Berlekamp code

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 \(GF(p)\) and then taking the subfield subcode of the corresponding code; see [3; Ch. 10.6].

Cousins

Alternant code— Berlekamp codes reduce to narrow-sense alternant codes for \(p=2\) [3; Ch. 10.6].
Reed-Solomon (RS) code— Berlekamp codes are obtained by first constructing an RS-like parity-check matrix out of a certain field extension of \(GF(p)\) and then taking the subfield subcode of the corresponding code; see [3; Ch. 10.6].

Member of code lists

Primary Hierarchy

Classical Domain

Ring Kingdom

Ring codeECC

\(q\)-ary code over \(\mathbb{Z}_q\)

\(q\)-ary additive code over \(\mathbb{Z}_q\)

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]: Berlekamp, Elwyn R. "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

Page edit log

Victor V. Albert (2024-08-09) — most recent

Cite as:

“Berlekamp code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/berlekamp

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