A Deterministic Algorithm for Computing the Weight Distribution of Polar Code

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. Thi...

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Vydáno v:IEEE transactions on information theory Ročník 70; číslo 5; s. 3175 - 3189
Hlavní autoři: Yao, Hanwen, Fazeli, Arman, Vardy, Alexander
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.05.2024
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Affine transformations
Algorithms
Automorphisms
Codes
Complexity
Complexity theory
Computation
Decoding
decreasing monomial codes
Indexes
Polar codes
Representations
Subgroups
Symbols
Upper bound
Upper bounds
weight distribution
ISSN:0018-9448, 1557-9654
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Shrnutí: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 <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, we can compute the weight distribution of any polar cosets in time <inline-formula> <tex-math notation="LaTeX">O(n^{2}) </tex-math></inline-formula>. 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 mixing 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 length <inline-formula> <tex-math notation="LaTeX">n=128 </tex-math></inline-formula>.
Bibliografie:ObjectType-Article-1
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2023.3243191