Efficient Construction of Successive Cancellation Decoding of Polar Codes using Logistic regression Algorithm

In this work, we present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distribution that arises in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, at code length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , we can compute the weight distribution of any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">polar cosets</i> in time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mixing factor</i> . To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> = 128.

Under successive cancellation (SC) decoding, polar codes are inferior to other codes of similar blocklength in terms of frame error rate. While more sophisticated decoding algorithms such as list- or stack-decoding partially mitigate this performance loss, they suffer from an increase in complexity. In this paper, we describe a new flavor of the SC decoder, called the SC flip decoder. Our algorithm preserves the low memory requirements of the basic SC decoder and adjusts the required decoding effort to the signal quality. In the waterfall region, its average computational complexity is almost as low as that of the SC decoder.

Polar coding is a recently introduced capacity-achieving code constructing method for binary-input discrete memoryless channels. We present a method to improve the finite-length performance of polar codes together with successive cancellation (SC) decoding by means of simple and short inner block codes. Example simulations show an improvement of about 0.3 dB.

Since polar codes are capacity-achieving codes, there have been various research works devoted to devising efficient implementation methods as well as improving error-correction performance. Since quantization is a critical implementation issue, in this paper, a nonuniform quantization method is proposed for the successive cancellation (SC) decoder of polar codes, which finds quantization boundary values based on the analysis of various quantization levels over the additive white Gaussian noise channel. Since low computational complexity, high reliability, and efficient memory management are required in the next-generation communication and memory systems, 2-4 bit precision levels are mainly considered in the proposed nonuniform quantization method. Depending on the presence of erasure, quantization levels are divided into three types, and the message alphabets and update rules are derived for each type. Also, a construction method of polar codes suitable for the proposed nonuniform-quantized SC decoder is proposed, which simultaneously determines the information set and the quantization boundary values based on the density evolution analysis and an upper bound of the block error probability. To determine quantization boundary values, a multivariate objective function is defined and an iterative coarse-to-fine search algorithm to minimize this objective function is proposed. In addition, a scaling method of quantizer output values is proposed when the number of quantization levels of quantizer is smaller than the number of quantization levels of decoder. Finally, simulation results confirm that the proposed nonuniform-quantized SC decoder shows better error-correction performance, lower decoding complexity, and higher memory efficiency compared to the best known uniform-quantized SC decoder.

Polar codes under cyclic redundancy check-aided successive cancellation list (CA-SCL) decoding can outperform the turbo codes and the LDPC codes when code lengths are configured to be several kilobits. In order to reduce the decoding complexity, a novel tree-pruning scheme for the SCL/CA-SCL decoding algorithms is proposed in this letter. In each step of the decoding procedure, the candidate paths with metrics less than a threshold are dropped directly to avoid the unnecessary computations for the path searching on the descendant branches of them. Given a candidate path, an upper bound of the path metric of its descendants is proposed to determined how much pruning this candidate path would affect frame error rate (FER) performance. By utilizing this upper bounding technique and introducing a dynamic threshold, the proposed scheme deletes as many as possible of the redundant candidate paths while keeping the performance deterioration in a tolerant region, thus it is much more efficient than the existing pruning scheme. With only a negligible loss of FER performance, the computational complexity of the proposed pruned decoding scheme is only about 40% of the standard algorithm in the low signal-to-noise ratio (SNR) region (where the FER under CA-SCL decoding is about 0.1 ~ 0.001), and it can be very close to that of the successive cancellation (SC) decoder in the moderate- and high-SNR regions.

We explore the relationship between polar and RM codes and we describe a coding scheme which improves upon the performance of the standard polar code at practical block lengths. Our starting point is the experimental observation that RM codes have a smaller error probability than polar codes under MAP decoding. This motivates us to introduce a family of codes that "interpolates" between RM and polar codes, call this family C-inter = {C-alpha : alpha is an element of [0, 1]}, where C-alpha vertical bar(alpha=1) is the original polar code, and C-alpha vertical bar(alpha=0) is an RM code. Based on numerical observations, we remark that the error probability under MAP decoding is an increasing function of a. MAP decoding has in general exponential complexity, but empirically the performance of polar codes at finite block lengths is boosted by moving along the family C-inter even under low-complexity decoding schemes such as, for instance, belief propagation or successive cancellation list decoder. We demonstrate the performance gain via numerical simulations for transmission over the erasure channel as well as the Gaussian channel.

