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Dual linear code over \(R\)

Description

For any linear code \(C\) over a ring \(R\), the dual code is the set of strings that are orthogonal to the codewords of \(C\) under some inner product.

Protection

For linear codes over a finite commutative Frobenius ring \(R\), the dual code is linear and satisfies \(|C||C^\perp|=|R|^n\) [1; Secs. 6.4.1 and 6.5].

Cousin

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
\(R\)-linear codeECC
Parents
\(R\)-linear code
Dual linear code over \(R\)
Children
Dual linear codeSelf-dual linear
Self-dual code over \(R\)Self-dual linear
Dual code over \(\mathbb{Z}_q\)

References

[1]
S. T. Dougherty, “Codes over rings.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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Zoo Code ID: dual_over_rings

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

Cite as:

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

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