Research proof index

A linear code LC of length ι and dimension dim over a field F is MDS if any dim columns of the generator matrix whose rows are an F-basis of LC are linearly independent.[GRS25] Equivalently, a linear code is MDS if and only if its generator matrix is MDS. Note: the hypothesis 0 < dim LC is necessary because for a trivial code (dim = 0),...

If a generator matrix is MDS with at least one row, then the code it generates is MDS.

If a linear code generated by a full-rank matrix is MDS, then the matrix is MDS.

Every nonzero codeword in a code generated by a k × n MDS matrix has Hamming weight greater or equal to n - k + 1.

If G has full row rank, then every nonzero vector maps to a nonzero codeword.