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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 codeConstant-weight Combinatorial design Self-dual additive Linear \(q\)-ary Gray Self-dual linear QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible AG Evaluation OA Perfect Nearly perfect Perfect binary GRM MDS GRS Balanced
Galois fields are rings. Codes over algebraic number fields have also been studied [1–3].
\(q\)-ary code over \(\mathbb{Z}_q\)Constant-weight Combinatorial design Nearly perfect Perfect binary Linear code over \(\mathbb{Z}_q\) Gray
\(R\)-linear codeLinear \(q\)-ary Evaluation MDS GRS GRM QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible Self-dual linear Linear code over \(\mathbb{Z}_q\) Gray

References

[1]
H. W. Lenstra, “Codes from algebraic number fields”, Mathematics and computer science II, Fundamental contributions in the Netherlands since 4, 94-104 (1945)
[2]
M. A. Tsfasman, “Global fields, codes and sphere packings”, Astérisque 198, 199-200 (1992)
[3]
V. Guruswami, “Constructions of codes from number fields”, IEEE Transactions on Information Theory 49, 594 (2003) DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)

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Zoo Code ID: rings_into_rings

Cite as:
“Ring code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/rings_into_rings, arXiv:2606.11484
BibTeX:
@incollection{eczoo_rings_into_rings,
title={Ring 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/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.), 2026. https://errorcorrectionzoo.org/c/rings_into_rings, arXiv:2606.11484

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