Dual code over \(R\) 

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 \(R\).