### Abstract

The capacity of a graph is defined as the rate of exponential growth of independent sets in the strong powers of the graph. In the strong power an edge connects two sequences if at each position their letters are equal or adjacent. We consider a variation of the problem where edges in the power graphs are removed between sequences which differ in more than a fraction δ of coordinates. The proposed generalization can be interpreted as the problem of determining the highest rate of zero undetected-error communication over a link with adversarial noise, where only a fraction δ of symbols can be perturbed and only some substitutions are allowed. We derive lower bounds on achievable rates by combining graph homomorphisms with a graph-theoretic generalization of the Gilbert-Varshamov bound. We then give an upper bound, based on Delsarte's linear programming approach, which combines Lovász' theta function with the construction used by McEliece et al. for bounding the minimum distance of codes in Hamming spaces.

Original language | English (US) |
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Title of host publication | Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 1331-1335 |

Number of pages | 5 |

ISBN (Electronic) | 9781509018062 |

DOIs | |

State | Published - Aug 10 2016 |

Event | 2016 IEEE International Symposium on Information Theory, ISIT 2016 - Barcelona, Spain Duration: Jul 10 2016 → Jul 15 2016 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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Volume | 2016-August |

ISSN (Print) | 2157-8095 |

### Other

Other | 2016 IEEE International Symposium on Information Theory, ISIT 2016 |
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Country | Spain |

City | Barcelona |

Period | 7/10/16 → 7/15/16 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Information Systems
- Modeling and Simulation
- Applied Mathematics

### Cite this

*Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory*(pp. 1331-1335). [7541515] (IEEE International Symposium on Information Theory - Proceedings; Vol. 2016-August). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISIT.2016.7541515

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*Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory.*, 7541515, IEEE International Symposium on Information Theory - Proceedings, vol. 2016-August, Institute of Electrical and Electronics Engineers Inc., pp. 1331-1335, 2016 IEEE International Symposium on Information Theory, ISIT 2016, Barcelona, Spain, 7/10/16. https://doi.org/10.1109/ISIT.2016.7541515

**Rate-distance tradeoff for codes above graph capacity.** / Cullina, Daniel; Dalai, Marco; Polyanskiy, Yury.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Rate-distance tradeoff for codes above graph capacity

AU - Cullina, Daniel

AU - Dalai, Marco

AU - Polyanskiy, Yury

PY - 2016/8/10

Y1 - 2016/8/10

N2 - The capacity of a graph is defined as the rate of exponential growth of independent sets in the strong powers of the graph. In the strong power an edge connects two sequences if at each position their letters are equal or adjacent. We consider a variation of the problem where edges in the power graphs are removed between sequences which differ in more than a fraction δ of coordinates. The proposed generalization can be interpreted as the problem of determining the highest rate of zero undetected-error communication over a link with adversarial noise, where only a fraction δ of symbols can be perturbed and only some substitutions are allowed. We derive lower bounds on achievable rates by combining graph homomorphisms with a graph-theoretic generalization of the Gilbert-Varshamov bound. We then give an upper bound, based on Delsarte's linear programming approach, which combines Lovász' theta function with the construction used by McEliece et al. for bounding the minimum distance of codes in Hamming spaces.

AB - The capacity of a graph is defined as the rate of exponential growth of independent sets in the strong powers of the graph. In the strong power an edge connects two sequences if at each position their letters are equal or adjacent. We consider a variation of the problem where edges in the power graphs are removed between sequences which differ in more than a fraction δ of coordinates. The proposed generalization can be interpreted as the problem of determining the highest rate of zero undetected-error communication over a link with adversarial noise, where only a fraction δ of symbols can be perturbed and only some substitutions are allowed. We derive lower bounds on achievable rates by combining graph homomorphisms with a graph-theoretic generalization of the Gilbert-Varshamov bound. We then give an upper bound, based on Delsarte's linear programming approach, which combines Lovász' theta function with the construction used by McEliece et al. for bounding the minimum distance of codes in Hamming spaces.

UR - http://www.scopus.com/inward/record.url?scp=84985898319&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84985898319&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2016.7541515

DO - 10.1109/ISIT.2016.7541515

M3 - Conference contribution

AN - SCOPUS:84985898319

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 1331

EP - 1335

BT - Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory

PB - Institute of Electrical and Electronics Engineers Inc.

ER -