On the minimum distance of elliptic curve codes

TL;DR

This paper presents a combinatorial method to explicitly determine the minimum distance of elliptic curve algebraic geometry codes when the evaluation set is sufficiently large, also strengthening the MDS conjecture for these codes.

Contribution

It provides a simple explicit formula for the minimum distance of ECAG codes with large evaluation sets, using a new sieving technique, and improves the MDS conjecture for these codes.

Findings

Explicit minimum distance formula for ECAG codes with large evaluation sets

New combinatorial sieving technique applied to coding theory

Stronger version of the MDS conjecture for ECAG codes

Abstract

Computing the minimum distance of a linear code is one of the fundamental problems in algorithmic coding theory. Vardy [14] showed that it is an \np-hard problem for general linear codes. In practice, one often uses codes with additional mathematical structure, such as AG codes. For AG codes of genus $0$ (generalized Reed-Solomon codes), the minimum distance has a simple explicit formula. An interesting result of Cheng [3] says that the minimum distance problem is already \np-hard (under \rp-reduction) for general elliptic curve codes (ECAG codes, or AG codes of genus $1$). In this paper, we show that the minimum distance of ECAG codes also has a simple explicit formula if the evaluation set is suitably large (at least $2/3$ of the group order). Our method is purely combinatorial and based on a new sieving technique from the first two authors [8]. This method also proves a significantly stronger version of the MDS (maximum distance separable) conjecture for ECAG codes.