We analyze interleaved concatenation schemes of polar codes with outer binary BCH codes and convolutional codes. We show that both BCH-polar and Conv-polar codes can have a frame error rate that decays exponentially with the code length for all rates up to capacity, which is a substantial improvement in the error exponent over stand-alone polar codes. Interleaved concatenation with long constraint length convolutional codes is an effective way to leverage the fact that polarization increases the cutoff rate of the channel. Simulation results show that Conv-polar codes when decoded with the proposed soft-output multistage iterative decoding algorithm can outperform stand-alone polar codes decoded with successive cancellation or belief propagation decoding. It may be comparable to stand-alone polar codes with list decoding in the high SNR regime. In addition to this, we show that the proposed concatenation scheme requires lower memory and decoding complexity in comparison to belief propagation and list decoding of polar codes. Practically, the scheme enables rate compatible outer codes which ease hardware implementation. Our results suggest that the proposed method may strike a better balance between performance and complexity compared to existing methods in the finite-length regime.

In this paper, we study basic properties of linear network error correction codes, their construction and error correction capability for various kinds of errors. Our discussion is restricted to the single-source multicast case. We define the minimum distance of a network error correction code. This plays the same role as it does in classical coding theory. We construct codes that can correct errors up to the full error correction capability specified by Singleton bound for network error correction codes recently established by Cai and Yeung. We propose a decoding principle for network error correction codes, based on which we introduce two decoding algorithms and analyze their performance. We formulate the global kernel error correction problem and characterize the error correction capability of codes for this kind of error.

Polar codes have received increasing attention in the past decade, and have been selected for the next generation of the wireless communication standard. Most research on polar codes has focused on codes constructed from a 2×2 polarization matrix, called binary kernel: codes constructed from binary kernels have code lengths that are bound to powers of 2. A few recent works have proposed construction methods based on multiple kernels of different dimensions, not only binary ones, allowing code lengths different from powers of 2. In this paper, we design and implement the first multi-kernel successive cancellation polar code decoder in literature. It can decode any code constructed with binary and ternary kernels: the architecture, sized for a maximum code length N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> , is fully flexible in terms of code length, code rate, and kernel sequence. The decoder can achieve a frequency of over 1 GHz in 65 nm CMOS technology, and a throughput of 615 Mb/s. The area occupation ranges between 0.11 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> for N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> = 256 and 2.01 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> for N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> = 4096. Implementation results show an unprecedented degree of flexibility: with N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> = 4096, up to 55 code lengths can be decoded with the same hardware, along with any kernel sequence and code rate.

In this paper, we investigate the performance of Polar codes, Turbo and Low-Density Parity-Check (LDPC) codes on a practical Visible Light Communication (VLC) link, in terms of the Bit-Error-Ratio (BER) vs. SNR separately. The considered parameters are the number of iterations in Turbo and LDPC and list size in the Polar codes respectively. Then, we compare three methods together with the best value of parameters of each one. To give a fair comparison, we use the same length of code as input and the rate of all codes is 1/2. The results show that the Polar codes perform better than Turbo and LDPC codes on VLC based applications.

Convolutionally precoded polar codes known as polarization-adjusted convolutional (PAC) codes are a promising variant of polar codes for short block lengths. The precoding in PAC codes has demonstrated an effective reduction in the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This reduction potentially improves the error correction performance significantly. From a codeword formation perspective, this reduction has limitations. Capitalizing on the understanding of the decomposition of minimum-weight codewords, this paper proposes a new coding scheme called reverse PAC (RPAC) codes that can effectively reduce minimum-weight codewords more than in PAC codes. Additionally, we propose a look-ahead list decoding for the RPAC codes, which maintains the same order of complexity as list decoding in PAC codes. Numerical results demonstrate that RPAC codes achieve significant improvements in block error rate over polar and PAC codes, especially in high-rate short-code scenarios where existing schemes are less effective.

