Dual additive code 

Description

For any \(q\)-ary additive code \(C\), the dual additive (or orthogonal additive) code is \begin{align} C^\perp = \{ y\in GF(q)^{n} ~|~ x \star y=0 \forall x\in C\}, \tag*{(1)}\end{align} where the trace inner product is \(x\star y = \sum_{i=1}^n \text{tr}(x_i y_i)\) for coordinates \(x_i,y_i\), and the trace maps elements of the field \(GF(q)\) with \(q=p^m\) to elements of \(GF(p)\) as \begin{align} \text{tr}(\gamma)=\sum_{k=0}^{m-1}\gamma^{p^{k}}~. \tag*{(2)}\end{align}

A code that is contained in its dual,\(C \subseteq C^\perp\), is called self-orthogonal additive or weakly self-dual additive. A code that contains its dual, \(C^\perp \subseteq C\), is called dual-containing additive. A code that is equal to its dual, \(C^\perp = C\), is called self-dual additive. A code is dual-containing additive iff its dual is self-orthogonal additive.

An alternative definition of dual substitutes the trace inner product for the trace-Hermitian inner product, \(x\star y \to \sum_{i=1}^n \text{tr}(x_i y^{p}_i)\). Another extension for when \(q=p^2\), relevant to certain stabilizer codes and reducing to the trace-Hermitian case for \(q=4\), is the trace-alternating inner product, \begin{align} x\star y \to \sum_{i=1}^{n}\text{tr}\left(\frac{x_{i}y_{i}^{\sqrt{q}}-x_{i}^{\sqrt{q}}y_{i}}{\alpha-\alpha^{q}}\right)~, \tag*{(3)}\end{align} where \(\{1,\alpha\}\) is a basis of \(GF(q)\) over \(GF(\sqrt{q})\). Self-dual additive codes with respect to the trace-Hermitian (trace-alternating) inner product are called trace Hermitian (trace-alternating) self-dual additive; similar definitions hold for self-orthogonal additive and dual-containing additive.

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References

[1]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
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Zoo Code ID: dual_additive

Cite as:
“Dual additive code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dual_additive
BibTeX:
@incollection{eczoo_dual_additive, title={Dual additive code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/dual_additive} }
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“Dual additive code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dual_additive

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/dual/dual_additive.yml.