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

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 and Joint Source-Channel Decoding

In this dissertation, we mainly address two issues: 1. improving the finite-length performance of capacity-achieving polar codes; 2. use polar codes to efficiently exploit the source redundancy to improve the reliability of the data storage system. In the first part of the dissertation, we propose interleaved concatenation schemes of polar codes with outer binary BCH and convolutional codes to improve the finite-length performance of polar codes. For asymptotically long blocklength, we show that our schemes achieve exponential error decay rate which is much larger than the sub-exponential decay rate of standalone polar codes. In practice we show by simulation that our schemes outperform stand-alone polar codes decoded with successive cancellation or belief propagation decoding. The performance of concatenated polar and convolutional codes can be comparable to stand-alone polar codes with list decoding in the high signal to noise ratio regime. In addition to this, we show that the proposed concatenation schemes require lower memory and decoding complexity in comparison to belief propagation and list decoding of polar codes. With the proposed schemes, polar codes are able to strike a good balance between performance, memory and decoding complexity. The second part of the dissertation is devoted to improving the decoding performance of polar codes where there is leftover redundancy after source compression. We focus on language-based sources, and propose a joint-source channel decoding scheme for polar codes. We show that if the language decoder is modeled as erasure correcting outer block codes, the rate of inner polar codes can be improved while still guaranteeing a vanishing probability of error. The improved rate depends on the frozen bit distribution of polar codes and we provide a formal proof for the convergence of that distribution. Both lower bound and maximum improved rate analysis are provided. To compare with the non-iterative joint list decoding scheme for polar codes, we study a joint iterative decoding scheme with graph codes. In particular, irregular repeat accumulate codes are exploited because of low encoding/decoding complexity and capacity achieving property for the binary erasure channel. We propose how to design optimal irregular repeat accumulate codes with different models of language decoder. We show that our scheme achieves improved decoding thresholds. A comparison of joint polar decoding and joint irregular repeat accumulate decoding is given

Symbol-Based Successive Cancellation List Decoder for Polar Codes

Polar codes is promising because they can provably achieve the channel capacity while having an explicit construction method. Lots of work have been done for the bit-based decoding algorithm for polar codes. In this paper, generalized symbol-based successive cancellation (SC) and SC list decoding algorithms are discussed. A symbol-based recursive channel combination relationship is proposed to calculate the symbol-based channel transition probability. This proposed method needs less additions than the maximum-likelihood decoder used by the existing symbol-based polar decoding algorithm. In addition, a two-stage list pruning network is proposed to simplify the list pruning network for the symbol-based SC list decoding algorithm.Comment: Accepted by 2014 IEEE Workshop on Signal Processing Systems (SiPS

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