Hard Problems of Algebraic Geometry Codes

TL;DR

This paper proves that key computational problems like minimum distance and maximum likelihood decoding are NP-hard for algebraic geometry codes, specifically those based on elliptic curves, highlighting their computational complexity.

Contribution

It demonstrates NP-hardness of these problems for algebraic geometry codes, a class of natural codes, using reductions from NP-complete problems with randomized algorithms.

Findings

Minimum distance problem is NP-hard for algebraic geometry codes.

Maximum likelihood decoding is NP-hard for algebraic geometry codes.

Codes based on elliptic curves with positive rates are computationally hard.

Abstract

The minimum distance is one of the most important combinatorial characterizations of a code. The maximum likelihood decoding problem is one of the most important algorithmic problems of a code. While these problems are known to be hard for general linear codes, the techniques used to prove their hardness often rely on the construction of artificial codes. In general, much less is known about the hardness of the specific classes of natural linear codes. In this paper, we show that both problems are NP-hard for algebraic geometry codes. We achieve this by reducing a well-known NP-complete problem to these problems using a randomized algorithm. The family of codes in the reductions are based on elliptic curves. They have positive rates, but the alphabet sizes are exponential in the block lengths.