Linear nonbinary covering codes and saturating sets in projective spaces

Abstract

Let $\mathcal A$R,q denote a family of covering codes, inwhich the covering radius $R$ and the size $q$ of theunderlying Galois field are fixed, while the code length tendsto infinity. The construction of families with small asymptoticcovering densities is a classical problem in the area ofCovering Codes. In this paper, infinite sets of families $\mathcal A$R,q,where $R$ is fixed but $q$ ranges over an infinite set of primepowers are considered, and the dependence on $q$ of theasymptotic covering densities of $\mathcal A$R,q isinvestigated. It turns out that for the upper limit $\mu$q*(R,$\mathcal A$R,q) of the covering density of$\mathcal A$R,q, the best possibility is$\mu$q*(R,$\mathcal A$R,q)=$O(q)$. The main achievement of thepresent paper is the construction of optimal infinitesets of families $\mathcal A$R,q, that is, sets of familiessuch that relation $\mu$q*(R,$\mathcal A$R,q)=$O(q)$holds, for any covering radius $R\geq 2$. We first showed that for a given $R$, to obtain optimalinfinite sets of families it is enough to construct $R$infinite families $\mathcal A$R,q(0),$\mathcal A$R,q(1), $\ldots$, $\mathcal A$R,q(R-1) such that, for all $u\geq u$0,the family $\mathcal A$R,q($\gamma$) contains codes ofcodimension $r$u$=Ru + \gamma$ and length $f$q($\gamma$)($r$u)where $f$q($\gamma$)$(r)=O(q$(r-R)/R$)$ and$u$0 is a constant. Then, we were able to construct thenecessary families $\mathcal A$R,q($\gamma$) for anycovering radius $R\geq 2$, with $q$ ranging over the (infinite)set of $R$-th powers. A result of independent interest is thatin each of these families $\mathcal A$R,q($\gamma$), thelower limit of the covering density is bounded from above by aconstant independent of $q$. The key tool in our investigation is the design of new smallsaturating sets in projective spaces over finite fields, whichare used as the starting point for the $q$m-concatenatingconstructions of covering codes. A new concept of $N$-foldstrong blocking set is introduced. As a result of ourinvestigation, many new asymptotic and finite upper bounds onthe length function of covering codes and on the smallest sizesof saturating sets, are also obtained. Updated tables for theseupper bounds are provided. An analysis and a survey of theknown results are presented.