Performance and complexity of ψ-unitary QC-LDGM codes

In this paper we assess, through theoretical arguments and numerical simulations, the performance and complexity of a novel family of quasi-cyclic low-density generator matrix (QC-LDGM) codes. The design of such codes, denoted as ψ-unitary codes, is based on a class of binary circulant matrices recently introduced by the authors. Such matrices can have sparse inverse though being neither identity nor permutation matrices. For this reason, ψ-unitary codes can be LDGM codes while avoiding the penalization in minimum distance due to the usage of identity blocks, as occurs in classic LDGM codes. So, they can represent a trade-off in terms of performance and complexity between classic LDGM codes and non-LDGM codes.