arXiv subjects

Alexander A. Davydov

Publications and source records attributed to Alexander A. Davydov.

39 records · Page 3Linked to original sources

arXiv2009

Some combinatorial aspects of constructing bipartite-graph codes

We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its vertices are associated with local constraints. We significantly extend the region of parameters m,n for which there exist an n-regular bipartite graph with 2m vertices and without 4-cycles. In that way we essentially increase the region of lengths and rates of the corresponding bipartite-graph codes. Many new matrices are either circulant or consist of circulant submatrices: this provides code parity-check matrices consisting of circulant submatrices, and hence quasi-cyclic bipartite-graph codes with simple implementation.

math.CO↗

arXiv2009

Linear nonbinary covering codes and saturating sets in projective spaces

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.

math.CO↗

arXiv2009

New inductive constructions of complete caps in $PG(N,q)$, $q$ even

Some new families of small complete caps in $PG(N,q)$, $q$ even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this paper provide an improvement on the currently known upper bounds on the size of the smallest complete cap in $PG(N,q),$ $N\geq 4,$ for all $q\geq 2^{3}.$ In particular, substantial improvements are obtained for infinite values of $q$ square, including $ q=2^{2Cm},$ $C\geq 5,$ $m\geq 3;$ for $q=2^{Cm},$ $C\geq 5,$ $m\geq 9,$ with $C,m$ odd; and for all $q\leq 2^{18}.$

math.CO↗