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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 the message set is a module over the alphabet and that encoding functions a module-homomorphisms [1].

Notes

See Ref. [2] 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 Tanner \(q\)-ary LDPC Divisible
Linear \(q\)-ary codes are \(GF(q)\)-linear.
Dual linear code over \(R\)Self-dual additive 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]
Sole, P. ed., 2009. Codes over rings (Vol. 6). World Scientific.
[3]
S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings (Springer International Publishing, 2017) DOI
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Zoo Code ID: rings_linear

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

“\(R\)-linear code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/rings_linear

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