On the Upper Bounds of MDS Codes

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).