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On the number of error correcting codes

Part of: Graph theory

Published online by Cambridge University Press:  09 June 2023

Dingding Dong ,
Nitya Mani Open the ORCID record for Nitya Mani [Opens in a new window]  and
Yufei Zhao
Dingding Dong
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
Nitya Mani*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Yufei Zhao
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
Corresponding author: Nitya Mani; Email: ddong@math.harvard.edu
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Abstract

We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$, where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (\log n)^{-2/3})$.

Keywords

error correcting codesgraph theorycontainers

MSC classification

Primary: 05C30: Enumeration in graph theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 5 , September 2023 , pp. 819 - 832
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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Footnotes

Mani was supported by the NSF Graduate Research Fellowship Program and a Hertz Graduate Fellowship.

Zhao was supported by NSF CAREER award DMS-2044606, a Sloan Research Fellowship, and the MIT Solomon Buchsbaum Fund.

References

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