Minimal codes from hypersurfaces in even characteristic

TL;DR

This paper constructs new minimal linear codes from hypersurfaces in even characteristic, analyzes their weight distributions, and introduces the concept of cutting gap to study their geometric properties.

Contribution

It introduces a new family of minimal codes from algebraic hypersurfaces related to quasi-Hermitian varieties in even characteristic, with explicit weight distributions and a novel geometric measure called cutting gap.

Findings

New infinite family of minimal codes, mostly with few weights.

Complete weight distribution for codes in dimensions 3 and 4.

Introduction of cutting gap to analyze non-hyperplane subspace intersections.

Abstract

The setting of projective systems can be used to study the parameters of a projective linear code $\mathcal{C}$. This can be done by considering the intersections of the point set $\Omega$ defined by the columns of a generating matrix for $\mathcal{C}$ with the hyperplanes of a projective space. In particular, $\mathcal{C}$ is minimal if $\Omega$ is cutting, i.e., every hyperplane is spanned by its intersection with $\Omega$. Minimal linear codes have important applications for secret sharing schemes and secure two-party computation. In this article we first investigate the properties of some algebraic hypersurfaces $\mathcal{V}_{\varepsilon}^r$ related to certain quasi-Hermitian varieties of $\mathrm{PG}(r,q^2)$, with $q=2^e$, $e>1$ odd. These varieties give rise to a new infinite family of linear codes which are minimal except for $r=3$ and $e\equiv 1 \pmod 4$. In the case $r \in \{3,4\}$, we exhibit codes having at most 6 non-zero weights whose we provide the complete list. As a byproduct, we obtain $(r+1)$-dimensional codes with just $3$ non-zero weights. We point out that linear codes with few weights are also important in authentication codes and association schemes. In the last part of the paper we consider an extension of the notion of being cutting with respect to subspaces other than hyperplanes and introduce the definition of cutting gap in order to characterize and measure what happens when this property is not satisfied. Finally, we then apply these notions to Hermitian codes and to the codes related to $\mathcal{V}_\varepsilon^r$ discussed before.