Polar Codes with Mixed-Kernels

A generalization of the polar coding scheme called mixed-kernels is introduced. This generalization exploits several homogeneous kernels over alphabets of different sizes. An asymptotic analysis of the proposed scheme shows that its polarization properties are strongly related to the ones of the constituent kernels. Simulation of finite length instances of the scheme indicate their advantages both in error correction performance and complexity compared to the known polar coding structures

Recursive Descriptions of Polar Codes

Polar codes are recursive general concatenated codes. This property motivates a recursive formalization of the known decoding algorithms: Successive Cancellation, Successive Cancellation with Lists and Belief Propagation. Using such description allows an easy development of these algorithms for arbitrary polarizing kernels. Hardware architectures for these decoding algorithms are also described in a recursive way, both for Arikan's standard polar codes and for arbitrary polarizing kernels

Polar Subcodes

An extension of polar codes is proposed, which allows some of the frozen symbols, called dynamic frozen symbols, to be data-dependent. A construction of polar codes with dynamic frozen symbols, being subcodes of extended BCH codes, is proposed. The proposed codes have higher minimum distance than classical polar codes, but still can be efficiently decoded using the successive cancellation algorithm and its extensions. The codes with Arikan, extended BCH and Reed-Solomon kernel are considered. The proposed codes are shown to outperform LDPC and turbo codes, as well as polar codes with CRC.Comment: Accepted to IEEE JSAC special issue on Recent Advances In Capacity Approaching Code

Large Kernel Polar Codes with efficient Window Decoding

In this paper, we modify polar codes constructed with some 2^t x 2^t polarization kernels to reduce the time complexity of the window decoding. This modification is based on the permutation of the columns of the kernels. This method is applied to some of the kernels constructed in the literature of size 16 and 32, with different error exponents and scaling exponents such as eNBCH kernel. It is shown that this method reduces the complexity of the window decoding significantly without affecting the performance

A Comparative Study of Polar Code Constructions for the AWGN Channel

We present a comparative study of the performance of various polar code constructions in an additive white Gaussian noise (AWGN) channel. A polar code construction is any algorithm that selects $K$ best among $N$ possible polar bit-channels at the design signal-to-noise-ratio (design-SNR) in terms of bit error rate (BER). Optimal polar code construction is hard and therefore many suboptimal polar code constructions have been proposed at different computational complexities. Polar codes are also non-universal meaning the code changes significantly with the design-SNR. However, it is not known which construction algorithm at what design-SNR constructs the best polar codes. We first present a comprehensive survey of all the well-known polar code constructions along with their full implementations. We then propose a heuristic algorithm to find the best design-SNR for constructing best possible polar codes from a given construction algorithm. The proposed algorithm involves a search among several possible design-SNRs. We finally use our algorithm to perform a comparison of different construction algorithms using extensive simulations. We find that all polar code construction algorithms generate equally good polar codes in an AWGN channel, if the design-SNR is optimized.Comment: 9 pages, submitted, under revision of an IEEE journa

Polar Codes With Higher-Order Memory

We introduce the design of a set of code sequences $\{ {\mathscr C}_{n}^{(m)} : n\geq 1, m \geq 1 \}$, with memory order $m$ and code-length $N=O(\phi^n)$, where $\phi \in (1,2]$ is the largest real root of the polynomial equation $F(m,\rho)=\rho^m-\rho^{m-1}-1$ and $\phi$ is decreasing in $m$. $\{ {\mathscr C}_{n}^{(m)}\}$ is based on the channel polarization idea, where $\{ {\mathscr C}_{n}^{(1)} \}$ coincides with the polar codes presented by Ar\i kan and can be encoded and decoded with complexity $O(N \log N)$. $\{ {\mathscr C}_{n}^{(m)} \}$ achieves the symmetric capacity, $I(W)$, of an arbitrary binary-input, discrete-output memoryless channel, $W$, for any fixed $m$ and its encoding and decoding complexities decrease with growing $m$. We obtain an achievable bound on the probability of block-decoding error, $P_e$, of $\{ {\mathscr C}_{n}^{(m)} \}$ and showed that $P_e = O (2^{-N^\beta} )$ is achievable for $\beta < \frac{\phi-1}{1+m(\phi-1)}$.Comment: 15 pages, 7 figure

