Efficient Progressive Edge-Growth Algorithm Based on Chinese Remainder Theorem

Progressive edge-growth (PEG) algorithm construction builds a Tanner graph, or equivalently a parity-check matrix, for an LDPC code by establishing edges between the symbol nodes and the check nodes in an edge-by-edge manner and maximizing the girth in a greedy fashion. This approach is simple but the complexity of the PEG algorithm scale is O(nm), where n is the number of symbol nodes and m is the number of check nodes. We deal with this problem by construct a base matrix H
_b
of size m
_b
× n
_b
with the PEG algorithm and simultaneously expand this base matrix into a parity-check matrix H of size mx n via the the Chinese remainder theorem (CRT), where m ≫ m
_b
and n ≥ n
_b
. The size of the base matrix is expanded without decreasing the girth. For convenience, the PEG and CRT combined algorithm is referred to as the PEG-CRT algorithm in this paper. Since a smaller matrix is constructed with the PEG algorithm and the complexity of the CRT computation is negligible compared to the PEG algorithm, the complexity of the whole code construction process is reduced. Furthermore, the proposed algorithm has a potential advantage of saving storage space by storing a smaller matrix H
_b
and expanding it to H "on-the-fly" in hardware. The expanded matrix H preserves the important properties of base matrix such as large girth, flexible code rate and low density. The complexity analysis shows that the complexity of the PEG-CRT algorithm does not grow with the code length n. Simulation results show that compared with the PEG LDPC codes of length n
_b
, the expanded PEG-CRT LDPC codes have better bit error rate (BER) performance with the iterative decoding. It is also shown that compared with PEG LDPC codes of length n, which constructed with higher complexities, the PEG-CRT codes have similar BER performance.