Binary Polar Codes Research Articles

On the Non-Approximate Successive Cancellation Decoding of Binary Polar Codes With Medium Kernels

On the Non-Approximate Successive Cancellation Decoding of Binary Polar Codes With Medium Kernels

Nonbinary polar coding with low decoding latency and complexity

In this paper, we propose a new class of nonbinary polar codes, where the symbol-level polarization is achieved by using a 2 × 2 q-ary matrix 10β1 as the kernel. Under bit-level code construction, some partially-frozen symbols exist, where the frozen bits in these symbols can be used as active-check bits to facilitate the decoder. The encoder/decoder of the proposed codes has a similar structure to the original binary polar codes, admitting an easily configurable and flexible implementation, which is an obvious advantage over the existing nonbinary polar codes based on Reed-Solomon (RS) codes. A low-complexity decoding method is also introduced, in which only more competitive symbols are considered rather than the whole q symbols in the finite field. To support high spectral efficiency, we also present, in addition to the single level coded modulation scheme with field-matched modulation order, a mixed multilevel coded modulation scheme with arbitrary modulation in order to trade off the latency against complexity. Simulation results show that our proposed nonbinary polar codes exhibit comparable performance with the RS4-based polar codes and outperform binary polar codes with low decoding latency, suggesting a potential application for future ultra-reliable and low-latency communications (URLLC).

Research on encoding and decoding of non-binary polar codes over GF(2m)

Binary Polar Codes (BPCs) have advantages of high-efficiency and capacity-achieving but suffer from large latency due to the Successive-Cancellation List (SCL) decoding. Non-Binary Polar Codes (NBPCs) have been investigated to obtain the performance gains and reduce latency under the implementation of parallel architectures for multi-bit decoding. However, most of the existing works only focus on the Reed-Solomon matrix-based NBPCs and the probability domain-based non-binary polar decoding, which lack flexible structure and have a large computation amount in the decoding process, while little attention has been paid to general non-binary kernel-based NBPCs and Log-Likelihood Ratio (LLR) based decoding methods. In this paper, we consider a scheme of NBPCs with a general structure over GF(2m). Specifically, we pursue a detailed Monte-Carlo simulation implementation to determine the construction for proposed NBPCs. For non-binary polar decoding, an SCL decoding based on LLRs is proposed for NBPCs, which can be implemented with non-binary kernels of arbitrary size. Moreover, we propose a Perfect Polarization-Based SCL (PPB-SCL) algorithm based on LLRs to reduce decoding complexity by deriving a new update function of path metric for NBPCs and eliminating the path splitting process at perfect polarized (i.e., highly reliable) positions. Simulation results show that the bit error rate of the proposed NBPCs significantly outperforms that of BPCs. In addition, the proposed PPB-SCL decoding obtains about a 40% complexity reduction of SCL decoding for NBPCs.

Binary Linear Codes With Optimal Scaling: Polar Codes With Large Kernels

We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Our proof is based on the construction and analysis of binary polar codes with large kernels. When communicating reliably at rates within ε > 0 of capacity, the code length n often scales as O(1/ε <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</sup> ), where the constant μ is called the scaling exponent. It is known that the optimal scaling exponent is μ = 2, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the 2×2 kernel) on the BEC is μ = 3.63. This falls far short of the optimal scaling guaranteed by random codes. Our main contribution is a rigorous proof of the following result: for the BEC, there exist <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> × <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> binary kernels, such that polar codes constructed from these kernels achieve scaling exponent μ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> ) that tends to the optimal value of 2 as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> grows. We furthermore characterize precisely how large <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> needs to be as a function of the gap between μ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> ) and 2. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity O(n) and encoding/decoding complexity O(nlogn).

Near-Optimal Finite-Length Scaling for Polar Codes Over Large Alphabets

For any prime power $q$ , Mori and Tanaka introduced a family of $q$ -ary polar codes based on the $q\,\,\times \,\,q$ Reed–Solomon polarization kernels. For transmission over a $q$ -ary erasure channel, they also derived a closed-form recursion for the erasure probability of each effective channel. In this paper, we use that expression to analyze the finite-length scaling of these codes on the $q$ -ary erasure channel with erasure probability $\epsilon \in (0,1)$ . Our primary result is that for any $\gamma >0$ and $\delta >0$ , there is a $q_{0}$ , such that for all $q\geq q_{0}$ , the fraction of effective channels with erasure rate at most $N^{-\gamma }$ is at least $1-\epsilon -O(N^{-1/2+\delta })$ , where $N=q^{n}$ is the blocklength. Since this fraction cannot be larger than $1-\epsilon -O(N^{-1/2})$ , this establishes near-optimal finite-length scaling for this family of codes. Our approach can be seen as an extension of a similar analysis for binary polar codes by Hassani, Alishahi, and Urbanke. A similar analysis is also considered for $q$ -ary polar codes with $m \times m$ polarizing matrices. This separates the effect of the alphabet size from the effect of the matrix size. If the polarizing matrix at each stage is drawn independently and uniformly from the set of invertible $m \times m$ matrices, then the linear operator associated with the Lyapunov function analysis can be written in the closed form. To prove near-optimal scaling for polar codes with fixed $q$ as $m$ increases, however, two technical obstacles remain. Thus, we conclude by stating two concrete mathematical conjectures that, if proven, would imply near-optimal scaling for fixed $q$ .

