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

Abstract

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 ℓq(r, R). In this work, for q an arbitrary prime power, we obtain the following new constructive upper bounds on ℓq(3t + 1, 3):• ℓq(r, 3) ⪅ 3 √ 18 • q (r−3)/3 • 3 ln q, r = 3t + 1, t ≥ 1, q large enough.