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When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?

Authors Dean Doron , Jonathan Mosheiff , Mary Wootters

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Author Details

Dean Doron
  • Ben-Gurion University of the Negev, Beersheba, Israel
Jonathan Mosheiff
  • Ben-Gurion University of the Negev, Beersheba, Israel
Mary Wootters
  • Stanford University, CA, USA

Funding

  • Doron, Dean: Supported in part by NSF-BSF grant #2022644.
  • Mosheiff, Jonathan: Supported by an Alon Fellowship.
  • Wootters, Mary: Partially supported by NSF grants CCF-2231157 and CCF-2133154.

Acknowledgements

We thank Amnon Ta-Shma for helpful and interesting discussions, and collaboration at the beginning of this work. We thank Arya Mazumdar for pointing out [Barg et al., 2001] and for helping us understand its implications. This work was done partly while the authors were visiting the Simons Institute for the Theory of Computing.

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Dean Doron, Jonathan Mosheiff, and Mary Wootters. When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.53

Abstract

The Gilbert-Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate ε² has relative distance at least 1/2 - O(ε) with high probability. However, it is a major challenge to construct explicit codes with similar parameters.
One hope to derandomize the Gilbert-Varshamov construction is with code concatenation: We begin with a (hopefully explicit) outer code 𝒞_out over a large alphabet, and concatenate that with a small binary random linear code 𝒞_in. It is known that when we use independent small codes for each coordinate, then the result lies on the GV bound with high probability, but this still uses a lot of randomness. In this paper, we consider the question of whether code concatenation with a single random linear inner code 𝒞_in can lie on the GV bound; and if so what conditions on 𝒞_out are sufficient for this.
We show that first, there do exist linear outer codes 𝒞_out that are "good" for concatenation in this sense (in fact, most linear codes codes are good). We also provide two sufficient conditions for 𝒞_out, so that if 𝒞_out satisfies these, 𝒞_out∘𝒞_in will likely lie on the GV bound. We hope that these conditions may inspire future work towards constructing explicit codes 𝒞_out.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Error-correcting codes
  • Concatenated codes
  • Derandomization
  • Gilbert-Varshamov bound

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