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√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√q, these are the shortest q-ary MDS codes with covering radius r - 1.

In this paper, algebraic-geometric (AG) codes associated to a recently discovered class of maximal curves are investigated. As a result, some linear codes with better parameters with respect to the previously known ones are discovered, and 70 improvements on MinT's tables ("Tables of optimal parameters for linear codes,'' University of Salzburg, Salzburg, Austria) are obtained.

Some new infinite families of short quasi-perfect linear codes are described. Such codes provide improvements on the currently known upper bounds on the minimal length of a quasi-perfect [n,n-m,4]q-code when either 1) q=16, m ges 5, m odd, or 2) q=2i, 7 les i les 15, m ges 4, or 3) q=22lscr, lscr ges 8, m ges 5, m odd. As quasi-perfect [n,n-m,4]q-codes and complete n-caps in projective spaces PG(m-1,q) are equivalent objects, new upper bounds on the size of the smallest complete cap in PG(m-1,q) are obtained