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Dual code over \(\mathbb{Z}_q\)

Description

For any linear code \(C\) over \(\mathbb{Z}_q\), the dual code is the set of \(q\)-ary strings over \(\mathbb{Z}_q\) that are orthogonal to the codewords of \(C\) under the standard inner product modulo \(q\).

The dual code over \(\mathbb{Z}_q\) is \begin{align} C^\perp = \{ y\in (\mathbb{Z}_q)^{n} ~|~ x \cdot y=0 \mod q \forall x\in C\}~. \tag*{(1)}\end{align}

Member of code lists

Primary Hierarchy

Classical Domain
Ring Kingdom
Ring codeECC
\(q\)-ary code over \(\mathbb{Z}_q\)ECC
Linear code over \(\mathbb{Z}_q\)ECC
Parents
Linear code over \(\mathbb{Z}_q\)
Dual linear code over \(R\)ECC
Dual code over \(\mathbb{Z}_q\)
Children
Self-dual code over \(\mathbb{Z}_q\)
Dual code over \(\mathbb{Z}_4\)
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Zoo Code ID: dual_over_zq

Cite as:
“Dual code over \(\mathbb{Z}_q\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/dual_over_zq
BibTeX:
@incollection{eczoo_dual_over_zq, title={Dual code over \(\mathbb{Z}_q\)}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/dual_over_zq} }
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Permanent link:
https://errorcorrectionzoo.org/c/dual_over_zq

Cite as:

“Dual code over \(\mathbb{Z}_q\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/dual_over_zq

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