Flexible and Low-Complexity Encoding and Decoding of Systematic Polar Codes

In this work, we present hardware and software implementations of flexible polar systematic encoders and decoders. The proposed implementations operate on polar codes of any length less than a maximum and of any rate. We describe the low-complexity, highly parallel, and flexible systematic-encoding algorithm that we use and prove its correctness. Our hardware implementation results show that the overhead of adding code rate and length flexibility is little, and the impact on operation latency minor compared to code-specific versions. Finally, the flexible software encoder and decoder implementations are also shown to be able to maintain high throughput and low latency.Comment: Submitted to IEEE Transactions on Communications, 201

Partial Sums Generation Architecture for Successive Cancellation Decoding of Polar Codes

Polar codes are a new family of error correction codes for which efficient hardware architectures have to be defined for the encoder and the decoder. Polar codes are decoded using the successive cancellation decoding algorithm that includes partial sums computations. We take advantage of the recursive structure of polar codes to introduce an efficient partial sums computation unit that can also implements the encoder. The proposed architecture is synthesized for several codelengths in 65nm ASIC technology. The area of the resulting design is reduced up to 26% and the maximum working frequency is improved by ~25%.Comment: Submitted to IEEE Workshop on Signal Processing Systems (SiPS)(26 April 2012). Accepted (28 June 2013

Universal Polar Decoding with Channel Knowledge at the Encoder

Polar coding over a class of binary discrete memoryless channels with channel knowledge at the encoder is studied. It is shown that polar codes achieve the capacity of convex and one-sided classes of symmetric channels

Achieving the Uniform Rate Region of General Multiple Access Channels by Polar Coding

We consider the problem of polar coding for transmission over $m$-user multiple access channels. In the proposed scheme, all users encode their messages using a polar encoder, while a multi-user successive cancellation decoder is deployed at the receiver. The encoding is done separately across the users and is independent of the target achievable rate. For the code construction, the positions of information bits and frozen bits for each of the users are decided jointly. This is done by treating the polar transformations across all the $m$ users as a single polar transformation with a certain \emph{polarization base}. We characterize the resolution of achievable rates on the dominant face of the uniform rate region in terms of the number of users $m$ and the length of the polarization base $L$. In particular, we prove that for any target rate on the dominant face, there exists an achievable rate, also on the dominant face, within the distance at most $\frac{(m-1)\sqrt{m}}{L}$ from the target rate. We then prove that the proposed MAC polar coding scheme achieves the whole uniform rate region with fine enough resolution by changing the decoding order in the multi-user successive cancellation decoder, as $L$ and the code block length $N$ grow large. The encoding and decoding complexities are $O(N \log N)$ and the asymptotic block error probability of $O(2^{-N^{0.5 - \epsilon}})$ is guaranteed. Examples of achievable rates for the $3$-user multiple access channel are provided

A New Class of Multiple-rate Codes Based on Block Markov Superposition Transmission

Hadamard transform~(HT) as over the binary field provides a natural way to implement multiple-rate codes~(referred to as {\em HT-coset codes}), where the code length $N=2^p$ is fixed but the code dimension $K$ can be varied from $1$ to $N-1$ by adjusting the set of frozen bits. The HT-coset codes, including Reed-Muller~(RM) codes and polar codes as typical examples, can share a pair of encoder and decoder with implementation complexity of order $O(N \log N)$. However, to guarantee that all codes with designated rates perform well, HT-coset coding usually requires a sufficiently large code length, which in turn causes difficulties in the determination of which bits are better for being frozen. In this paper, we propose to transmit short HT-coset codes in the so-called block Markov superposition transmission~(BMST) manner. At the transmitter, signals are spatially coupled via superposition, resulting in long codes. At the receiver, these coupled signals are recovered by a sliding-window iterative soft successive cancellation decoding algorithm. Most importantly, the performance around or below the bit-error-rate~(BER) of $10^{-5}$ can be predicted by a simple genie-aided lower bound. Both these bounds and simulation results show that the BMST of short HT-coset codes performs well~(within one dB away from the corresponding Shannon limits) in a wide range of code rates