On Near-MDS Elliptic Codes

TL;DR

This paper investigates the properties of near MDS elliptic codes, providing non-extendability results and a simpler construction method, thereby advancing understanding of their length and extendability beyond classical MDS code limits.

Contribution

It presents new non-extendability results for near MDS elliptic codes and offers a simpler construction method for certain known codes.

Findings

Near MDS elliptic codes can have length greater than q+1.

Some near MDS elliptic codes cannot be extended to longer codes.

A new, simpler construction method for certain near MDS elliptic codes.

Abstract

The main conjecture on maximum distance separable (MDS) codes states that, execpt for some special cases, the maximum length of a q-ary linear MDS code is q+1. This conjecture does not hold true for near maximum distance separable codes because of the existence of q-ary near MDS elliptic codes having length bigger than q+1. An interesting related question is whether a near MDS elliptic code can be extended to a longer near MDS code. Our results are some non-extendability results and an alternative and simpler construction for certain known near MDS elliptic codes.