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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| Published in: | IEEE transactions on information theory Vol. 70; no. 5; pp. 3175 - 3189 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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IEEE
01.05.2024
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | 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>. |
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| AbstractList | 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 [Formula Omitted], we can compute the weight distribution of any polar cosets in time [Formula Omitted]. 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 [Formula Omitted]. 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>. |
| Author | Vardy, Alexander Fazeli, Arman Yao, Hanwen |
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| References | ref13 ref12 ref15 ref14 ref31 ref30 ref11 ref10 ref32 ref2 ref1 ref16 ref19 ref18 Arikan (ref25) 2019 Niu (ref17) 2019 ref24 Cormen (ref23) 2009 ref20 ref21 ref28 ref27 ref29 ref8 ref7 ref9 ref4 ref3 ref6 (ref26) 2023 ref5 MacWilliams (ref22) 1977; 16 |
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| SubjectTerms | 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 |
| Title | A Deterministic Algorithm for Computing the Weight Distribution of Polar Code |
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