Low-Rate Repeat-Zigzag-Hadamard Codes

Abstract

Abstract-In this paper, we propose a new class of low-rate error correction codes called repeat-zigzag-Hadamard (RZH) codes featuring simple encoder and decoder structures, and flexible coding rate. RZH codes are serially concatenated turbo-like codes where the outer code is a repetition code and the inner code is a punctured zigzag-Hadamard (ZH) code. By analyzing the code structure of RZH codes, we prove that both systematic and nonsystematic RZH codes are good codes, in the sense that for an RZH code ensemble, there exists a positive number 0 such that for any binary-input memoryless channel whose Bhattacharyya noise parameter is less than 0, the average block error probability of maximum-likelihood (ML) decoding approaches zero. Two decoding algorithms-serial and parallel decoders for RZH codes-are proposed. We then employ the extrinsic information transfer (EXIT) chart technique to design irregular RZH codes. Results show that the optimized irregular RZH codes exhibit a performance that is very close to capacity in the low-rate regime.

Spatially Coupled PLDPC-Hadamard Convolutional Codes

“…The gaps of the SC-PLDPCH-CC (with I = 80 and BER of 10 −5 ) to the PEXIT threshold (−1.37 dB) and to the ultimate Shannon limit (−1.59 dB) are about 0.07 dB and 0.29 dB, respectively. In the same figure, we can also observe that under the same latency, SC-PLDPCH-CC using I = 75 processors outperforms PLDPCH-BC using I BC = 150 layered decoding iterations (equivalent to I max = 300 standard decoding iterations, i.e., the blue curve with symbol "•") by about 0.05 dB at a BER of 10 [16], rate-0.0189 THC [9], rate-0.018 IRZHC [46] and rate-0.022 LDPCH-BC [14]. Information lengths are around 65536 for all codes.…”

“…Then the corresponding BER is evaluated. Under comparable information lengths and code rates, we compare the BER performance of our proposed codes not only with that of the underlying PLDPCH-BC, but also with those of state-ofthe-art codes, namely LDPC-Hadamard block code (LDPCH-BC) [14], turbo-Hadamard code (THC) [9], irregular zigzag-Hadamard code (IRZHC) [45], [46] (whenever BER results are available). These long codes with good error performance under very low E b /N 0 values can potentially be applied to deep space communications where the signal from the space probe to Earth is extremely weak; and to embed/transmit hidden information over an ordinary wireless communication link.…”

“…7, we further redraw the BER curves of rate-0.0189 THC [9,Fig. 11], rate-0.018 IRZHC [46,Fig. 14] and rate-0.022 LDPCH-BC [14,Fig.…”

Raptor Codes in the Low SNR Regime

“…6. For the sake of comparison, the rate efficiency of a few existing repeat ZigZag Hadamard codes [23] have been also plotted in Fig. 6.…”

“…Parallel-concatenated zigzag-Hadamard (PCZH) codes are also proposed in [22] which offer the bit-error-rate (BER) of 10 −5 at E b /N 0 = −1.2 dB, which is only around 0.4 dB away from the Shannon limit. In [23], a new class of lowrate codes called repeat-zigzag-Hadamard (RZH) codes, were proposed which are essentially serially concatenated turbo-like codes where the outer code is a repetition code and the inner code is a punctured ZH code. Low-rate LDPC codes with simple protograph structure was also proposed in [24], which can be viewed as hybrid concatenations of simple modules such as accumulators, repetition codes, differentiators, and punctured single parity check codes.…”