List Decoding of Polar Codes

Channel coding is an important instrument used in communication to correct errors that occur on channels. It is interesting to find the best-suited channel code for different communication systems. Polar codes have been in the spotlight lately for their simple structure and performance when in combination with list decoding and cyclic redundancy check code. Polar codes have a recursive structure that makes them interesting to implement in hardware, and they have lately been chosen as a standard for short code communication in 5G to correct bit errors. However, polar codes by themselves are shown to work poorly for practical block lengths, and it is therefore of interest to research them further. This thesis investigates polar codes with a suggested combination of list decoding and CRC. The combination is shown to improve short polar codes enough to compete with the best-known channel codes today for short block lengths. This thesis investigates why this combination works so well with polar codes. The focus lies on the selection of frozen bits in polar codes, in comparison with the similar Reed-Muller codes, and on the size and bit-placement of the CRCs. All investigations focus on codes with length 128 bits and code rate 0.5. We find that a slightly modified frozen bit selection can result in huge performance changes of polar codes. We also find how the use of a list decoder with a large list size improves Reed-Muller codes such that they challenge polar codes both with and without added CRCs. We study if a long CRC is preferred, or if the code performance can be improved by dividing it into several shorter CRCs spread out over the polar code. Results from different modifications to polar codes are presented and discussed.Polar codes have recently been selected as standard to be implemented in 5G for short messages, but they only perform well for short messages after some modifications. This thesis compares variations of those modifications. Assume that you want to send data between two devices. The optimal outcome would be that the receiver receives your message unmodified. A problem that occurs when data is transmitted is that transmission channels are subject to noise, which can result in bit errors in your data. Channel codes are used to solve this issue. They add redundancy to your message in shape of more bits before it gets transmitted on the channel, in a way such that the receiver can use these added bits to detect or correct errors that occurred on the channel. Polar codes are recently discovered channel codes. They have many desirable properties, such as low error rates and a low complexity encoder and decoder. They are fast, do not need much computing power, and are simple to implement in hardware. Unfortunately, the codes do not perform well for short messages of up to a few thousand bits. However, recent research found that this could be changed if the code is combined with a more complex list decoder and another channel code called CRC. The CRC is added to aid the decoder in its last step to find the correct message in a list. Polar codes have recently been selected to be used as a standard in 5G for short messages. This thesis investigates how this relatively poorly performing code can be improved enough to compete with the best codes for short code communication, focusing on 128-bit codes. Polar codes polarize channels so that some become reliable and other unreliable. The set of reliable channels is used to send data on, and all other are called frozen. With the improved polar codes, three variables are not uniquely specified. They are the selection of frozen bits, the list decoder size, and the CRC polynomial. We investigate how these three variables change the code performance. The frozen bit selection is compared with that of the similar Reed-Muller code. Results include the observation that the Reed-Muller codes under some circumstances perform better than polar codes in combination with list decoding and CRC. We also observe that the selection of frozen bits is crucial for finding the best performing short polar code, but not trivial. The CRC is constructed to detect long burst errors, but we do not know if that is the type of errors that occur in the polar code, and therefore not if a CRC is an optimal code to use in the list decoder. Interesting results show that two shorter CRCs spread out over the decoder sometimes improve the code compared to one stronger CRC at the end. Results and conclusions can be used when constructing polar codes for implementation in 5G. The divided CRC can, for example, be used to compensate for a lower complexity decoder. Conclusions include that polar codes should be tested and compared before implementation since finding the best polar code is not trivial for short codes. Further research should include a closer look at CRCs

Partitioned Successive-Cancellation List Decoding of Polar Codes

Successive-cancellation list (SCL) decoding is an algorithm that provides very good error-correction performance for polar codes. However, its hardware implementation requires a large amount of memory, mainly to store intermediate results. In this paper, a partitioned SCL algorithm is proposed to reduce the large memory requirements of the conventional SCL algorithm. The decoder tree is broken into partitions that are decoded separately. We show that with careful selection of list sizes and number of partitions, the proposed algorithm can outperform conventional SCL while requiring less memory.Comment: 4 pages, 6 figures, to appear at IEEE ICASSP 201

