Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on non-commuting permutation matrices. We also show encoder and decoder implementations for these codes.