Constructions and families of nonbinary linear codes with covering radius 2 (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 codes with covering radius 3

Designs, Codes and Cryptography, 2010

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by q (r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on q (r, 3). General constructions are given and upper bounds on q (r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.

Further results on the covering radius of codes

IEEE Transactions on Information Theory, 1986

A number of upper and lower bounds are obtained for K( n, R), the minimal number of codewords in any binary code of length n and covering radius R. Several new constructions are used to derive the upper bounds, including an amalgamated direct sum construction for nonlinear codes. This construction works best when applied to normal codes, and we give some new and stronger conditions which imply that a linear code is normal. An upper bound is given for the density of a covering code over any alphabet, and it is shown that K(n + 2,R + 1) 5 K(n, R) holds for sufficiently large n.

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.

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.

New Quaternary Linear Codes with Covering Radius 2

Finite Fields and Their Applications, 2000

A new quaternary linear code of length 19, codimension 5, and covering radius 2 is found in a computer search using tabu search, a local search heuristic. Starting from this code, which has some useful partitioning properties, di!erent lengthening constructions are applied to get an in"nite family of new, record-breaking quaternary codes of covering radius 2 and odd codimension. An algebraic construction of covering codes over alphabets of even characteristic is also given.

New Linear Codes with Covering Radius 2 and Odd Basis

1999

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 r ≤ 24, R = 2, and q = 3, 5 is given.

New Bounds for Linear Codes of Covering Radius 2

2017

The length function \(\ell _q(r,R)\) is the smallest length of a q-ary linear code of covering radius R and codimension r. New upper bounds on \(\ell _q(r,2)\) are obtained for odd \(r\ge 3\). In particular, using the one-to-one correspondence between linear codes of covering radius 2 and saturating sets in the projective planes over finite fields, we prove that \begin{aligned} \ell _q(3,2)\le \sqrt{q(3\ln q+\ln \ln q)}+\sqrt{\frac{q}{3\ln q}}+3 \end{aligned}$$ and then obtain estimations of \(\ell _q(r,2)\) for all odd \(r\ge 5\). The new upper bounds are smaller than the previously known ones. Also, the new bounds hold for all q, not necessary large, whereas the previously best known estimations are proved only for q large enough.