Polar-Coded GFDM Systems

Generalized frequency division multiplexing (GFDM) is a block-based non-orthogonal multicarrier modulation scheme proposed for 5G PHY layer. In this paper, to efficiently coordinate the features of interference existing in the GFDM system, we propose a theoretical framework of the polar-coded GFDM (PC-GFDM) system, which allows jointly optimizing the combination of binary polar coding and GFDM modulation. The original GFDM channel is decomposed into multiple bit polarized channels by using a two-stage channel transform. The general modulation partition is performed in the first stage and the bit polarization transform is done in the second stage. Specifically, two schemes are considered for the first stage channel transform, multilevel coding (MLC) and bit-interleaved coded modulation (BICM). Based on the theorem we have proved in this paper that the capacities of channels corresponding to all transmitted symbols are identical, the MLC based PC-GFDM system is designed to optimize the combining of polar codes and GFDM system. Then the BICM based PC-GFDM system is designed to reduce the complexity and processing latency, which yields the suboptimal performance. Simulation results indicate that the proposed PC-GFDM systems significantly outperform the existing turbo-coded GFDM systems because of the joint design between the polar coding and the GFDM modulation.

이진 극 부호를 위한 16-QAM 성상도 최적화

이진 극 부호를 위한 16-QAM 성상도 최적화

Concatenated Coding Schemes for High Areal Density Bit-Patterned Media Magnetic Recording

This paper proposes a new concatenated error correction coding scheme for 2-D magnetic recording (TDMR). The designed concatenated coding scheme consists of outer codes as binary polar codes and an inner code as non-binary low-density parity-check (LDPC) code. These error-correcting codes (ECCs) are considered to achieve large coding gains with small bandwidth expansion in the presence of intersymbol interference and inter-track interference. Polar codes have attracted much attention as one of the ECCs that provably achieve the optimal symmetric capacity of memoryless channels with an explicit construction and are decoded by the successive-cancellation decoding algorithm, whose computational complexity is comparatively low. Using systematic polar codes, we are able to design proper low-complexity ECCs with run-length-limited constraints for any TDMR schemes based on the selection criterion (which is called channel polarization ) for choosing parity and information symbols by the magnitude of the approximately calculated mutual information amount regarding each recorded symbol in a codeword. Concretely, the concatenated coding schemes are designed for four-track and three-track recording, and the block error rate (BER) performances of 2-D partial response equalized signal processing schemes with proposed concatenated ECCs for bit-patterned media recording are evaluated. As a result, it shows that the BER performances of the proposed schemes are superior to that of the conventional scheme using a 1-D high rate binary LDPC code.

Polar Codes for Covert Communications over Asynchronous Discrete Memoryless Channels.

This paper introduces an explicit covert communication code for binary-input asynchronous discrete memoryless channels based on binary polar codes, in which legitimate parties exploit uncertainty created by both the channel noise and the time of transmission to avoid detection by an adversary. The proposed code jointly ensures reliable communication for a legitimate receiver and low probability of detection with respect to the adversary, both observing noisy versions of the codewords. Binary polar codes are used to shape the weight distribution of codewords and ensure that the average weight decays as the block length grows. The performance of the proposed code is severely limited by the speed of polarization, which in turn controls the decay of the average codeword weight with the block length. Although the proposed construction falls largely short of achieving the performance of random codes, it inherits the low-complexity properties of polar codes.

Binary polar codes are optimised codes for bitwise multistage decoding

Polar codes are considered the latest major breakthrough in coding theory. Polar codes were introduced by Ar{\i}kan in 2008. In this letter, we show that the binary polar codes are the same as the optimized codes for bitwise multistage decoding (OCBM), which have been discovered before by Stolte in 2002. The equivalence between the techniques used for the constructions and decodings of both codes is established.

Multiple Folding for Successive Cancelation Decoding of Polar Codes

Polar coding is known as the first provably capacity-achieving coding scheme under low-complexity suboptimal successive cancelation decoding (SCD). The large error-correction capability of finite-length polar codes is mostly achieved with relatively long codes. SCD is the conventional decoder for polar codes and exhibits a quasi-linear complexity in terms of the code length. Practical decoder schemes with low latency are important for high-speed polar coding applications. In this letter, we propose a nonbinary multiple folded SCD scheme to reduce the decoding latency of standard binary polar codes. Multiple foldings were first proposed to improve the efficiency of folded tree maximum-likelihood decoder for Kronecker product-based codes. By successively applying the folding operation κ times on the SCD, for a code length N, the latency is reduced from 2N - 1 to (N/2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">κ-1</sup> ) - 1 time slots, assuming full parallelization. We show that multiple folded SCD can be effectively implemented for up to κ = 3 foldings due to memory limitations. This decoder achieves exactly the same performance of the original SCD with significantly reduced latency.

Polar-Coded Modulation

A framework is proposed that allows for a joint description and optimization of both binary polar coding and 2m-ary digital pulse-amplitude modulation (PAM) schemes. For the latter, the multilevel coding (MLC) approach as well as bit-interleaved coded modulation (BICM) are considered. The conceptual equivalence of polar coding and multilevel coding is covered in detail. Based on an alternative characterization of the channel polarization phenomenon, rules for the optimum choice of the labeling in coded modulation schemes employing polar codes are developed. Simulation results regarding the error performance of the proposed schemes on the AWGN channel are included.

Performance Limits and Practical Decoding of Interleaved Reed-Solomon Polar Concatenated Codes

A scheme for concatenating the recently invented polar codes with non-binary MDS codes, as Reed-Solomon 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 $2^{-N^{0.5-\epsilon}}$, the frame error probability of Arikan's polar codes. The proposed concatenated polar codes and Arikan's polar codes are also compared for transmission over channels with erasure bursts. We provide a sufficient condition on the length of erasure burst which guarantees failure of the polar decoder. On the other hand, it is shown that the parameters of the concatenated polar code can be set in such a way that the capacity-achieving properties of polar codes are preserved. 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.