Lowest density MDS codes over extension alphabets

Let F be a finite field and b be a positive integer. A construction is presented of codes over the alphabet F^b with the following three properties: i) the codes are maximum-distance separable (MDS) over F^b, ii) they are linear over F, and iii) they have systematic generator and parity-check matrices over F with the smallest possible number of nonzero entries. Furthermore, for the case F=GF(2), the construction is the longest possible among all codes that satisfy properties i)-iii).