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Berlekamp code[1][2; Ch. 9]

Description

A linear \(p\)-ary code (for prime \(p\)) that has Lee distance 5 and whose construction resembles that of RS codes. It is obtained by first constructing an RS-like parity-check matrix out of a certain field extension of \(\mathbb{F}_p\) and then taking the subfield subcode of the corresponding code; see [3; Ch. 10.6].

Cousins

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
\(q\)-ary code over \(\mathbb{Z}_q\)ECC
Linear code over \(\mathbb{Z}_q\)ECC
Parents
Linear code over \(\mathbb{Z}_q\)
Linear \(q\)-ary codeECC
Constacyclic codeECC
Berlekamp codes are negacyclic [2; Ch. 9].
Berlekamp code

References

[1]
E. R. Berlekamp, “Negacyclic codes for the Lee metric”, Proceedings of the Conference on Combinatorial Mathematics and its Applications. Chapel Hill: University of North Carolina Press, 1968
[2]
E. R. Berlekamp, Algebraic Coding Theory (WORLD SCIENTIFIC, 2014) DOI
[3]
R. Roth, Introduction to Coding Theory (Cambridge University Press, 2006) DOI
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Zoo Code ID: berlekamp

Cite as:
“Berlekamp code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/berlekamp, arXiv:2606.11484
BibTeX:
@incollection{eczoo_berlekamp,
title={Berlekamp 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/berlekamp}
}
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Permanent link:
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Cite as:

“Berlekamp code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/berlekamp, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/over_zq/berlekamp.yml.