In this work, we introduce β-expansion, a notion borrowed from number theory, as a theoretical framework to study fast construction of polar codes based on a recursive structure of universal partial order (UPO) and polarization weight (PW) algorithm. We show that polar codes can be recursively constructed from UPO by continuously solving several polynomial equations at each recursive step. From these polynomial equations, we can extract an interval for β, such that ranking the synthetic channels through a closed- form β-expansion preserves the property of nested frozen sets, which is a desired feature for low- complex construction. In an example of AWGN channels, we show that this interval for β converges to a constant close to 1.1892 when the code block-length trends to infinity. Both asymptotic analysis and simulation results validate our theoretical claims.
In this work, we introduce β-expansion, a notion borrowed from number theory, as a theoretical framework to study fast construction of polar codes based on a recursive structure of universal partial order (UPO) and polarization weight (PW) algorithm. We show that polar codes can be recursively constructed from UPO by continuously solving several polynomial equations at each recursive step. From these polynomial equations, we can extract an interval for β, such that ranking the synthetic channels through a closed- form β-expansion preserves the property of nested frozen sets, which is a desired feature for low- complex construction. In an example of AWGN channels, we show that this interval for β converges to a constant close to 1.1892 when the code block-length trends to infinity. Both asymptotic analysis and simulation results validate our theoretical claims.
In 2008, Arikan introduced a powerful family of polar codes, which are the first codes to demonstrably achieve Shannon capacity. Polar codes are built on (and named for) the phenomenon of channel polarization: a mechanism for synthetically increasing and decreasing the reliabilities of a set of channels until some fraction are perfectly noiseless and the remaining channels are completely noisy. In Arikan’s initial publication, this is achieved with a simple recursive binary transform, and shortly thereafter, it was shown by Korada et al. that polarization is a very general phenomenon achieved by many different (not necessarily binary) recursive transforms. Transmission is straightforward once the reliable channels have been identified, but this task is difficult, both computationally and theoretically. Moreover, the guarantees of polar code performance come to fruition only as the code length goes to infinity. Polar codes do not compare favorably to, e.g., low-density parity-check codes at finite block lengths. Motivated by these concerns, this dissertation investigates graph-theoretic properties of polar codes in the context of belief propagation decoding. We first define the class of encoding trellises applicable to the polar coding framework and give a graph- theoretic criterion for polarization. A fundamental contribution is the introduction of the tensor-like join of trellises: a graph product which supplies a direct connection to the matrix construction of polar codes. We leverage the tensor-like join throughout as a mechanism for understanding both graph-theoretic structures and decoding outcomes. We build up a family of acyclic auxiliary graphs, called computation graphs, in order to study belief propagation analytically. In this context, we show that the tensor-like join is sufficient as a construction mechanism and moreover, that such constructions endow computation graphs with powerful structural properties. Following Wiberg, we extend the notion of a deviation as an indicator for decoding failure in the polar computation graph framework. We provide a recursive characterization of the deviation set for Arikan’s polar codes. Finally, looking to the future, we focus on further applications of the tools developed herein.
Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attains many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials. However, finding the right set of generating monomials for a polar code which optimises the decoding performances is a nontrivial task and is channel dependent. The purpose of this paper is to reveal some universal properties of these monomials. We will namely prove that there is a way to define a nontrivial (partial) order on monomials so that the monomials generating a polar code devised for a binary-input symmetric channel always form a decreasing set. We call such codes decreasing monomial codes. The fact that polar codes are decreasing monomial codes turns out to have rather deep consequences on their structure. Indeed, we show that decreasing monomial codes have a very large permutation group by proving that it contains a group called lower triangular affine group. Furthermore, the codewords of minimum weight correspond exactly to the orbits of the minimum weight codewords that are obtained from evaluations of monomials of the generating set. In particular, it gives an efficient way of counting the number of minimum weight codewords of a decreasing monomial code and henceforth of a polar code.
Polar codes constructed with large polarization kernels were recently proven to be able to achieve optimal finite-length scaling properties. However, straightforward decoding for large kernel polar codes introduces a complexity coefficient that is exponential to the kernel sizes, which makes such codes generally believed to be impractical. In this paper, we present a new method that decodes large kernel polar codes with a complexity coefficient that is polynomial to the kernel sizes. This could facilitate the implementation of large kernel polar codes for practical use in the future.Successive cancellation decoding for large kernel polar codes requires calculation on the probabilities of its bit channels. Similar to conventional polar codes, those bit channels follow a recursive relation, which make this calculation boil down to computing the probabilities for bit channels of a single polarization kernel. This kernel-level computation can be shown equivalent to soft-output maximum-likelihood (ML) decoding on a single bit of a linear block code. In our proposed method, we first use linear operations to represent the considered linear block code as a polar code with dynamically frozen bits, and then use a modified polar list decoder to get an approximate value on the soft-output of the desired bit. This method is motivated by the observation that at short block lengths, polar list decoding with a large enough list size can well-approximate ML decoding. The proposed low-complexity method allows us to decode polar codes constructed with a 64 × 64 polarization kernel with scaling exponent µ ≈ 2.87 for the first time.
Polar code is a low-complexity and capacity-achieving code with applications in channel coding, source coding and secrecy. One way to evaluate the error performance of the finite-length polar 'channel' code, for instance, is to use Bonferroni-type (e.g., union bound) bounds requiring the Hamming weight distribution. Currently, there are no low-complexity algorithms for the polar code's spectra. In this paper, we propose a recursive probabilistic weight distribution (PWD) expression for polar codes. As this expression is complex, two polynomial-time approximations and a tree-based approach have been presented for a faster implementation of the PWD method with much accuracy.
The problem of computing the Hamming weight distribution of linear codes is considered in this paper. A novel method to enumerate all codewords up to a certain Hamming weight for binary linear block codes in general and in particular for the punctured, shortened, precoded polar codes is introduced. The proposed approach performs a recursive decomposition of the codes using construction X4 that is typically used to combine codes of different lengths. This allows to enumerate the low-weight codewords of the overall code as combinations of the low-weight codewords of the component codes. Numerical results show that the proposed approach can efficiently compute the exact partial weight distribution of the 5G New Radio punctured/shortened polar codes with CRC11 and pure polar codes. In the former and latter cases, the low-weight codeword number is up to 106 and 108, respectively. Besides, randomly punctured and shortened polar codes and randomly precoded polar codes are also considered. To the best of the authors’ knowledge, this is the first method able to solve these problems.
In this paper, we propose an analysis of the automorphism group of polar codes, with the aim of designing codes tailored forautomorphism ensemble(AE) decoding. Using a novel description of polar codes as monomial codes through negative monomials, we prove the equivalence between the notion ofdecreasing monomial codesand the universal partial order (UPO) framework for polar codes; this property is widely believed to hold true but a formal proof was missing. We further provide a rigorous mathematical connection between code word permutations and affine transformations, an important link to understand the considered automorphisms. Based on this mathematical formalisms, we analyze the algebraic properties of theaffine automorphisms groupof polar codes, providing a novel description of its structure. We classify automorphisms such that all automorphisms in the same class lead to the same result under permutation decoding, which gives rise to the concept ofredundantautomorphisms. Mathematically this is achieved by introducing equivalence classes of affine automorphisms under AE-based decoding. For practical application, we provide an algorithm to compute representatives for the equivalence classes, such that one automorphism from each equivalence class can be selected for use in AE decoding. A numerical analysis of the error correction performance of AE decoding of polar codes, based on equivalence classes, concludes the paper.