2018/08/28 by Davydov, Alexander A., Marcugini, Stefano, Pambianco, Fernanda
#51E21 #51E22 #94B05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1808.09301
The length function ℓq(r,R) is the smallest length of a q -ary linear code of codimension r and covering radius R. In this work we obtain new constructive upper bounds on ℓq(r,R) for all R≥4, r=tR, t≥2, and also for all even R≥2, r=tR+(R)/(2), t≥1. The new bounds are provided by infinite families of new covering codes with fixed R and increasing codimension. The new bounds improve upon the known ones. We propose a general regular construction (called ``Line+Ovals'') of a minimal ρ-saturating ((ρ+1)q+1)-set in the projective space PG(2ρ+1,q) for all ρ≥0. Such a set corresponds to an [Rq+1,Rq+1-2R,3]qR locally optimal1 code of covering radius R=ρ+1. Basing on combinatorial properties of these codes regarding to spherical capsules1, we give constructions for code codimension lifting and obtain infinite families of new surface-covering1 codes with codimension r=tR, t≥2. In addition, we obtain new 1-saturating sets in the projective plane PG(2,q2) and, basing on them, construct infinite code families with fixed even radius R≥2 and codimension r=tR+(R)/(2), t≥1. (1 see the definitions in Section 1)