Fast Decoding of Multi-Kernel Polar Codes

Polar codes are a class of linear error correction codes which provably attain channel capacity with infinite codeword lengths. Finite length polar codes have been adopted into the 5th Generation 3GPP standard for New Radio, though their native length is limited to powers of 2. Utilizing multiple polarizing matrices increases the length flexibility of polar codes at the expense of a more complicated decoding process. Successive cancellation (SC) is the standard polar decoder and has time complexity $\mathcal{O}(N \log N)$ due to its sequential nature. However, some patterns in the frozen set mirror simple linear codes with low latency decoders, which allows for a significant reduction in SC latency by pruning the decoding schedule. Such fast decoding techniques have only previously been used for traditional Arikan polar codes, causing multi-kernel polar codes to be an impractical length-compatibility technique with no fast decoders available. We propose fast simplified successive cancellation decoding node patterns, which are compatible with polar codes constructed with both the Arikan and ternary kernels, and generalization techniques. We outline efficient implementations, made possible by imposing constraints on ternary node parameters. We show that fast decoding of multi-kernel polar codes has at least 72% reduced latency compared with an SC decoder in all cases considered where codeword lengths are (96, 432, 768, 2304).Comment: To appear in IEEE WCNC 2019 (Submitted September 25, 2018), 6 page

Construction and Decoding Algorithms for for Polar Codes based on 2×22\times2 Non-Binary Kernels

Polar codes based on $2\times2$ non-binary kernels are discussed in this work. The kernel over $\text{GF}(q)$ is selected by maximizing the polarization effect and using Monte-Carlo simulation. Belief propagation (BP) and successive cancellation (SC) based decoding algorithms are extended to non-binary codes. Additionally, a successive cancellation list (SCL) decoding with a pruned tree is proposed. Simulation results show that the proposed decoder performs very close to a conventional SCL decoder with significantly lower complexity

Explicit Polar Codes with Small Scaling Exponent

Herein, we focus on explicit constructions of $\ell\times\ell$ binary kernels with small scaling exponent for $\ell \le 64$. In particular, we exhibit a sequence of binary linear codes that approaches capacity on the BEC with quasi-linear complexity and scaling exponent $\mu < 3$. To the best of our knowledge, such a sequence of codes was not previously known to exist. The principal challenges in establishing our results are twofold: how to construct such kernels and how to evaluate their scaling exponent. In a single polarization step, an $\ell\times\ell$ kernel $K_\ell$ transforms an underlying BEC into $\ell$ bit-channels $W_1,W_2,\ldots,W_\ell$. The erasure probabilities of $W_1,W_2,\ldots,W_\ell$, known as the polarization behavior of $K_\ell$, determine the resulting scaling exponent $\mu(K_\ell)$. We first introduce a class of self-dual binary kernels and prove that their polarization behavior satisfies a strong symmetry property. This reduces the problem of constructing $K_\ell$ to that of producing a certain nested chain of only $\ell/2$ self-orthogonal codes. We use nested cyclic codes, whose distance is as high as possible subject to the orthogonality constraint, to construct the kernels $K_{32}$ and $K_{64}$. In order to evaluate the polarization behavior of $K_{32}$ and $K_{64}$, two alternative trellis representations (which may be of independent interest) are proposed. Using the resulting trellises, we show that $\mu(K_{32})=3.122$ and explicitly compute over half of the polarization behavior coefficients for $K_{64}$, at which point the complexity becomes prohibitive. To complete the computation, we introduce a Monte-Carlo interpolation method, which produces the estimate $\mu(K_{64})\simeq 2.87$. We augment this estimate with a rigorous proof that $\mu(K_{64})<2.97$.Comment: Add a reference to G. Trofimiuk and P. Trifonov's pape

Memory Management in Successive-Cancellation based Decoders for Multi-Kernel Polar Codes

Multi-kernel polar codes have recently been proposed to construct polar codes of lengths different from powers of two. Decoder implementations for multi-kernel polar codes need to account for this feature, that becomes critical in memory management. We propose an efficient, generalized memory management framework for implementation of successivecancellation decoding of multi-kernel polar codes. It can be used on many types of hardware architectures and different flavors of SC decoding algorithms. We illustrate the proposed solution for small kernel sizes, and give complexity estimates for various kernel combinations and code lengths.Comment: to appear in 2018 Asilomar Conference on Signals, Systems, and Computer