On the Upper Bounds of MDS Codes

TL;DR

This paper investigates the maximum lengths of MDS codes for various parameters, deriving new upper bounds that improve understanding of their limitations in coding theory.

Contribution

The paper introduces new upper bounds for the maximum length of MDS codes, expanding the theoretical limits for specific parameters.

Findings

Established that M_{q}(q-1) ≤ q + 2 when q ≡ 4 mod 6

Derived that M_{q}(q-2) ≤ q + 1 when q ≡ 4 mod 6

Proved that M_{q}(k) ≤ q + k - 3 for q=36(5s+1) and k=6,7

Abstract

Let $M_{q}(k)$ be the maximum length of MDS codes with parameters $q,k$. In this paper, the properties of $M_{q}(k)$ are studied, and some new upper bounds of $M_{q}(k)$ are obtained. Especially we obtain that $M_{q}(q-1)\leq q+2(q\equiv4(mod 6)), M_{q}(q-2)\leq q+1(q\equiv4(mod 6)), M_{q}(k)\leq q+k-3 (q=36(5s+1), s\in N$ and $ k=6,7).