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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
Parents
Binary code
\((u|u+v)\)-construction codeECC
Perfect binary codeNearly perfect Perfect OA \(t\)-design ECC
Vasilyev codes are perfect nonlinear binary codes and are inequivalent to any linear code.
Small-distance block codeECC
Vasilyev code

References

[1]
J. L. Vasilyev On nongroup close-packed codes (in Russian), Probl. Kibernet., 8 (1962), 337-339, translated in Probleme der Kibernetik 8 (1965), 375-378.
[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:
“Vasilyev code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/vasilyev
BibTeX:
@incollection{eczoo_vasilyev, title={Vasilyev code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/vasilyev} }
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Permanent link:
https://errorcorrectionzoo.org/c/vasilyev

Cite as:

“Vasilyev code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/vasilyev

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