Rate-distance tradeoff for codes above graph capacity

Daniel Cullina, Marco Dalai, Yury Polyanskiy

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

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 languageEnglish (US)
Title of host publicationProceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1331-1335
Number of pages5
ISBN (Electronic)9781509018062
DOIs
StatePublished - Aug 10 2016
Event2016 IEEE International Symposium on Information Theory, ISIT 2016 - Barcelona, Spain
Duration: Jul 10 2016Jul 15 2016

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2016-August
ISSN (Print)2157-8095

Other

Other2016 IEEE International Symposium on Information Theory, ISIT 2016
CountrySpain
CityBarcelona
Period7/10/167/15/16

Fingerprint

Linear programming
Substitution reactions
Trade-offs
Communication
Graph in graph theory
Graph Powers
Graph Homomorphisms
Theta Functions
Exponential Growth
Minimum Distance
Independent Set
Substitution
Adjacent
Lower bound
Upper bound
Zero
Generalization

All Science Journal Classification (ASJC) codes

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

Cite this

Cullina, D., Dalai, M., & Polyanskiy, Y. (2016). Rate-distance tradeoff for codes above graph capacity. In 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
Cullina, Daniel ; Dalai, Marco ; Polyanskiy, Yury. / Rate-distance tradeoff for codes above graph capacity. Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory. Institute of Electrical and Electronics Engineers Inc., 2016. pp. 1331-1335 (IEEE International Symposium on Information Theory - Proceedings).
@inproceedings{f6a3be2447f345c6ad5584e3b7fda899,
title = "Rate-distance tradeoff for codes above graph capacity",
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 Lov{\'a}sz' theta function with the construction used by McEliece et al. for bounding the minimum distance of codes in Hamming spaces.",
author = "Daniel Cullina and Marco Dalai and Yury Polyanskiy",
year = "2016",
month = "8",
day = "10",
doi = "10.1109/ISIT.2016.7541515",
language = "English (US)",
series = "IEEE International Symposium on Information Theory - Proceedings",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
pages = "1331--1335",
booktitle = "Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory",
address = "United States",

}

Cullina, D, Dalai, M & Polyanskiy, Y 2016, Rate-distance tradeoff for codes above graph capacity. in 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.

Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory. Institute of Electrical and Electronics Engineers Inc., 2016. p. 1331-1335 7541515 (IEEE International Symposium on Information Theory - Proceedings; Vol. 2016-August).

Research output: Chapter in Book/Report/Conference proceedingConference 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 -

Cullina D, Dalai M, Polyanskiy Y. Rate-distance tradeoff for codes above graph capacity. In Proceedings - ISIT 2016; 2016 IEEE International Symposium on Information Theory. Institute of Electrical and Electronics Engineers Inc. 2016. p. 1331-1335. 7541515. (IEEE International Symposium on Information Theory - Proceedings). https://doi.org/10.1109/ISIT.2016.7541515