New bounds for covering codes of radius 3 and codimension 3t+1

The smallest possible length of a q-ary linear code of covering radius R and codimension (redundancy) r is called the length function and is denoted by `q(r, R). In this work, for q an arbitrary prime power, we obtain the following new constructive upper bounds on l_q(3t + 1, 3): (Formula presented) For t = 1, we use a one-to-one correspondence between codes of covering radius 3 and codimension 4, and 2-saturating sets in the projective space PG(3, q). A new construction providing sets of small size is proposed. The codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called “qm-concatenating constructions”) to obtain infinite families of codes with radius 3 and growing codimension r = 3t+ 1, t ≥ 1. The new bounds are essentially better than the known ones.