On generator matrices of MDS codes (Corresp.)

It is shown that the family of
q
-ary generalized Reed-Solomon codes is identical to the family of
q
-ary linear codes generated by matrices of the form
[I|A]
, where
I
is the identity matrix, and
A
is 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, where
A
is a Hankel matrix. The parameters of the codes are
(n,k,d)
, for
1 \leq n \leq q+ 1, 1 \leq k \leq n
, and
d=n-k+1
.