New bounds for linear codes of covering radii 2 and 3

The length function l(q) (r, R) is the smallest length of a q-ary linear code of covering radius R and codimension r. In this work we obtain new upper bounds on l(q) (2t + 1, 2), l(q) (3t + 1, 3), l(q) (3t + 2, 3), t >= 1. In particular, we prove that l(q)(3, 2) <= root q(3 lnq + ln ln q) + root q/3 ln q + 3 for all q, l(q) (3, 2) <= 1.05 root 3q ln q for q <= 321007, l(q) (4, 3) < 2.8 (3 root) q ln q for q <= 6229, and l(q) (5, 3) < 3 (3)root q2 ln q for q <= 761. The new bounds on l(q) (2t + 1, 2), l(q) (3t + 1, 3), l(q) (3t + 2, 3), t > 1, are then obtained by lift-constructions. For q a non-square the new bound on l(q) (2t+ 1, 2) improves the previously known ones. For many values of q not equal (q')(3) and r not equal 3t we provide infinite families of [n, n-r](q)3 codes showing that l(q) (r, 3) approximate to c (3)root ln q . q((r-3)/3), where c is a universal constant. As far as it is known to the authors, such families have not been previously described in the literature.