On the Weights of General MDS Codes

The weight spectra of MDS codes of length $n$ and dimension $k$ over the arbitrary alphabets are studied. For all $q$-ary MDS codes of dimension $k$ containing the zero codeword, it is shown that all $k$ weights from $n$ to $n-k+1$ are realized. The remaining case $n=q+k-1$ is also determined. Additionally, we prove that all binary MDS codes are equivalent to linear MDS codes. The proofs are combinatorial, and self contained.