On the order bounds for one-point AG codes

TL;DR

This paper investigates a new bound for the minimum distance of one-point algebraic geometry codes, connecting it with existing bounds and extending its application to generalized Hamming weights.

Contribution

It introduces a new bound $d^*$ for one-point AG codes, relates it to the order bound, and extends it to generalized Hamming weights.

Findings

Bound $d^*$ improves minimum distance estimates.

Connection established between $d^*$ and the order bound.

Extended $d^*$ to generalized Hamming weights.

Abstract

The order bound for the minimum distance of algebraic geometry codes was originally defined for the duals of one-point codes and later generalized for arbitrary algebraic geometry codes. Another bound of order type for the minimum distance of general linear codes, and for codes from order domains in particular, was given in [H. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields and their Applications 14 (2008), pp. 92-123]. Here we investigate in detail the application of that bound to one-point algebraic geometry codes, obtaining a bound $d^*$ for the minimum distance of these codes. We establish a connection between $d^*$ and the order bound and its generalizations. We also study the improved code constructions based on $d^*$. Finally we extend $d^*$ to all generalized Hamming weights.