Construction of Rational Surfaces Yielding Good Codes

TL;DR

This paper constructs algebraic geometry codes on rational surfaces, estimating their minimum distance via point counting, and discovers new codes with record parameters over finite fields.

Contribution

It introduces new algebraic geometry codes from rational surfaces, including the first study of codes from forms of degree 3 on elliptic quadrics, and finds codes surpassing known bounds.

Findings

Codes with parameters [57,12,34] over F_7 and [91,18,53] over F_9 were discovered.

The minimum distance estimates are achieved using Weil and Homma-Kim bounds.

New classes of codes from rational surfaces outperform existing codes.

Abstract

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound "\`a la Weil" of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over $\mathbf{F}_7$ and a [91,18,53] code over $\mathbf{F}_9$ are discovered, these codes beat the best known codes up to now.