### Abstract

What is the maximum possible number, f_{3}(n), of vectors of length n over {0,1,2} such that the Hamming distance between every two is even? What is the maximum possible number, g^{3}(n), of vectors in {0,1,2} ^{n} such that the Hamming distance between every two is odd? We investigate these questions, and more general ones, by studying Xor powers of graphs, focusing on their independence number and clique number, and by introducing two new parameters of a graph G. Both parameters denote limits of series of either clique numbers or independence numbers of the Xor powers of G (normalized appropriately), and while both limits exist, one of the series grows exponentially as the power tends to infinity, while the other grows linearly. As a special case, it follows that f_{3}(n) = Θ(2^{n} ) whereas g_{3}(n)=Θ(n).

Original language | English (US) |
---|---|

Pages (from-to) | 13-33 |

Number of pages | 21 |

Journal | Combinatorica |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2007 |

### Fingerprint

### Keywords

- 05C69
- 94A15

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

*27*(1), 13-33. https://doi.org/10.1007/s00493-007-0042-5

**Codes and Xor graph products.** / Alon, Noga; Lubetzky, Eyal.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 27, no. 1, pp. 13-33. https://doi.org/10.1007/s00493-007-0042-5

}

TY - JOUR

T1 - Codes and Xor graph products

AU - Alon, Noga

AU - Lubetzky, Eyal

PY - 2007/2

Y1 - 2007/2

N2 - What is the maximum possible number, f3(n), of vectors of length n over {0,1,2} such that the Hamming distance between every two is even? What is the maximum possible number, g3(n), of vectors in {0,1,2} n such that the Hamming distance between every two is odd? We investigate these questions, and more general ones, by studying Xor powers of graphs, focusing on their independence number and clique number, and by introducing two new parameters of a graph G. Both parameters denote limits of series of either clique numbers or independence numbers of the Xor powers of G (normalized appropriately), and while both limits exist, one of the series grows exponentially as the power tends to infinity, while the other grows linearly. As a special case, it follows that f3(n) = Θ(2n ) whereas g3(n)=Θ(n).

AB - What is the maximum possible number, f3(n), of vectors of length n over {0,1,2} such that the Hamming distance between every two is even? What is the maximum possible number, g3(n), of vectors in {0,1,2} n such that the Hamming distance between every two is odd? We investigate these questions, and more general ones, by studying Xor powers of graphs, focusing on their independence number and clique number, and by introducing two new parameters of a graph G. Both parameters denote limits of series of either clique numbers or independence numbers of the Xor powers of G (normalized appropriately), and while both limits exist, one of the series grows exponentially as the power tends to infinity, while the other grows linearly. As a special case, it follows that f3(n) = Θ(2n ) whereas g3(n)=Θ(n).

KW - 05C69

KW - 94A15

UR - http://www.scopus.com/inward/record.url?scp=33847416221&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847416221&partnerID=8YFLogxK

U2 - 10.1007/s00493-007-0042-5

DO - 10.1007/s00493-007-0042-5

M3 - Article

AN - SCOPUS:33847416221

VL - 27

SP - 13

EP - 33

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -