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\((2^{m+1}-1,2^{2n-m},3)\) Vasilyev code[1]

Description

Member of an infinite \((2^{m+1}-1,2^{2n-m},3)\) family of perfect nonlinear codes for any \(m \geq 3\). Constructed by applying a modification of the \((u|u+v)\) construction to a perfect \((2^m-1,2^{n-m},3)\) code, not necessarily linear [2; pg. 77].

The automorphism group of these codes is always nontrivial [3].

Member of code lists

Primary Hierarchy

Classical Domain
Binary Kingdom
Binary codeECC
Nearly perfect codeECC
Perfect binary codePerfect OA ECC \(t\)-design
Parents
Perfect binary code
Vasilyev codes are perfect nonlinear binary codes and are inequivalent to any linear code.
\((u|u+v)\)-construction codeECC
Small-distance block codeECC
\((2^{m+1}-1,2^{2n-m},3)\) Vasilyev code

References

[1]
J. L. Vasilyev, “On nongroup close-packed codes”, Problemy Kibernetiki 8, 337-339 (1962); translated in Probleme der Kibernetik 8, 375-378 (1965)
[2]
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
[3]
S. V. Avgustinovich, F. I. Solov’eva, and O. Heden, “On the Structure of Symmetry Groups of Vasil’ev Codes”, Problems of Information Transmission 41, 105 (2005) DOI
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Zoo Code ID: vasilyev

Cite as:
“\((2^{m+1}-1,2^{2n-m},3)\) Vasilyev code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/vasilyev, arXiv:2606.11484
BibTeX:
@incollection{eczoo_vasilyev,
title={\((2^{m+1}-1,2^{2n-m},3)\) Vasilyev code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/vasilyev}
}
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Permanent link:
https://errorcorrectionzoo.org/c/vasilyev

Cite as:

“\((2^{m+1}-1,2^{2n-m},3)\) Vasilyev code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/vasilyev, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/nonlinear/vasilyev.yml.