Linear nonbinary covering codes and saturating sets in projective spaces

Details Linear nonbinary covering codes and saturating sets in projective spaces Linear nonbinary covering codes and saturating sets in projective spaces

2009-04-24

arxiv.org/abs/0904.3835v1

Mathematics

Combinatorics

Scientific paper

Let A_{R,q} denote a family of covering codes, in which the covering radius R and the size q of the underlying Galois field are fixed, while the code length tends to infinity. In this paper, infinite sets of families A_{R,q}, where R is fixed but q ranges over an infinite set of prime powers are considered, and the dependence on q of the asymptotic covering densities of A_{R,q} is investigated. It turns out that for the upper limit of the covering density of A_{R,q}, the best possibility is O(q). The main achievement of the present paper is the construction of asymptotic optimal infinite sets of families A_{R,q} for any covering radius R >= 2. We first showed that for a given R, to obtain optimal infinite sets of families it is enough to construct R infinite families A_{R,q}^{(0)},A_{R,q}^{(1)},...,A_{R,q}^{(R-1)} such that, for all u >= u_{0}, the family A_{R,q}^{(v)} contains codes of codimension r_{u}=Ru+v and length f_{q}^{v}(r_{u}) where f_{q}^{v}(r)=O(q^{(r-R)/R) and u_{0} is a constant. Then, we were able to construct the needed families A_{R,q}^{(v)} for any covering radius R >= 2, with q ranging over the (infinite) set of R-th powers. For each of these families A_{R,q}^{(v)}, the lower limit of the covering density is bounded from above by a constant independent of q.

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