Successive Cancellation Automorphism List Decoding of Polar Codes

The discovery of suitable automorphisms of polar codes gained a lot of attention by applying them in Automorphism Ensemble Decoding (AED) to improve the error-correction performance, especially for short block lengths. This paper introduces Successive Cancellation Automorphism List (SCAL) decoding of polar codes as a novel application of automorphisms in advanced Successive Cancellation List (SCL) decoding. Initialized with L permutations sampled from the automorphism group, a superposition of different noise realizations and path splitting takes place inside the decoder. In this way, the SCAL decoder automatically adapts to the channel conditions and outperforms the error-correction performance of conventional SCL decoding and AED. For a polar code of length 128, SCAL performs near Maximum Likelihood (ML) decoding with L=8, in contrast to M=16 needed decoder cores in AED. Application-Specific Integrated Circuit (ASIC) implementations in a 12 nm technology show that high-throughput, pipelined SCAL decoders outperform AED in terms of energy efficiency and power density, and SCL decoders additionally in area efficiency.Comment: 5 pages, 5 figures, submitted to IEEE for possible publicatio

CRC-Aided Belief Propagation List Decoding of Polar Codes

Although iterative decoding of polar codes has recently made huge progress based on the idea of permuted factor graphs, it still suffers from a non-negligible performance degradation when compared to state-of-the-art CRC-aided successive cancellation list (CA-SCL) decoding. In this work, we show that iterative decoding of polar codes based on the belief propagation list (BPL) algorithm can approach the error-rate performance of CA-SCL decoding and, thus, can be efficiently used for decoding the standardized 5G polar codes. Rather than only utilizing the cyclic redundancy check (CRC) as a stopping condition (i.e., for error-detection), we also aim to benefit from the error-correction capabilities of the outer CRC code. For this, we develop two distinct soft-decision CRC decoding algorithms: a Bahl-Cocke-Jelinek-Raviv (BCJR)-based approach and a sum product algorithm (SPA)-based approach. Further, an optimized selection of permuted factor graphs is analyzed and shown to reduce the decoding complexity significantly. Finally, we benchmark the proposed CRC-aided belief propagation list (CA-BPL) to state-of-the-art 5G polar codes under CA-SCL decoding and, thereby, showcase an error-rate performance not just close to the CA-SCL but also close to the maximum likelihood (ML) bound as estimated by ordered statistic decoding (OSD).Comment: Submitted to IEEE for possible publicatio

Partitioned List Decoding of Polar Codes: Analysis and Improvement of Finite Length Performance

Polar codes represent one of the major recent breakthroughs in coding theory and, because of their attractive features, they have been selected for the incoming 5G standard. As such, a lot of attention has been devoted to the development of decoding algorithms with good error performance and efficient hardware implementation. One of the leading candidates in this regard is represented by successive-cancellation list (SCL) decoding. However, its hardware implementation requires a large amount of memory. Recently, a partitioned SCL (PSCL) decoder has been proposed to significantly reduce the memory consumption. In this paper, we examine the paradigm of PSCL decoding from both theoretical and practical standpoints: (i) by changing the construction of the code, we are able to improve the performance at no additional computational, latency or memory cost, (ii) we present an optimal scheme to allocate cyclic redundancy checks (CRCs), and (iii) we provide an upper bound on the list size that allows MAP performance.Comment: 2017 IEEE Global Communications Conference (GLOBECOM

On Metric Sorting for Successive Cancellation List Decoding of Polar Codes

We focus on the metric sorter unit of successive cancellation list decoders for polar codes, which lies on the critical path in all current hardware implementations of the decoder. We review existing metric sorter architectures and we propose two new architectures that exploit the structure of the path metrics in a log-likelihood ratio based formulation of successive cancellation list decoding. Our synthesis results show that, for the list size of $L=32$, our first proposed sorter is $14\%$ faster and $45\%$ smaller than existing sorters, while for smaller list sizes, our second sorter has a higher delay in return for up to $36\%$ reduction in the area.Comment: To be presented in 2015 IEEE International Symposium on Circuits and Systems (ISCAS'2015