New bounds for linear codes of covering radius 3 and 2-saturating sets in projective spaces
Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco · 2019
The length function lq(r, R) is the smallest length of a q-ary linear code of covering radius R and codimension (redundancy) r. In this paper, we obtained new upper bounds on lq(r, 3), r = 3t+1 ≥ 4,andr = 3t+2 ≥ 5, t ≥ 1.For r = 4, 5 we use the one-to-one correspondence between [n, n-r]qR codes and (R-1)-saturating sets (e.g. complete arcs) in the projective space PG(r-1, q). Then, with the help of lift-constructions increasing r, we obtain new upper bounds on lq(3t + 1, 3), lq(3t + 2, 3). In particular, we show that p lq(r, 3)3√ ln q · q(r-3)/3, r = 3t + 1 ≥ 4, t ≥ 1, q ≥ 6553; p lq(r, 3)3√ ln q · q(r-3)/3, r = 3t + 2 ≥ 5, t ≥ 1, q ≤ 839. Also, in PG(3, q) we consider an iterative step-by-step construction of complete arcs and prove that uncovered points are evenly placed on the space. A natural conjecture on an estimate of the number of new covered points in every step is done. Under this conjecture, the following bounds for values of q, not limited from above, are proposed: lq(r, 3)3√ ln q · q(r-3)/3, r = 3t + 1 ≥ 4, t ≥ 1.