New covering codes of radius R, codimension tR and $$tR+\frac{R}{2}$$, and saturating sets in projective spaces

Abstract

The length function $$\ell _q(r,R)$$ 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 $$\ell _q(r,R)$$ for all $$R\ge 4$$ , $$r=tR$$ , $$t\ge 2$$ , and also for all even $$R\ge 2$$ , $$r=tR+\frac{R}{2}$$ , $$t\ge 1$$ . 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 $$\rho $$ -saturating $$((\rho +1)q+1)$$ -set in the projective space $$\mathrm {PG}(2\rho +1,q)$$ for all $$\rho \ge 0$$ . Such a set corresponds to an $$[Rq+1,Rq+1-2R,3]_qR$$ locally optimal code of covering radius $$R=\rho +1$$ . Basing on combinatorial properties of these codes regarding to spherical capsules, we give constructions for code codimension lifting and obtain infinite families of new surface-covering codes with codimension $$r=tR$$ , $$t\ge 2$$ . In addition, we obtain new 1-saturating sets in the projective plane $$\mathrm {PG}(2,q^2)$$ and, basing on them, construct infinite code families with fixed even radius $$R\ge 2$$ and codimension $$r=tR+\frac{R}{2}$$ , $$t\ge 1$$ .