On generator matrices of MDS codes (Corresp.)
Ron M. Roth, G. Seroussi · IEEE Transactions on Information Theory · 1985
It is shown that the family ofq-ary generalized Reed-Solomon codes is identical to the family ofq-ary linear codes generated by matrices of the form[I|A], whereIis the identity matrix, andAis a generalized Cauchy matrix. Using Cauchy matrices, a construction is shown of maximal triangular arrays over GF(q), which are constant along diagonals in a Hankel matrix fashion, and with the property that every square subarray is a nonsingular matrix. By taking rectangular subarrays of the described triangles, it is possible to construct generator matrices[I|A]of maximum distance separable codes, whereAis a Hankel matrix. The parameters of the codes are(n,k,d), for1 \leq n \leq q+ 1, 1 \leq k \leq n, andd=n-k+1.