Polar codes are error correction codes developed by Erdal Arikan which achieves channel capacity and its reduced complexity makes it more attractive and successful. In the past several decades we are searching for some techniques which will reduce the effect of noise to a very low value so that we can reproduce the transmitted signal more accurately. Erdal Arkan come up with a new concept which known as polar codes, here we use a channel polarization concept and hence named as polar codes. The discovery of polar code is standing as a milestone in coding theory and lot of researches are carrying out on this topic. Variety of schemes have been proposed over these years for the generation, encoding and decoding of polar codes. The important area of research is the decoder section and most widely used one is successive cancellation decoder[3]. Reduced complexity is the most attractive feature with an overall encoding and decoding complexity of O(NlogN) for a block size of N, which leads to its great success. Polar code concept is used in the most promising 5G technology[10]. In this work a complete implementation of the polar Encoder and decoder is discussed with its MATLAB and Verilog implementations

Polar codes are a new class of error correcting linear block codes, whose generator matrix is specified by the knowledge of transmission channel parameters, code length and code dimension. Moreover, regarding computational security, it is assumed that an attacker with a restricted processing power has unlimited access to the transmission media. Therefore, the attacker can construct the generator matrix of polar codes, especially in the case of binary erasure channels, on which this matrix can be easily constructed. In this paper, we introduce a novel method to keep the generator matrix of polar codes in secret in a way that the attacker cannot access the required information to decode the intended polar code. With the help of this method, a secret key cryptosystem is proposed based on non-systematic polar codes. In fact, the main objective of this study is to achieve an acceptable level of security and reliability through taking advantage of the special properties of polar codes. The analyses revealed that our scheme resists the typical attacks on the secret key cryptosystems based on linear block codes. In addition, by employing some efficient methods, the key length of the proposed scheme is decreased compared to that of the previous cryptosystems. Moreover, this scheme enjoys other advantages including high code rate, and proper error performance as well.

Space-time block code (STBC) based on orthogonal design is one of the space-time coding methods to achieve multi-input multi-output (MIMO) diversity gain. STBC can provide sufficient space diversity gain, but it can not provide coding gain. Polar codes based on channel polarization are the first class of provably capacity-achieving codes with low encoding and decoding complexity. In order to improve the bit error rate (BER) performance of STBC, a concatenated scheme with Polar codes and STBC, named Polar-STBC scheme, is proposed in this paper. The BER performance of Polar-STBC system is discussed by numerical simulation under different polar code length, different polar code rates, and different coding methods of STBC. The results show that the performance of Polar-STBC has a big improvement in comparison with that using STBC only. The coding gain of Polar-STBC system is about 5dB when the BER is 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> . Meanwhile, the longer code length and the lower code rate Polar codes have, the better BER performance the Polar-STBC system has. In addition, the different STBC coding methods have influence on the Polar-STBC system. The Polar-STBC system with more receiving antennas has a better BER performance in comparison with those with the same sending antennas and less receiving antennas. The coding gain of Polar-STBC system with 2 receiving antenna is 6.5dB improvement to the system with 1 receiving antenna when BER is 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> and the code length is 1024.

Polar codes have recently received a lot of attention because of their capacity-achieving performance and low encoding and decoding complexity. The performance of the successive cancellation decoder (SCD) of the polar codes highly depends on that of the partial-sum network (PSN) implementation. Hence, in this work, an efficient PSN architecture is proposed, based on the properties of polar codes. First, a new partial-sum updating algorithm and the corresponding PSN architecture are introduced which achieve a delay performance independent of the code length. Moreover, the area complexity is also reduced. Second, for a high-performance and area-efficient semi-parallel SCD implementation, a folded PSN architecture is presented to integrate seamlessly with the folded processing element architecture. This is achieved by using a novel folded decoding schedule. As a result, both the critical path delay and the area (excluding the memory for folding) of the semi-parallel SCD are approximately constant for a large range of code lengths. The proposed designs are implemented in both FPGA and ASIC and compared with the existing designs. Experimental result shows that for polar codes with large code length, the decoding throughput is improved by more than 1.05 times and the area is reduced by as much as 50.4%, compared with the state-of-the-art designs.