Evaluation codes from linear systems of conics

TL;DR

This paper investigates the properties of evaluation codes derived from linear systems of symmetric polynomials over finite fields, focusing on the even characteristic case, extending previous work on the odd characteristic case.

Contribution

It generalizes the Datta-Johnsen code by analyzing evaluation codes from low-dimensional linear systems of symmetric polynomials in even characteristic fields.

Findings

Extended the understanding of evaluation codes in even characteristic fields.

Provided new insights into the structure of codes from symmetric polynomials.

Connected the generalization to previous results in odd characteristic cases.

Abstract

The Datta-Johnsen code is an evaluation code where the linear combinations of elementary symmetric polynomials are evaluated on the set of all points with pairwise distinct coordinates in an affine space of dimension $\ge 2$ over a finite field $\mathbb{F}_q$. A generalization is obtained by taking a low dimensional linear system of symmetric polynomials. The odd characteristic case was the subject of a recent paper. Here, the even characteristic case is investigated.