Locally Optimal (Nonshortening) Linear Covering Codes and Minimal Saturating Sets in Projective Spaces (original) (raw)

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.

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.

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...

New Constructions of Covering Codes

Designs, Codes and Cryptography - DCC, 2001

Coveringcode constructions obtaining new codes from starting ones weredeveloped during last years. In this work we propose new constructionsof such kind. New linear and nonlinear covering codes and aninfinite families of those are obtained with the help of constructionsproposed. A table of new upper bounds on the length functionis given.

Linear codes with covering radius 2 and other new covering codes

IEEE Transactions on Information Theory, 1991

A recent paper shows that the matched-filter/tappeddelay-line structure is optimum not only for linear pulse-modulated signals and linear channel distortion, hut also for nonlinear finitealphabet pulse-modulation and some nonlinear channel distortion. This has important practical applications. Therefore, its connection with other work reported in the literature is brought to light in this note.

Constructions, families, and tables of binary linear covering codes

IEEE Transactions on Information Theory, 1994

&s&z-We present constructions and infinite families of binary linear covering codes with covering radii R = 2,3,4. Using these codes, we obtain a table of constructive ipper bounds on the length function lb, R) for r I 64 and R = 2.3.4. where l(r. R) is the smallest length of a binary linear code with given codimensioh r and covering radi& R. We 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. Zndex Terms-Covering radius, covering codes, binary linear codes. I. INTRODUCTION Covering codes are being extensively studied, see, e.g., [l]-[17]. We, consider binary linear covering codes.

Constructions and families of nonbinary linear codes with covering radius 2

IEEE Transactions on Information Theory, 1999

New constructions of linear nonbinary codes with covering radius R = 2 are proposed. They are in part modifications of earlier constructions by the author and in part are new. Using a starting code with R = 2 as a "seed" these constructions yield an infinite family of codes with the same covering radius. New infinite families of codes with R = 2 are obtained for all alphabets of size q 4 and all codimensions r 3 with the help of the constructions described. The parameters obtained are better than those of known codes. New estimates for some partition parameters in earlier known constructions are used to design new code families. Complete caps and other saturated sets of points in projective geometry are applied as starting codes. A table of new upper bounds on the length function for q = 4; 5; 7; R = 2; and r 24 is included.

Linear programming bounds for codes in infinite projective spaces

Journal of Geometry, 1997

We develop a technique for improving the universal linear programming bounds on the cardinality and the minimum distance of codes in projective spaces I FP n−1 . We firstly investigate test functions P j (m, n, s) having the property that P j (m, n, s) < 0 for some j if and only if the corresponding universal linear programming bound can be further improved by linear programming. Then we describe a method for improving the universal bounds. We also investigate the possibilities for attaining the first universal bounds.