Linear code over \(\mathbb{Z}_4\)

Description

A code that forms a subgroup of \(\mathbb{Z}_4^n\) under addition. More technically, linear codes over \(\mathbb{Z}_4\) are submodules of \(\mathbb{Z}_4^n\).

Linear codes over \(\mathbb{Z}_4\) can be defined in terms of a generator matrix \(G\), whose rows generate the \(k\)-dimensional codespace. Given a message (i.e., logical string) \(x\), the corresponding encoded codeword is \(G^T x\).

A code's generator matrix can be reduced via coordinate permutations to its standard form \begin{align} G=\left[\begin{array}{ccc} I_{k_{1}} & A & B\\ 0 & 2I_{k_{2}} & 2C \end{array}\right]~, \tag*{(1)}\end{align} where \(A,C\) are binary matrices, and where \(B\) is a quaternary matrix. Such a code encodes \(4^{k_1} 2^{k_2}\) codewords, consisting of \(k_1\) logical quaternary digits and \(k_2\) logical bits. It is called an \((n,4^{k_1} 2^{k_2})_{\mathbb{Z}_4}\) linear code over \(\mathbb{Z}_4\) of type \(4^{k_1} 2^{k_2}\). The generator matrix forms a basis when there are no logical bits, i.e., the submodule is free if and only if \(k_2=0\) [1].

Two linear codes over \(\mathbb{Z}_4\) are monomially equivalent if the codewords of one code can be mapped into those of the other under a permutation and coordinate sign flips. The monomial automorphism group of a linear code over \(\mathbb{Z}_4\) is the largest subgroup of coordinate permutations and sign flips that maps the code onto itself.