Algebraic-geometric codes from vector bundles and their decoding

TL;DR

This paper generalizes algebraic-geometric codes by using vector bundles instead of line bundles, enabling the construction of interleaved codes with extended decoding capabilities based on an extended algorithm.

Contribution

It introduces a novel class of algebraic-geometric codes derived from vector bundles, extending existing decoding algorithms to improve error correction.

Findings

Codes can correct errors up to t* - g/2

Decoding algorithm extended to new code class

Codes are interleaved versions of classical algebraic-geometric codes

Abstract

Algebraic-geometric codes can be constructed by evaluating a certain set of functions on a set of distinct rational points of an algebraic curve. The set of functions that are evaluated is the linear space of a given divisor or, equivalently, the set of section of a given line bundle. Using arbitrary rank vector bundles on algebraic curves, we propose a natural generalization of the above construction. Our codes can also be seen as interleaved versions of classical algebraic-geometric codes. We show that the algorithm of Brown, Minder and Shokrollahi can be extended to this new class of codes and it corrects any number of errors up to $t^{*} - g/2$, where $t^{*}$ is the designed correction capacity of the code and $g$ is the curve genus.