## Description

For any linear code \(C\) over a ring \(R\), the dual code over \(R\) is \begin{align} C^\perp = \{ y\in R^{n} ~|~ x \cdot y=0 \forall x\in C\}, \tag*{(1)}\end{align} where the ordinary, standard, or Euclidean inner product is \(x\cdot y = \sum_{i=1}^n x_i y_i\) for coordinates \(x_i,y_i\).

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

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## Page edit log

- Victor V. Albert (2022-07-22) — most recent

## Cite as:

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