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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

  • Dual additive code— Dual additive codes are additive analogues of dual linear codes over rings.

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), pp. 111-128 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.), 2026. https://errorcorrectionzoo.org/c/dual_over_rings, arXiv:2606.11484
BibTeX:
@incollection{eczoo_dual_over_rings,
title={Dual linear code over \(R\)},
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/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.), 2026. https://errorcorrectionzoo.org/c/dual_over_rings, arXiv:2606.11484

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