The length function l_q(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 l_q(r,3), r = 3t+1 ≥ 4, and r = 3t+2 ≥ 5, t ≥ 1. For r = 4,5 we usetheone-to-one correspondence between [n,n−r]qR codesand (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 l_q(3t + 1,3), l_q(3t + 2,3). 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: l_q(r,3) < 3 (ln q)^(1/3)·q^((r−3)/3), r = 3t + 1 ≥ 4, t ≥ 1.
New bounds for linear codes of covering radius 3 and 2-saturating sets in projective spaces
Stefano Marcugini;Fernanda Pambianco
2019
Abstract
The length function l_q(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 l_q(r,3), r = 3t+1 ≥ 4, and r = 3t+2 ≥ 5, t ≥ 1. For r = 4,5 we usetheone-to-one correspondence between [n,n−r]qR codesand (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 l_q(3t + 1,3), l_q(3t + 2,3). 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: l_q(r,3) < 3 (ln q)^(1/3)·q^((r−3)/3), r = 3t + 1 ≥ 4, t ≥ 1.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
