## Description

A \(q\)-ary LDPC code whose parity-check matrix is constructed using the lifting procedure applied to the incidence matrix of a sparse graph called, in this context, a protograph. An ability to assign non-binary edge weight called edge scaling can also be used in code construction.

Lifting: Given the incidence matrix \(A\) of a protograph, each non-zero 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 non-zero entries in the top row are replaced by the exchange permutation while the bottom non-zero entry is replaced by the trivial permutation.

## Protection

## Parent

## Child

## References

- [1]
- A. Marinoni, P. Savazzi, and R. D. Wesel, “Protograph-based q-ary LDPC codes for higher-order modulation”, 2010 6th International Symposium on Turbo Codes & Iterative Information Processing (2010) DOI
- [2]
- D. Divsalar and L. Dolecek, “Enumerators for protograph-based ensembles of nonbinary LDPC codes”, 2011 IEEE International Symposium on Information Theory Proceedings (2011) DOI
- [3]
- K. Huang et al., “Performance comparison of non-binary LDPC block and spatially coupled codes”, 2014 IEEE International Symposium on Information Theory (2014) DOI
- [4]
- L. Dolecek et al., “Non-Binary Protograph-Based LDPC Codes: Enumerators, Analysis, and Designs”, IEEE Transactions on Information Theory 60, 3913 (2014) DOI
- [5]
- C. A. Kelley and J. L. Walker, “LDPC codes from voltage graphs”, 2008 IEEE International Symposium on Information Theory (2008) DOI
- [6]
- L. W. Beineke et al., editors , Topics in Topological Graph Theory (Cambridge University Press, 2009) DOI
- [7]
- D. Divsalar and L. Dolecek, “On the typical minimum distance of protograph-based non-binary LDPC codes”, 2012 Information Theory and Applications Workshop (2012) DOI

## Page edit log

- Victor V. Albert (2023-05-04) — most recent

## Cite as:

“\(q\)-ary protograph LDPC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/q-ary_protograph_ldpc