On the Covering Radius of MDS Codes

For a linear maximum distance separable (MDS) code with redundancy r, the covering radius is either r or r - 1. However, for r > 3, few examples of q-ary linear MDS codes with radius r - 1 are known, including the Reed-Solomon codes with length q + 1. In this paper, for redundancies r as large as 12 root q, infinite families of q-ary MDS codes with covering radius r - 1 and length less than q + 1 are constructed. These codes are obtained from algebraic-geometric codes arising from elliptic curves. For most pairs (r, q) with r <= 12 root q, these are the shortest q-ary MDS codes with covering radius r - 1.