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

TL;DR

This paper introduces new upper bounds for the length function of q-ary linear codes with specific covering radii and codimensions, using novel constructions of covering codes and saturating sets in projective spaces.

Contribution

It provides new constructive upper bounds for the length function for all R≥4 and specific codimension forms, along with a general regular construction of saturating sets in projective spaces.

Findings

New bounds improve upon existing ones for covering codes.

Infinite families of surface-covering codes are constructed.

New 1-saturating sets in projective planes are identified.

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 $R\ge4$, $r=tR$, $t\ge2$, and also for all even $R\ge2$, $r=tR+\frac{R}{2}$, $t\ge1$. 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 $\rho\ge0$. Such a set corresponds to an $[Rq+1,Rq+1-2R,3]_qR$ locally optimal$^1$ code of covering radius $R=\rho+1$. Basing on combinatorial properties of these codes regarding to spherical capsules$^1$, we give constructions for code codimension lifting and obtain infinite families of new surface-covering$^1$ codes with codimension $r=tR$, $t\ge2$. 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 $R\ge2$ and codimension $r=tR+\frac{R}{2}$, $t\ge1$. ($^1$ see the definitions in Section 1)