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\(R\)-linear code

Description

A code of length \(n\) over a ring \(R\) is \(R\)-linear if it is a submodule of \(R^n\). Axiomatically, one can define such a code by assuming that the message set is a module over the alphabet and that encoding functions are module homomorphisms [1].

For finite commutative Frobenius rings, the standard inner product, duality, and MacWilliams identities extend from the field case [2; Secs. 6.4 and 6.5]. There is a standard form for codes over finite chain rings with maximal ideals [3; Thm. 2.12].

Notes

See Ref. [4] for an introduction.See book [3] for an introduction.

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
Parents
Ring code
Linear code over \(G\)ECC
\(R\)-linear codes are linear over \(G=R\) since rings and submodules are Abelian groups under addition.
\(R\)-linear code
Children
Linear \(q\)-ary codeGray Evaluation MDS GRS Self-dual linear GRM QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible
Linear \(q\)-ary codes are \(\mathbb{F}_q\)-linear.
Dual linear code over \(R\)Self-dual linear
Linear code over \(\mathbb{Z}_q\)Gray

References

[1]
A. E.F. Jr. and H. F. Mattson, “Error-correcting codes: An axiomatic approach”, Information and Control 6, 315 (1963) DOI
[2]
S. T. Dougherty, “Codes over rings”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 111-128 DOI
[3]
S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings (Springer International Publishing, 2017) DOI
[4]
P. Sole, ed., Codes over rings, vol. 6 (World Scientific, 2009)
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Zoo Code ID: rings_linear

Cite as:
“\(R\)-linear code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/rings_linear, arXiv:2606.11484
BibTeX:
@incollection{eczoo_rings_linear,
title={\(R\)-linear 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_linear}
}
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Cite as:

“\(R\)-linear code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/rings_linear, arXiv:2606.11484

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