New informations on the structure of the functional codes defined by forms of degree $h$ on non-degenerate Hermitian varieties in $\mathbb{P}^{n(\mathbb{F}_q)$}

TL;DR

This paper investigates the structure and weight distribution of functional codes of degree h on non-degenerate Hermitian varieties in projective space, providing new divisibility conditions, explicit weight listings for surfaces, and conjectures on minimum distance.

Contribution

It introduces a divisibility condition for codeword weights, explicitly lists the first five weights for Hermitian surfaces, and proposes conjectures on minimum distance and weight distribution for higher-dimensional cases.

Findings

Established a divisibility condition for codeword weights.

Listed the first five weights and codewords for Hermitian surfaces.

Proposed conjectures on minimum distance and weight distribution for general Hermitian varieties.

Abstract

We study the functional codes of order $h$ defined by G. Lachaud on $\mathcal{X} \subset {\mathbb{P}}^n(\mathbb{F}_q)$ a non-degenerate Hermitian variety. We give a condition of divisibility of the weights of the codewords. For $\mathcal{X}$ a non-degenerate Hermitian surface, we list the first five weights and the corresponding codewords and give a positive answer on a conjecture formulated on this question. The paper ends with a conjecture on the minimum distance and the distribution of the codewords of the first $2h+1$ weights of the functional codes for the functional codes of order $h$ on $\mathcal{X} \subset {\mathbb{P}}^n(\mathbb{F}_q)$ a non-singular Hermitian variety.