Exact bounds on the sizes of covering codes

Maria Axenovich, Z. Füredi

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

A code script C sign is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from script C sign. The minimum size of such a script C sign is denoted by c r(X). Answering a question of Hämäläinen et al., we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n0(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}n. We prove that cr(Bn(0, r + 2)) = Σ1≤i≤r+1 (2 ⌊(n+i-1)/(r+1)⌋) + ⌊n/(r + 1)⌋ and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.

Original languageEnglish
Pages (from-to)21-38
Number of pages18
JournalDesigns, Codes, and Cryptography
Volume30
Issue number1
DOIs
Publication statusPublished - Aug 2003

Fingerprint

Covering Codes
Hamming distance
Ball
Radius
Constant Weight Codes
Hamming Distance
Hypergraph
Covering
Partition
Cover

Keywords

  • Binary codes
  • Covering radius
  • Supersaturated hypergraphs
  • Turán theorem

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Applied Mathematics

Cite this

Exact bounds on the sizes of covering codes. / Axenovich, Maria; Füredi, Z.

In: Designs, Codes, and Cryptography, Vol. 30, No. 1, 08.2003, p. 21-38.

Research output: Contribution to journalArticle

Axenovich, Maria ; Füredi, Z. / Exact bounds on the sizes of covering codes. In: Designs, Codes, and Cryptography. 2003 ; Vol. 30, No. 1. pp. 21-38.
@article{f6d3b3d8ee3e4a2ead1747d98220bb9e,
title = "Exact bounds on the sizes of covering codes",
abstract = "A code script C sign is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from script C sign. The minimum size of such a script C sign is denoted by c r(X). Answering a question of H{\"a}m{\"a}l{\"a}inen et al., we show further connections between Tur{\'a}n theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n0(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}n. We prove that cr(Bn(0, r + 2)) = Σ1≤i≤r+1 (2 ⌊(n+i-1)/(r+1)⌋) + ⌊n/(r + 1)⌋ and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.",
keywords = "Binary codes, Covering radius, Supersaturated hypergraphs, Tur{\'a}n theorem",
author = "Maria Axenovich and Z. F{\"u}redi",
year = "2003",
month = "8",
doi = "10.1023/A:1024703225079",
language = "English",
volume = "30",
pages = "21--38",
journal = "Designs, Codes, and Cryptography",
issn = "0925-1022",
publisher = "Springer Netherlands",
number = "1",

}

TY - JOUR

T1 - Exact bounds on the sizes of covering codes

AU - Axenovich, Maria

AU - Füredi, Z.

PY - 2003/8

Y1 - 2003/8

N2 - A code script C sign is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from script C sign. The minimum size of such a script C sign is denoted by c r(X). Answering a question of Hämäläinen et al., we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n0(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}n. We prove that cr(Bn(0, r + 2)) = Σ1≤i≤r+1 (2 ⌊(n+i-1)/(r+1)⌋) + ⌊n/(r + 1)⌋ and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.

AB - A code script C sign is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from script C sign. The minimum size of such a script C sign is denoted by c r(X). Answering a question of Hämäläinen et al., we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n0(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}n. We prove that cr(Bn(0, r + 2)) = Σ1≤i≤r+1 (2 ⌊(n+i-1)/(r+1)⌋) + ⌊n/(r + 1)⌋ and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.

KW - Binary codes

KW - Covering radius

KW - Supersaturated hypergraphs

KW - Turán theorem

UR - http://www.scopus.com/inward/record.url?scp=0041928148&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041928148&partnerID=8YFLogxK

U2 - 10.1023/A:1024703225079

DO - 10.1023/A:1024703225079

M3 - Article

VL - 30

SP - 21

EP - 38

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

IS - 1

ER -