Systematic Luby Transform Codes and Their Soft Decoding

Abstract

-Luby Transform codes (LT) were originally designed for the Binary Erasure Channel (BEC) encountered owing to randomly dropped packets in the statistical multiplexing aided classic wireline-based Internet, where transmitted packets are not affected by the fading or noise of the propagation environment of the wireless Internet. For the sake of transmitting data over the BEC routinely encountered in statistical multiplexing aided wireless Internet -style scenarios, we applied the belief propagation algorithm for decoding LT codes and designed a novel version of LT codes, which we refer to as systematic LT codes. When using soft decoding of the proposed systematic LT code, the decoding process becomes capable of preventing the potentially avalanche-like inter-packet error propagation. For example, the systematic LT(1000,3000) code achieved a BER below 10 −5 at E b /N 0 = 3.5dB after six decoding iterations. An even lower E b /N 0 of 2.7dB was required, when using a longer systematic LT(10000,30000) code for transmission over the AWGN channel. In the combined BEC-AWGN channel the BER recorded at the output of the systematic LT(1000,3000) code was about 10 −5 at E b /N 0 = 4.5dB, when encounter an erasure probability of P e = 0.1.

Amplify-forward and decode-forward cooperation relying on systematic Luby transform coded Hybrid Automatic-Repeat-reQuest

“…Hence, we can analyse SLT codes with the aid of EXIT charts [27]. According to [6], an EXIT chart may be formulated based on the LLR messages being passed between the SLT message nodes and parity check nodes. Let Q ′ and R ′ denote the LLR information passed from the message nodes to the check nodes and vice versa.…”

“…For example, systematic Luby transform (SLT) codes [6] are powerful erasure-filling codes and hence they substantially reduce the number of HARQ retransmissions. However, the coding scheme has not been thoroughly investigated, especially when collaborating with other system's component, such as the demodulator, using softinformation exchanging.…”

“…In order to mitigate the deleterious effects of error propagation, LT codes have been frequently combined with classic physicallayer FEC codes [23,24]. The idea of combining classic FEC codes with LT codes by directly amalgamating them was proposed in [6], where systematically concatenated parity bits were incorporated in order to create the family of SLT codes.…”

“…If not, the LT decoding process will be continued in an iterative fashion, until the output codeword becomes legitimate or the maximum affordable number of iterations is exhausted. In order to improve the achievable performance, the LT code's degree distribution is created by expanding its (K × N )-element generator matrix A with the aid of attaching a unity matrix having a size of (K × K ), resulting in a systematic code [6]. Particularly, if we have a generator matrix…”

Fountain Codes in Broadcast Channel: Transmission and Joint BP Decoding Algorithm

“…However, LT codes and Raptor codes in prior works [10,11] are not originally designed systematically. As pointed out in [12], the performance of non-systematic LT codes is degraded on noisy channels because of low density of the generator matrix, when using the BP decoding algorithm. Ref.…”