics with the maximal $65$ nodes. It develops and employs $8$-divisible $[n,12,24]_2$ codes, together with the Pless power moments and computer-aided classification, to prove the uniqueness of the code for sextics with $65$ nodes and to describe its automorphism group and weight enumerator, along with the structure of its residual codes. The paper also discusses septics, presenting current numerical bounds $99\le \mu(7)\le 104$ and a candidate $[96,10,44]_2$ code, while underscoring the need for additional coding-theoretic constraints to improve upper bounds. Overall, it highlights the deep link between algebraic geometry of nodal surfaces and binary coding theory and outlines practical directions for tightening node-count bounds via associated codes and generalized MacWilliams-type techniques.

Abstract

To each nodal hypersurface one can associate a binary linear code. Here we show that the binary linear code associated to sextics in $\mathbb{P}^3$ with the maximum number of $65$ nodes, as e.g. the Barth sextic, is unique. We also state possible candidates for codes that might be associated with a hypothetical septic attaining the currently best known upper bound for the maximum number of nodes.

Nodal surfaces in $\mathbb{P}^3$ and coding theory

TL;DR

This work investigates the binary codes associated to nodal hypersurfaces in , focusing on sextics with the maximal nodes. It develops and employs -divisible codes, together with the Pless power moments and computer-aided classification, to prove the uniqueness of the code for sextics with nodes and to describe its automorphism group and weight enumerator, along with the structure of its residual codes. The paper also discusses septics, presenting current numerical bounds and a candidate code, while underscoring the need for additional coding-theoretic constraints to improve upper bounds. Overall, it highlights the deep link between algebraic geometry of nodal surfaces and binary coding theory and outlines practical directions for tightening node-count bounds via associated codes and generalized MacWilliams-type techniques.

Abstract

To each nodal hypersurface one can associate a binary linear code. Here we show that the binary linear code associated to sextics in with the maximum number of nodes, as e.g. the Barth sextic, is unique. We also state possible candidates for codes that might be associated with a hypothetical septic attaining the currently best known upper bound for the maximum number of nodes.

Paper Structure

This paper contains 4 sections, 7 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniqueness of the code associated to a sextic with $\mathbf{65}$ nodes
  4. Conclusion and outlook

Key Result

Lemma 1

Let $C$ be a binary $8$-divisible linear code with minimum distance $d\ge 24$, dimension $k=12$ and effective length $n\le 65$, then $a_{40}\ge 1$ and $n\ge 63$.

Theorems & Definitions (12)

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