A note on multiple coverings of the farthest-off points (original) (raw)
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Further results on multiple coverings of the farthest-off points
Multiple coverings of the farthest-off points ($(R,\mu)$-MCF codes) and the corresponding (rho,mu)(\rho,\mu)(rho,mu)-saturating sets in projective spaces PG(N,q)PG(N,q)PG(N,q) are considered. We propose and develop some methods which allow us to obtain new small (1,mu)(1,\mu)(1,mu)-saturating sets and short (2,mu)(2,\mu)(2,mu)-MCF codes with mu\mumu-density either equal to 1 (optimal saturating sets and almost perfect MCF-codes) or close to 1 (roughly 1+1/cq1+1/cq1+1/cq, cge1c\ge1cge1). In particular, we provide new algebraic constructions and some bounds. Also, we classify minimal and optimal (1,mu)(1,\mu)(1,mu)-saturating sets in PG(2,q)PG(2,q)PG(2,q), qqq small.
Multiple coverings of the farthest-off points with small density from projective geometry
Advances in Mathematics of Communications, 2015
In this paper we deal with the special class of covering codes consisting of multiple coverings of the farthest-off points (MCF). In order to measure the quality of an MCF code, we use a natural extension of the notion of density for ordinary covering codes, that is the µ-density for MCF codes; a generalization of the length function for linear covering codes is also introduced. Our main results consist in a number of upper bounds on such a length function, obtained through explicit constructions, especially for the case of covering radius R = 2. A key tool is the possibility of computing the µ-length function in terms of Projective Geometry over finite fields. In fact, linear (R, µ)-MCF codes with parameters [n, n − r, d]qR have a geometrical counterpart consisting of special subsets of n points in the projective space P G(n − r − 1, q). We introduce such objects under the name of (ρ, µ)-saturating sets and we provide a number of example and existence results. Finally, Almost Perfect MCF (APMCF) codes, that is codes for which each word at distance R from the code belongs to exactly µ spheres centered in codewords, are considered and their connections with uniformly packed codes, two-weight codes, and subgroups of Singer groups are pointed out.
Linear nonbinary covering codes and saturating sets in projective spaces
Advances in Mathematics of Communications, 2011
Let AR,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. The construction of families with small asymptotic covering densities is a classical problem in the area Covering Codes.
New covering codes of radius R, codimension tR and tR+R/2, and saturating sets in projective spaces
2018
The length function ℓ_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 ℓ_q(r,R) for all R>4, r=tR, t>2, and also for all even R>2, r=tR+R/2, t>1. 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 ρ-saturating ((ρ+1)q+1)-set in the projective space PG(2ρ+1,q) for all ρ>0. Such a set corresponds to an [Rq+1,Rq+1-2R,3]_qR locally optimal^1 code of covering radius R=ρ+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>2. In addition, we obtain new 1-saturating sets in the projective plane PG(2,q^2) and, basing...
IEEE Transactions on Information Theory, 2005
A concept of locally optimal (LO) linear covering codes is introduced in accordance with the concept of minimal saturating sets in projective spaces over finite fields. An LO code is nonshortening in the sense that one cannot remove any column from a parity-check matrix without increasing the code covering radius. Several -concatenating constructions of LO covering codes are described. Taking a starting LO code as a "seed", such constructions produce an infinite family of LO codes with the same covering radius. The infinite families of LO codes are designed using minimal saturating sets as starting codes. New upper bounds on the length function are given. New extremal and classification problems for linear covering codes are formulated and investigated, in particular, the spectrum of possible lengths of LO codes including the greatest possible length. The complete computer classification of the minimal saturating sets in small geometries and of the corresponding LO codes is obtained.
Constructions and families of covering codes and saturated sets of points in projective geometry
IEEE Transactions on Information Theory, 1995
Simplified expression for the expected error span recovery for variable length codes," ht. Abstract-In a recent paper by this author, constructions of linear binary covering codes are considered. In this work, constructions and techniques of the earlier paper are developed and modified for q-ary linear nonbinary covering codes, q 2 3, and new constructions are proposed. The described constructions design an infinite family of codes with covering radius R based on a starting code of the same covering radius. For arbitrary R 2 2, q 1 3, new infinite families of nonbinary covering codes with "good" parameters are obtained with the help of an iterative process when constructed codes are the starting codes for the following steps. The table of upper bounds on the length function for codes with q = 3, R = 2, 3, and codimension up to 24 is given. We propose to use saturated sets of points in projective geometries over finite fields as parity check matrices of starting codes. New saturated sets are obtained.
On upper bounds on the smallest size of a saturating set in a projective plane
2015
In a projective plane Π _q (not necessarily Desarguesian) of order q, a point subset S is saturating (or dense) if any point of Π _q∖ S is collinear with two points in S. Using probabilistic methods, the following upper bound on the smallest size s(2,q) of a saturating set in Π _q is proved: s(2,q)≤ 2√((q+1) (q+1))+2 2√(q q). We also show that for any constant c> 1 a random point set of size k in Π _q with 2c√((q+1)(q+1))+2< k<q^2-1/q+2 q is a saturating set with probability greater than 1-1/(q+1)^2c^2-2. Our probabilistic approach is also applied to multiple saturating sets. A point set S⊂Π_q is (1,μ)-saturating if for every point Q of Π _q∖ S the number of secants of S through Q is at least μ , counted with multiplicity. The multiplicity of a secant ℓ is computed as #(ℓ ∩ S)2. The following upper bound on the smallest size s_μ(2,q) of a (1,μ)-saturating set in Π_q is proved: s_μ(2,q)≤ 2(μ +1)√((q+1) (q+1))+2 2(μ +1)√( q q) for 2≤μ≤√(q). By using inductive constructions, u...
Linear codes with covering radius 2, 3 and saturating sets in projective geometry
Ieee Transactions on Information Theory, 2004
Infinite families of linear codes with covering radius = 2, 3 and codimension + 1 are constructed on the base of starting codes with codimension 3 and 4. Parity-check matrices of the starting codes are treated as saturating sets in projective geometry that are obtained by computer search using projective properties of objects. Upper bounds on the length function and on the smallest sizes of saturating sets are given.
On saturating sets in projective spaces
Journal of Combinatorial Theory, Series A, 2003
Minimal saturating sets in projective spaces PGðn; qÞ 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 PGð2; qÞ; qp587; are constructed by computer. r