The Weights in MDS Codes

TL;DR

This paper investigates the weight distribution of MDS codes over finite fields, demonstrating that, except for specific cases, these codes contain all possible weights within a certain range, using the dual code's covering radius.

Contribution

It establishes that most MDS codes have a complete set of weights in a specific interval, confirming a conjecture with explicit exceptions.

Findings

Most MDS codes contain all weights from n-k+1 to n.

Explicit exceptions are identified where the weight set differs.

The proof utilizes the covering radius of the dual code.

Abstract

The weights in MDS codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n-k+1 to n. The proof uses the covering radius of the dual code