Tables, bounds and graphics of short linear codes with covering radius 3 and codimension 4 and 5

The length function `q(r, R) is the smallest length of a q-ary linear code of covering radius R and codimension r. In this work, by computer search in wide regions of q, we obtained short [n, n− 4, 5]q3 quasiperfect MDS codes and [n, n− 5, 5]q3 quasiperfect Almost MDS codes with covering radius R = 3. The new codes imply the following upper bounds: `q(4, 3) < 2.8 3 √ q ln q for 8 ≤ q ≤ 3323 and q = 3511, 3761, 4001; `q(5, 3) < 3 3 √ q2 ln q for 5 ≤ q ≤ 563. For r 6= 3t and q 6= (q′)3, the new bounds have the form `q(r, 3) < c 3 √ ln q · q(r−3)/3, c is a universal constant, r = 4, 5. ∗The research of D. Bartoli, S. Marcugini, and F. Pambianco was supported in part by Ministry for Education, University and Research of Italy (MIUR) (Project “Geometrie di Galois e strutture di incidenza”) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA INDAM). The research of A.A. Davydov was carried out at the IITP RAS at the expense of the Russian Foundation for Sciences (project 14-50-00150). This work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC Kurchatov Institute, http://ckp.nrcki.ru/. 1 ar X iv :1 71 2. 07 07 8v 1 [ cs .I T ] 1 9 D ec 2 01 7 As far as it is known to the authors, such bounds have not been previously described in the literature. In computer search, we use the leximatrix algorithm to obtain parity check matrices of codes. The algorithm is a version of the recursive g-parity check algorithm for greedy codes.