Sequential decoding of polar codes with arbitrary binary kernel

Miloslavskaya, Vera; Trifonov, Peter

doi:10.1109/itw.2014.6970857

Cited by 23 publications

(19 citation statements)

References 13 publications

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“…Note that we cannot easily verify the properties of a given ℓ × ℓ kernel K, since computing the polarization behavior of K is an NP-hard problem [8,38]. Some preliminary results on constructing good kernels are given in [8,21,22] for ℓ 16, and it would be interesting to extend these results to larger values of ℓ.…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

“…This effectively changes the decoding complexity from O(n log n) to O(2 ℓ n log n). The structure of a kernel may help mitigate this exponential blow-up with ℓ, and interesting results along these lines can be found in [5,21,22]. However, a general approach to reducing the successive-cancellation decoding complexity of large kernels is currently lacking.…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

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Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels

Hassani

Mondelli

Vardy

2018

2018 IEEE Information Theory Workshop (ITW)

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We prove that, at least for the binary erasure channel, the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but, in fact, do so under the best possible scaling of their block length as a function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Specifically, for any fixed δ > 0, we exhibit binary linear codes that ensure reliable communication at rates within ε > 0 of capacity with block length n = O(1/ε 2+δ ), construction complexity Θ(n), and encoding/decoding complexity Θ(n log n).Our proof is based on the construction and analysis of binary polar codes with large kernels. It was recently shown that, for all binary-input symmetric memoryless channels, conventional polar codes (based on a 2 × 2 kernel) allow reliable communication at rates within ε > 0 of capacity with block length, construction, encoding and decoding complexity all bounded by a polynomial in 1/ε. In particular, this means that the block length n scales as O(1/ε µ ), where the constant µ is called the scaling exponent. It is furthermore known that the optimal scaling exponent is µ = 2, and it is achieved by random linear codes. However, for general channels, the decoding complexity of random linear codes is exponential in the block length. As far as conventional polar codes, their scaling exponent depends on the channel, and for the binary erasure channel it is given by µ = 3.63. This falls far short of the optimal scaling guaranteed by random codes.Our main contribution is a rigorous proof of the following result: there exist ℓ × ℓ binary kernels, such that polar codes constructed from these kernels achieve scaling exponent µ(ℓ) that tends to the optimal value of 2 as ℓ grows. We furthermore characterize precisely how large ℓ needs to be as a function of the gap between µ(ℓ) and 2. The resulting binary codes maintain the beautiful recursive structure of conventional polar codes, and thereby achieve construction complexity Θ(n) and encoding/decoding complexity Θ(n log n). This implies that block length, construction, encoding, and decoding complexity are all linear or quasi-linear in 1/ε 2 , which meets the information-theoretic lower bound.

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Section: Discussion and Open Problemsmentioning

confidence: 99%

Section: Discussion and Open Problemsmentioning

confidence: 99%

Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels

Hassani

Mondelli

Vardy

2018

2018 IEEE Information Theory Workshop (ITW)

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“…with BEC scaling exponent µ(K 1 ) = 3.346 [8]. Furthermore, to minimize the size of decoding windows, we derived another kernel K 2 = P σ K 1 , were P σ is a permutation matrix corresponding to permutation σ = [0, 1, 2, 7, 3, 4, 5, 6,9,10,11,12,8,13,14,15], with scaling exponent µ(K 2 ) = 3.45. Both kernels have polarization rate 0.51828.…”

Section: Efficient Processing Of 16 × 16 Kernelsmentioning

confidence: 99%

“…Polar codes with large kernels were shown to provide asymptotically optimal scaling exponent [7]. Many kernels with various properties were proposed [6], [8], [9], [10], but, to the best of our knowledge, no efficient decoding algorithms for kernels with polarization rate greater than 0.5 were presented, except [11], where an approximate algorithm was introduced. Therefore, polar codes with large kernels are believed to be impractical due to very high decoding complexity.…”

Section: Introductionmentioning

confidence: 99%

Efficient decoding of polar codes with some 16×16 kernels

Trofimiuk

Trifonov

2018

2018 IEEE Information Theory Workshop (ITW)

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A decoding algorithm for polar codes with binary 16 × 16 kernels with polarization rate 0.51828 and scaling exponents 3.346 and 3.450 is presented. The proposed approach exploits the relationship of the considered kernels and the Arikan matrix to significantly reduce the decoding complexity without any performance loss. Simulation results show that polar (sub)codes with 16 × 16 kernels can outperform polar codes with Arikan kernel, while having lower decoding complexity.

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“…The same transformation can be performed with W . That is, one obtains (9) and the recursion in (6) becomes…”

Section: Decoding Of Polar Codesmentioning

confidence: 99%

Algebraic Matching Techniques for Fast Decoding of Polar Codes with Reed-Solomon Kernel

Trifonov

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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We propose to reduce the decoding complexity of polar codes with non-Arikan kernels by employing a (near) ML decoding algorithm for the codes generated by kernel rows. A generalization of the order statistics algorithm is presented for soft decoding of Reed-Solomon codes. Algebraic properties of the Reed-Solomon code are exploited to increase the reprocessing order. The obtained algorithm is used as a building block to obtain a decoder for polar codes with Reed-Solomon kernel.

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Sequential decoding of polar codes with arbitrary binary kernel

Cited by 23 publications

References 13 publications

Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels

Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels

Efficient decoding of polar codes with some 16×16 kernels

Algebraic Matching Techniques for Fast Decoding of Polar Codes with Reed-Solomon Kernel

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