Residus de 2-formes differentielles sur les surfaces algebriques et applications aux codes correcteurs d'erreurs

TL;DR

This paper introduces a new differential construction of error-correcting codes on algebraic surfaces, explores their properties, and investigates their relationships with functional codes, revealing differences from the curve case.

Contribution

It generalizes the differential code construction from curves to algebraic surfaces and analyzes their properties and relations with functional codes.

Findings

Differential codes on surfaces can differ from the curve case, not always being orthogonal to functional codes.

Under certain conditions, these codes can be expressed as sums of differential codes.

Studying Bertini-type problems can provide insights into code parameters.

Abstract

The theory of algebraic-geometric codes has been developed in the beginning of the 80's after a paper of V.D. Goppa. Given a smooth projective algebraic curve X over a finite field, there are two different constructions of error-correcting codes. The first one, called "functional", uses some rational functions on X and the second one, called "differential", involves some rational 1-forms on this curve. Hundreds of papers are devoted to the study of such codes. In addition, a generalization of the functional construction for algebraic varieties of arbitrary dimension is given by Y. Manin in an article of 1984. A few papers about such codes has been published, but nothing has been done concerning a generalization of the differential construction to the higher-dimensional case. In this thesis, we propose a differential construction of codes on algebraic surfaces. Afterwards, we study the properties of these codes and particularly their relations with functional codes. A pretty surprising fact is that a main difference with the case of curves appears. Indeed, if in the case of curves, a differential code is always the orthogonal of a functional one, this assertion generally fails for surfaces. Last observation motivates the study of codes which are the orthogonal of some functional code on a surface. Therefore, we prove that, under some condition on the surface, these codes can be realized as sums of differential codes. Moreover, we show that some answers to some open problems "a la Bertini" could give very interesting informations on the parameters of these codes.