Upper bounds on the length function for covering codes
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The length function ℓ_q(r,R) is the smallest length of a q-ary linear code with codimension (redundancy) r and covering radius R. In this work, new upper bounds on ℓ_q(tR+1,R) are obtained in the following forms: (a) ℓ_q(r,R)≤ cq^(r-R)/R·√(ln q), R≥3, r=tR+1, t≥1, (a) q is an arbitrary prime power, c is independent of q. (b) ℓ_q(r,R)< 4.5Rq^(r-R)/R·√(ln q), R≥3, r=tR+1, t≥1, (b) q is an arbitrary prime power, q is large enough. In the literature, for q=(q')^R with q' a prime power, smaller upper bounds are known; however, when q is an arbitrary prime power, the bounds of this paper are better than the known ones. For t=1, we use a one-to-one correspondence between [n,n-(R+1)]_qR codes and (R-1)-saturating n-sets in the projective space PG(R,q). A new construction of such saturating sets providing sets of small size is proposed. Then the [n,n-(R+1)]_qR codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "q^m-concatenating constructions") for covering codes to obtain infinite families of codes with growing codimension r=tR+1, t≥1.
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