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Ring code

Root code for the Ring Kingdom

Description

Encodes \(K\) states (codewords) in \(n\) coordinates over a finite ring \(R\).

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Parents
Block codeECC
Ring codes are block codes with \(\Sigma=R\).
Finite-dimensional error-correcting code (ECC)ECC
Group-alphabet codeECC
A ring \(R\) is an Abelian group under addition.
Ring code
Children
\(q\)-ary codeSelf-dual additive Linear \(q\)-ary Gray Evaluation Self-dual linear QR Projective geometry Tanner \(q\)-ary LDPC Divisible AG Editing OA Perfect Nearly perfect Perfect binary GRM Combinatorial design MDS GRS Balanced
Galois fields are rings under addition. Codes over algebraic number fields have also been studied [1–3].
Pentacode
\(q\)-ary code over \(\mathbb{Z}_q\)\(q\)-ary linear code over \(\mathbb{Z}_q\)
\(R\)-linear codeLinear \(q\)-ary Gray Evaluation MDS GRS GRM QR Projective geometry Tanner \(q\)-ary LDPC Divisible Self-dual additive Self-dual linear \(q\)-ary linear code over \(\mathbb{Z}_q\)

References

[1]
Lenstra, Hendrik W. "Codes from algebraic number fields." Mathematics and computer science II, Fundamental contributions in the Netherlands since 4 (1945): 94-104.1
[2]
Tsfasman, Michael A. "Global fields, codes and sphere packings." Journees Arithmetiques 1989,-” Asterisque 198 (1992): 199-200.
[3]
V. Guruswami, “Constructions of codes from number fields”, IEEE Transactions on Information Theory 49, 594 (2003) DOI
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Zoo Code ID: rings_into_rings

Cite as:
“Ring code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/rings_into_rings
BibTeX:
@incollection{eczoo_rings_into_rings, title={Ring code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/rings_into_rings} }
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Permanent link:
https://errorcorrectionzoo.org/c/rings_into_rings

Cite as:

“Ring code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/rings_into_rings

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