Codes from surfaces with small Picard number

TL;DR

This paper investigates Goppa-type evaluation codes from algebraic surfaces with small Picard number, analyzing how geometric properties influence code parameters and providing examples with improved minimum distances.

Contribution

It extends previous work by analyzing the impact of small Picard number and sectional genus on code performance, offering new bounds and examples.

Findings

Surfaces with small Picard number do not always produce good codes.

The sectional genus significantly affects the minimum distance.

Examples surpassing known bounds in Grassl's tables are provided.

Abstract

Extending work of M. Zarzar, we evaluate the potential of Goppa-type evaluation codes constructed from linear systems on projective algebraic surfaces with small Picard number. Putting this condition on the Picard number provides some control over the numbers of irreducible components of curves on the surface and hence over the minimum distance of the codes. We find that such surfaces do not automatically produce good codes; the sectional genus of the surface also has a major influence. Using that additional invariant, we derive bounds on the minimum distance under the assumption that the hyperplane section class generates the N\'eron-Severi group. We also give several examples of codes from such surfaces with minimum distance better than the best known bounds in Grassl's tables.