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.
@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2024.53, author = {Doron, Dean and Mosheiff, Jonathan and Wootters, Mary}, title = {{When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {53:1--53:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.53}, URN = {urn:nbn:de:0030-drops-210467}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.53}, annote = {Keywords: Error-correcting codes, Concatenated codes, Derandomization, Gilbert-Varshamov bound} }
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