### 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 > n_{0}(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 c_{r}(B_{n}(0, r + 2)) = Σ_{1≤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 language | English |
---|---|

Pages (from-to) | 21-38 |

Number of pages | 18 |

Journal | Designs, Codes, and Cryptography |

Volume | 30 |

Issue number | 1 |

DOIs | |

Publication status | Published - Aug 2003 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Designs, Codes, and Cryptography*,

*30*(1), 21-38. https://doi.org/10.1023/A:1024703225079

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

Research output: Contribution to journal › Article

*Designs, Codes, and Cryptography*, vol. 30, no. 1, pp. 21-38. https://doi.org/10.1023/A:1024703225079

}

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 -