Alexander A. Davydov

Linear codes with covering radius 2 and other new covering codes

Infinite families of linear binary codes with covering radius R=2 and minimum distance d=3 and d=4 are given. Using the constructed codes with d=3, R=2, families of covering codes with R>2 are obtained. The parameters of many constructed codes with R >

On sizes of complete arcs in PG(2,q)

New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853 = 23. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms. Also new constructions of complete arcs are proposed. These constructions form families of k-arcs in PG(2,q) containing arcs of all sizes k in a region k_{min} 1367. New sizes of complete arcs in PG(2,q) are presented for 169 <= q <= 349 and q=1013,2003.

On saturating sets in projective spaces

Minimal saturating sets in projective spaces PG(n, q) are considered. Estimates and exact values of some extremal parameters are given. In particular the greatest cardinality of a minimal 1-saturating set has been determined. A concept of saturating density is introduced. It allows to obtain new lower bounds for the smallest minimal saturating sets. A number of exhaustive results for small q are obtained. Many new small 1-saturating sets in PG(2, q), q≤ 587, are constructed by computer.

On Saturating Sets in Small Projective Geometries

A set of points, S?PG(r, q), is said to be ? -saturating if, for any point x?PG(r, q), there exist ?+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q,? ). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction presented proves that k(5, q, 2) ? 3 q+ 1 forq= 2, q? 4. We further give an upper bound onk (?+ 1, pm, ?).

Recursive constructions of complete caps

We present three constructions of complete caps in PG(d; q), q odd, where complete caps in a projective space of smaller dimension are involved. We thereby obtain new series of upper bounds on n2(d; q), the smallest number of points in a complete cap in PG(d; q). The constructions show that for k ? 0, n2(k +1 ; 3) 6 2n2(k; 3); n2(4k +2 ;q ) 6 q 2k+1 + n2(2k; q) for q ? 5 an odd prime power; and n2(4k +2 ;q ) 6 q 2k+1 − (q +1 ) +n2(2k; q )+ n2(2 ;q ) for q ? 9 an odd prime power. c � 2001 Elsevier Science B.V. All rights reserved. MSC: 51E22

Constructions, families, and tables of binary linear covering codes

Presents constructions and infinite families of binary linear covering codes with covering radii R=2,3,4. Using these codes, the authors obtain a table of constructive upper bounds on the length function l(r,R) for r/spl les/64 and R=2,3,4, where l(r, R) is the smallest length of a binary linear code with given codimension r and covering radius R. They obtain also upper bounds on l(r, R) for r=21, 28, R=5. Parameters of the constructed codes are better than parameters of previously known codes. >

A new algorithm and a new type of estimate for the smallest size of complete arcs in PG(2,q)

Abstract In this work we summarize some recent results to be included in a forthcoming paper [Bartoli, D., A. A. Davydov, S. Marcugini and F. Pambianco, New types of estimate for the smallest size of complete arcs in a finite Desarguesian projective plane, preprint]. We propose a new type of upper bound for the smallest size t 2 ( 2 , q ) of a complete arc in the projective plane P G ( 2 , q ) . We put t 2 ( 2 , q ) = d ( q ) q ln q , where d ( q ) 1 is a decreasing function of q. The case d ( q ) α / ln β q + γ , where α , β , γ are positive constants independent of q, is considered. It is shown that t 2 ( 2 , q ) ( 2 / ln 1 10 q + 0.32 ) q ln q if q ⩽ 54881 , q prime, or q ∈ R , where R is a set of 34 values in the region 55001 … 110017 . Moreover, our results allow us to conjecture that this estimate holds for all q. An algorithm FOP using any fixed order of points in P G ( 2 , q ) is proposed for constructing complete arcs. The algorithm is based on an intuitive postulate that P G ( 2 , q ) contains a sufficient number of relatively small complete arcs. It is shown that the type of order on the points of P G ( 2 , q ) is not relevant.

New Linear Codes with Covering Radius 2 and Odd Basis

On the way of generalizing recent results by Cock and the second author, it is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R=2. These constructions (together with a lengthening construction by the first author) give new infinite families of linear covering codes with codimension r=2k+1 (the case q=3, r=4k+1 was considered earlier). New code families with r=4k are also obtained. An updated table of upper bounds on the length function for linear codes with ≤ 24, R=2, and q=3,5 is given.

A note on multiple coverings of the farthest-off points

Abstract n this work we summarize some recent results, to be included in a forthcoming paper [D. Bartoli, A. A. Davydov, M. Giulietti, S. Marcugini, and F. Pambianco, Multiple coverigns of the farthest-off points with small density from projective geometry, preprint]. We define μ-density as a characteristic of quality for the kind of coverings codes called multiple coverings of the farthest-off points (MCF). A concept of multiple saturating sets ( ( ρ , μ ) -saturating sets) in projective spaces P G ( N , q ) is introduced. A fundamental relationship of these sets with MCF is showed. Bounds for the smallest possible cardinality of ( 1 , μ ) -saturating sets are obtained. Constructions of small ( 1 , μ ) -saturating sets improving the probabilistic bound are proposed.

The minimum order of complete caps in

It has been verified that in