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Error-Correcting Graph Codes

Authors Swastik Kopparty , Aditya Potukuchi , Harry Sha

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Author Details

Swastik Kopparty
  • Department of Mathematics and Department of Computer Science, University of Toronto, Canada
Aditya Potukuchi
  • Department of Electrical Engineering and Computer Science, York University, Toronto, Canada
Harry Sha
  • Department of Mathematics and Department of Computer Science, University of Toronto, Canada

Funding

  • Kopparty, Swastik: Research supported by an NSERC Discovery Grant.
  • Potukuchi, Aditya: Research supported by an NSERC Discovery Grant.

Acknowledgements

We are grateful to Mike Saks, Shubhangi Saraf and Pat Devlin for valuable discussions.

Cite As Get BibTex

Swastik Kopparty, Aditya Potukuchi, and Harry Sha. Error-Correcting Graph Codes. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 67:1-67:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.67

Abstract

In this paper, we construct Error-Correcting Graph Codes. An error-correcting graph code of distance δ is a family C of graphs, on a common vertex set of size n, such that if we start with any graph in C, we would have to modify the neighborhoods of at least δ n vertices in order to obtain some other graph in C. This is a natural graph generalization of the standard Hamming distance error-correcting codes for binary strings. 
Yohananov and Yaakobi were the first to construct codes in this metric. We extend their work by showing  
1) Combinatorial results determining the optimal rate vs distance trade-off nonconstructively. 
2) Graph code analogues of Reed-Solomon codes and code concatenation, leading to positive distance codes for all rates and positive rate codes for all distances. 
3) Graph code analogues of dual-BCH codes, yielding large codes with distance δ = 1-o(1). This gives an explicit "graph code of Ramsey graphs". 
Several recent works, starting with the paper of Alon, Gujgiczer, Körner, Milojević, and Simonyi, have studied more general graph codes; where the symmetric difference between any two graphs in the code is required to have some desired property. Error-correcting graph codes are a particularly interesting instantiation of this concept.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Coding theory
Keywords
  • Graph codes
  • explicit construction
  • concatenation codes
  • tensor codes

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References

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