Further results on multiple coverings of the farthest-off points (original) (raw)
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A note on multiple coverings of the farthest-off points
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In this work we summarize some recent results, to be included in a forthcoming paper [1]. 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.
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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.
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.
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.
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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.
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.
Classification of minimal 1-saturating sets in PG(2,q)PG(2,q)PG(2,q), qleq23q\leq 23qleq23
Minimal 1−saturating sets in the projective plane P G(2, q) are considered. They correspond to covering codes which can be applied to many branches of combinatorics and information theory, as data compression, compression with distortion, broadcasting in interconnection network, write-once memory or steganography (see [3] and [2]). The full classification of all the minimal 1-saturating sets in P G(2, 9) and P G(2, 11) and the classification of minimal 1-saturating sets of smallest size in P G(2, q), 16 ≤ q ≤ 23 are given. These results have been found using a computer-based exhaustive search that exploits projective equivalence properties.
Discrete Mathematics, 1989
Let F be a set of f points in a finite projective geometry PG(t, q) of t dimensions where t 2 2, f 3 1 and q is a prime power. If (a) IF n HI 3 m for any hyperplane H in PG(t, q) and (b) IF fl H( = m for some hyperplane H in PG(t, q), then F is said to be an {f, m; t, q}-minhyper (or an {f, m; t, q}minihyper) where m 30 and IAl denotes the number of points in the set A. The concept of a min . hyper (called a minihyper) has been introduced by Hamada and Tamari [22]. In the special case t = 2, an {f, m; 2, q}-min . hyper F is called an m-blocking set if F contains no l-flat in PG(2, q).