# 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.

## Key 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.