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arXiv:1601.06215 · IEEE International Symposium on Information Theory (ISIT)

Algebraic properties of polar codes from a new polynomial formalism

Magali Bardet, Vlad Drăgoi, Ayoub Otmani, Jean–Pierre Tillich

Published Jul 1, 2016• 155 citations
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Abstract

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.

DOI
10.1109/isit.2016.7541295

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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.
In this article, we introduce a new family of polar codes from evaluation codes, called polar decreasing monomial-Cartesian codes, and prove that families of polar codes with multiple kernels over certain symmetric channels can be viewed as polar decreasing monomial-Cartesian codes. This offers a unified treatment for such codes over any finite field. We define decreasing monomial-Cartesian codes as evaluation codes obtained from a set of monomials closed under divisibility over a Cartesian product and determine their parameters (length, dimension, and minimum distance). We show that the dual of a decreasing monomial-Cartesian code is monomially equivalent to a decreasing monomial-Cartesian code. Polar decreasing monomial-Cartesian codes are then obtained by utilizing decreasing monomial-Cartesian codes whose sets of monomials are closed with respect to a partial order. We prove that any sequence of invertible matrices over an arbitrary field satisfying certain conditions polarizes any channel that is symmetric over the field.
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 lengthn, we can compute the weight distribution of anypolar cosetsin timeO(n2). 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 themixing factor. 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 lengthn= 128.
The weight distribution of an error correction code is a critical determinant of its error-correcting performance. In the case of polar codes, the minimum weight wmin(equal to the minimum distanced) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight 1.5wmin. The key insight for code design is that the enumeration of codewords with weight wmin and 1.5wminrelies on the set of maximum degree monomials. This set corresponds to the indices of minimum weight rows of the polar transformGNbelonging to the information setI. Consequently, reducing the cardinality of this set can lead to a reduction of the number of codewords in both weight categories.
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.
On the Weight Distribution of Weights Less than 2wmin⁡2w_{\min}2wmin​ in Polar Codes
The number of low-weight codewords is critical to the performance of error-correcting codes. In 1970, Kasami and Tokura characterized the codewords of Reed-Muller (RM) codes whose weights are less than 2wmin⁡2w_{\min}2wmin​, where wmin⁡w_{\min}wmin​ represents the minimum weight. In this paper, we extend their results to decreasing polar codes. We present the closed-form expressions for the number of codewords in decreasing polar codes with weights less than 2wmin⁡2w_{\min}2wmin​. Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.