On the Concatenations of Polar Codes and Non-binary LDPC Codes

An interleaved concatenation scheme of polar codes with non-binary low-density parity check (NBLDPC) codes is proposed in this paper to improve the error-correcting performance of polar codes with finite code length. The information blocks of inner polar codes are split into several information sub-blocks, and several segment successive cancellation list (SSCL) decoders are carried out in parallel for all inner polar codes. Moreover, for a better error-correcting performance, an improved SCL decoder with a selective extension is proposed for the concatenated polar codes, which will be referred to selective extended segment SCL (SES-SCL) decoder. The SESSCL decoder uses soft information of some unreliable information sub-blocks for the decoding of subsequent sub-blocks so as to mitigate the error propagation of premature hard decision of S-SCL decoder. Simulation results show that NBLDPC-polar codes can outperform Reed Solomon (RS)-polar codes. NBLDPCpolar codes with the proposed SES-SCL algorithm can also be comparable to pure polar codes with cyclic redundancy check aided successive cancellation list (CA-SCL) decoding with list size L = 4 in the high SNR, but require lower decoding storage. Therefore, NBLDPC-polar codes may strike a better balance between memory space and performance compared to the state-of-art schemes in the finite length regime

On the Construction and Decoding of Concatenated Polar Codes

A scheme for concatenating the recently invented polar codes with interleaved block codes is considered. By concatenating binary polar codes with interleaved Reed-Solomon codes, we prove that the proposed concatenation scheme captures the capacity-achieving property of polar codes, while having a significantly better error-decay rate. We show that for any $\epsilon > 0$, and total frame length $N$, the parameters of the scheme can be set such that the frame error probability is less than $2^{-N^{1-\epsilon}}$, while the scheme is still capacity achieving. This improves upon 2^{-N^{0.5-\eps}}, the frame error probability of Arikan's polar codes. We also propose decoding algorithms for concatenated polar codes, which significantly improve the error-rate performance at finite block lengths while preserving the low decoding complexity

Concatenated Polar Codes

Polar codes have attracted much recent attention as the first codes with low computational complexity that provably achieve optimal rate-regions for a large class of information-theoretic problems. One significant drawback, however, is that for current constructions the probability of error decays sub-exponentially in the block-length (more detailed designs improve the probability of error at the cost of significantly increased computational complexity \cite{KorUS09}). In this work we show how the the classical idea of code concatenation -- using "short" polar codes as inner codes and a "high-rate" Reed-Solomon code as the outer code -- results in substantially improved performance. In particular, code concatenation with a careful choice of parameters boosts the rate of decay of the probability of error to almost exponential in the block-length with essentially no loss in computational complexity. We demonstrate such performance improvements for three sets of information-theoretic problems -- a classical point-to-point channel coding problem, a class of multiple-input multiple output channel coding problems, and some network source coding problems

An efficient length- and rate-preserving concatenation of polar and repetition codes

We improve the method in \cite{Seidl:10} for increasing the finite-lengh performance of polar codes by protecting specific, less reliable symbols with simple outer repetition codes. Decoding of the scheme integrates easily in the known successive decoding algorithms for polar codes. Overall rate and block length remain unchanged, the decoding complexity is at most doubled. A comparison to related methods for performance improvement of polar codes is drawn.Comment: to be presented at International Zurich Seminar (IZS) 201

Polar Coding for the Large Hadron Collider: Challenges in Code Concatenation

In this work, we present a concatenated repetition-polar coding scheme that is aimed at applications requiring highly unbalanced unequal bit-error protection, such as the Beam Interlock System of the Large Hadron Collider at CERN. Even though this concatenation scheme is simple, it reveals significant challenges that may be encountered when designing a concatenated scheme that uses a polar code as an inner code, such as error correlation and unusual decision log-likelihood ratio distributions. We explain and analyze these challenges and we propose two ways to overcome them.Comment: Presented at the 51st Asilomar Conference on Signals, Systems, and Computers, November 201

Belief Propagation Decoding of Polar Codes on Permuted Factor Graphs

We show that the performance of iterative belief propagation (BP) decoding of polar codes can be enhanced by decoding over different carefully chosen factor graph realizations. With a genie-aided stopping condition, it can achieve the successive cancellation list (SCL) decoding performance which has already been shown to achieve the maximum likelihood (ML) bound provided that the list size is sufficiently large. The proposed decoder is based on different realizations of the polar code factor graph with randomly permuted stages during decoding. Additionally, a different way of visualizing the polar code factor graph is presented, facilitating the analysis of the underlying factor graph and the comparison of different graph permutations. In our proposed decoder, a high rate Cyclic Redundancy Check (CRC) code is concatenated with a polar code and used as an iteration stopping criterion (i.e., genie) to even outperform the SCL decoder of the plain polar code (without the CRC-aid). Although our permuted factor graph-based decoder does not outperform the SCL-CRC decoder, it achieves, to the best of our knowledge, the best performance of all iterative polar decoders presented thus far.Comment: in IEEE Wireless Commun. and Networking Conf. (WCNC), April 201