Distance bounds for algebraic geometric codes

TL;DR

This paper reviews and unifies various methods for improving the minimum distance bounds of algebraic geometric codes, providing simplified proofs and a common theoretical framework for existing bounds.

Contribution

It offers short proofs for known bounds and introduces unifying theorems that generalize the Feng-Rao bound for one-point codes.

Findings

Unified framework for floor and order bounds

Simplified proofs for existing bounds

Generalizations of the Feng-Rao bound

Abstract

Various methods have been used to obtain improvements of the Goppa lower bound for the minimum distance of an algebraic geometric code. The main methods divide into two categories and all but a few of the known bounds are special cases of either the Lundell-McCullough floor bound or the Beelen order bound. The exceptions are recent improvements of the floor bound by Guneri-Stichtenoth-Taskin, and Duursma-Park, and of the order bound by Duursma-Park and Duursma-Kirov. In this paper we provide short proofs for all floor bounds and most order bounds in the setting of the van Lint and Wilson AB method. Moreover, we formulate unifying theorems for order bounds and formulate the DP and DK order bounds as natural but different generalizations of the Feng-Rao bound for one-point codes.