New bounds for covering codes of radius 3 and codimension 3t+1
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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 ℓ_q(r,R). In this work, for q an arbitrary prime power, we obtain the following new constructive upper bounds on ℓ_q(3t+1,3): ℓ_q(r,3)⪅√(k)· q^(r-3)/3·√(ln q), r=3t+1, t≥1, q≥⌈𝒲(k)⌉, 18 <k≤20.339, 𝒲(k) is a decreasing function of k ; ℓ_q(r,3)⪅√(18)· q^(r-3)/3·√(ln q), r=3t+1, t≥1, q large enough. For t = 1, we use a one-to-one correspondence between codes of covering radius 3 and codimension 4, and 2-saturating sets in the projective space PG(3,q). A new construction providing sets of small size is proposed. The codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called “q^m-concatenating constructions”) to obtain infinite families of codes with radius 3 and growing codimension r = 3t + 1, t≥1. The new bounds are essentially better than the known ones.
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