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Cooperative decoder design for non-binary LDPC code with coefficients selection

Yang Yu, Wen Chen, Jun Li, Benoît Geller

Published Dec 1, 2013• 2 citations
Publisher

Abstract

In this paper we design novel decoders for non-binary low density parity check (LDPC) codes. For a non-binary LDPC code C over the field Fqof size q for some q > 0, we propose two novel cooperative decoders, each composed of a binary component decoder and a q-ary component decoder in a concatenated manner, to obtain excellent decoding performance. Specifically to reduce the complexity of the cooperative decoders, we design a hybrid q-ary component decoder. Then, we propose an algorithm to construct the parity check matrix to eliminate the bit-level cycles. Simulations show that the decoding performance of the proposed algorithm approaches the capacity limit within 0.2dB at BER= 10−4.

DOI
10.1109/glocom.2013.6831346

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In this paper we design novel decoders for non-binary low density parity check (LDPC) codes. For a non-binary LDPC code C over the field Fqof size q for some q > 0, we propose two novel cooperative decoders, each composed of a binary component decoder and a q-ary component decoder in a concatenated manner, to obtain excellent decoding performance. Specifically to reduce the complexity of the cooperative decoders, we design a hybrid q-ary component decoder. Then, we propose an algorithm to construct the parity check matrix to eliminate the bit-level cycles. Simulations show that the decoding performance of the proposed algorithm approaches the capacity limit within 0.2dB at BER= 10−4.
The equivalent binary parity check matrices for the binary images of the cycle-free non-binary LDPC codes have numerous bit-level cycles. In this paper, we show how to transform these binary parity check matrices into their cycle-free forms. It is shown that the proposed methodology can be adopted not only for the binary images of non-binary LDPC codes but also for a large class of binary LDPC codes. Specifically, we present an extended ppp-reducible (EPR) LDPC code structure to eliminate the bit-level cycles. For the non-binary LDPC codes with short length symbol-level cycles, the EPR-LDPC codes can largely avoid the corresponding short length bit-level cycles. As to the decoding of the EPR-LDPC codes, we propose a hybrid hard-decision decoder and a hybrid parallel decoder for binary symmetric channel and binary input Gaussian channel, respectively. A simple code optimization algorithm for these binary decoders is also provided. Simulations show the comparative results and justify the advantages, i.e., better performance and lower decoding complexity, of the proposed binary constructions.
By presenting an approximated performance-complexity tradeoff (PCT) algorithm, a low-complexity non-binary low density parity check (LDPC) code over q-ary-input symmetric-output channel is designed in this manuscript which converges faster than the threshold-optimized non-binary LDPC codes in the low error rate regime. We examine our algorithm by both hard and soft decision decoders. Moreover, simulation shows that the approximated PCT algorithm has accelerated the convergence process by 30% regarding the number of the decoding iterations.
In this thesis, we propose low-complexity designing algorithms for non-binary low-density parity-check (NB-LDPC) codes over GF(q) with low error floor and low-complexity decoding algorithms for binary LDPC codes. We first propose a low-complexity designing algorithm for for NB-LDPC codes over GF(q) based on message-passing algorithms. Short cycles in an NB-LDPC code may be even more harmful to its performance if they do not satisfy the so-called full rank condition (FRC). This is because they may induce low-weight codewords or absorbing sets in that case. Thus, it is crucial to count the number of short cycles not satisfying the FRC as well as the number of short cycles for analyzing the performance of an NB-LDPC code. In order to count those cycles, we first develop a novel message-passing algorithm and identify how it is related to the FRC.We then propose a low-complexity algorithmfor counting the number of short cycles not satisfying the FRC in an NB-LDPC code, as well as the number of short cycles. The proposed algorithm is not only theoretically validated, but its computational and storage complexities are also computed. It does not require cycle enumeration with high complexity, and therefore is still feasible even in the case of large code length, say ≤ 5000, unlike the conventional methods. Based on it, we also provide a locally optimal rule to choose a nonzero element for a given edge and propose a low-complexity algorithm for designing an NB-LDPC code with low error floor. Furthermore, several design examples are presented to verify the validity of the proposed design algorithm. Secondly, we propose a low-complexity decoding algorithm for binary LDPC codes over the binary symmetric channel (BSC). The min-sum (MS) algorithm is a low-complexity decoding algorithm for binary LDPC codes and has lower computational complexity than the sum-product algorithm (SPA). Since the MS algorithm is an approximation of the SPA, it has performance degradation. Furthermore, reducing the number of bits for representing messages to two in a decoding algorithm for further reducing computational and storage complexities, causes significant performance losses. In order to minimize the performance degradation and computational and storage complexities, we propose a low-complexity decoding algorithm which uses several decoders with two-bit precision for binary LDPC codes over the BSC. The proposed algorithm adaptively changes decoders according to the number of unsatisfied check nodes at each iteration. Compared with the SPA using floating-point operations, the proposed decoding algorithm greatly reduces computational and storage complexities with negligible performance losses. Numerical results show that the proposed decoding algorithm has similar performance to the SPA with floating-point operations over the BSC. In the last part of this thesis, low-complexity decoding algorithms of binary LDPC codes for the core layer at fixed receivers in ATSC 3.0 are proposed. Gallager’s decoding algorithms …
Non-binary LDPC code can achieve better error correcting performance than its binary counterpart especially when the code length is short or moderate. A novel Non-Binary LDPC code is presented in this paper to replace the traditional binary standard in 802.11n system. The proposed Parity Check Matrix (PCM) is optimized by eliminating short-girths and low weight code-words. Partially-parallel quasi-cyclic structure is adopted for the encoder and the check matrix is quasi diagonal line structure. Encoding process directly utilizes check matrix rather than generator matrix to generate check sequence. For decoding process, the (Forward Backward Extended Minimum Sum (FB-EMS) algorithm is adopted. The performance of the proposed Non-binary LDPC code over GF(16) shows more than 1dB coding gain at 10^(-6) BER compared to the traditional binary standard with same bit length for 802.11n standard. Encoder structure and decoder algorithm have been also presented. For practical purposes, this Non-binary quasi-cyclic LDPC code with low complexity makes our proposed system very attractive.
The finite-length design of non-binary low-density parity-check (LDPC) codes for the q-ary erasure channel under maximum a posteriori (MAP) decoding is addressed. A low-complexity MAP decoding algorithm is reviewed for which a code design strategy is proposed. In particular, it is illustrated how a judicious code design permits to find a trade-off between performance in terms of codeword error rate (CER) and decoding complexity. As an example, the performance curve of a short (400, 200) code on the memoryless 4-ary erasure channel tightly approaches the Singleton bound at least down to a CER of 10-8.
In this letter, we present a hybrid iterative decoder for non-binary low density parity check (LDPC) codes over binary erasure channel (BEC), based on which the recursion of the erasure probability is derived to design non-binary LDPC codes with convergence-optimized degree distributions. The resulting one-step decoding tree is cycle-free and achieves lower decoding complexity. Experimental studies show that the proposed convergence-optimization algorithm accelerates the convergence process by 33%.