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. 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 to (q')^3 and r not equal to 3t we provide infinite families of [n, n−r]_q 3 codes showing that L_q (r, 3) ≈ c (ln q)^(1/3) · q^((r−3)/3), where c is a universal constant. As far as it is known to the authors, such families have notbeen previously described in the literature.
New bounds for linear codes of covering radii 2 and 3
Daniele Bartoli;Massimo Giulietti;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 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. 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 to (q')^3 and r not equal to 3t we provide infinite families of [n, n−r]_q 3 codes showing that L_q (r, 3) ≈ c (ln q)^(1/3) · q^((r−3)/3), where c is a universal constant. As far as it is known to the authors, such families have notbeen previously described in the literature.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
