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

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}

Protection

For \(q=4\), the dual of a code \(C=(n,4^{k_1} 2^{k_2})_{\mathbb{Z}_4}\) is \(C^{\perp} = (n,4^{n-k_1-k_2} 2^{k_2})\), whose generator matrix can be written in terms of the standard form of \(C\) [1; Prop. 1.2].