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

TL;DR

This paper investigates the unique binary linear code associated with sextic hypersurfaces in projective 3-space having the maximum number of nodes, and explores potential codes for hypothetical septic hypersurfaces reaching known node bounds.

Contribution

It proves the uniqueness of the code for the Barth sextic and proposes candidate codes for septic hypersurfaces at the current maximum node bounds.

Findings

The code for the Barth sextic is unique.

Candidates for codes of septic hypersurfaces are identified.

Insights into the relationship between nodal hypersurfaces and associated codes.

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.