\(q\)-ary protograph LDPC code[1–4]

Lifting: Given the incidence matrix \(A\) of a protograph, each nonzero entry is replaced by a sum of \(\ell\)-dimensional permutation matrices while each zero entry is replaced by the \(\ell\)-dimensional zero matrix. The resulting matrix is called a lift of \(A\). The permutation matrices can be chosen randomly or deterministically, with a deterministic lift also called a permutation voltage assignment in the theory of theory of voltage graphs [5,6].

The matrices can come from a group \(G\) or its group algebra \(\mathbb{F}_q G\), in which case the lift is often called a \(G\)-lift. In this case, matrix entries of a \(\mathbb{F}_q\)-valued matrix \(A\) are substitited with matrices forming the regular representation of \(\mathbb{F}_q G\) according to some rule.

For example, the lift of a binary two-dimensional incidence matrix using two-dimensional permutation matrices associated with the group \(\mathbb{Z}_2\) is as follows: \begin{align} \begin{pmatrix}1 & 1\\ 0 & 1 \end{pmatrix}\to\left(\begin{smallmatrix}0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{smallmatrix}\right)~. \tag*{(1)}\end{align} Here, the two nonzero entries in the top row are replaced by the exchange permutation while the bottom nonzero entry is replaced by the trivial permutation.