On the scaling exponent of binary polarization kernels

Fazeli, Arman; Vardy, Alexander

doi:10.1109/allerton.2014.7028536

Cited by 58 publications

(60 citation statements)

References 4 publications

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“…To minimize complexity of proposed approach (13) one needs to find kernels with small decoding windows while preserving required polarization rate (> 0.5 in our case) and scaling exponent. By computer search, based on heuristic algorithm presented in [8], we found a 16 × 16 kernel 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1…”

Section: Efficient Processing Of 16 × 16 Kernelsmentioning

confidence: 99%

“…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%

“…The overall complexity is given by 135 operation. Let s be a FHT of the vector s, where s i = S , y i+8 )), i ∈[8]. Observe that we can compute arg max i∈[8] |s i | with 7 operations and obtain v 3 , v 5 , v 6 , v 7 which gives us max R(v 7…”

mentioning

confidence: 99%

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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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Section: Efficient Processing Of 16 × 16 Kernelsmentioning

confidence: 99%

Section: Efficient Processing Of 16 × 16 Kernelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Efficient decoding of polar codes with some 16×16 kernels

Trofimiuk

Trifonov

2018

2018 IEEE Information Theory Workshop (ITW)

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“…In particular, the scaling exponent is characterized in [56,57] and is shown to be between three and four (in contrast to an exponent of two for random codes), with further details available in [58]. Other works have also studied the effect of using kernels that have dimension greater than two, such as in [59][60][61]. Such approaches allow one to achieve a probability of error that decays faster than exponential in the square root of the block length, in fact almost exponential for arbitrary large kernels, but to the expense of a significant increase in complexity (leaving dimension two the most relevant dimension for practical applications).…”

Section: Polar Martingalementioning

confidence: 99%

Entropies of Weighted Sums in Cyclic Groups and an Application to Polar Codes

Abbé

Madiman

2017

Entropy

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Abstract:In this note, the following basic question is explored: in a cyclic group, how are the Shannon entropies of the sum and difference of i.i.d. random variables related to each other? For the integer group, we show that they can differ by any real number additively, but not too much multiplicatively; on the other hand, for Z/3Z, the entropy of the difference is always at least as large as that of the sum. These results are closely related to the study of more-sums-than-differences (i.e., MSTD) sets in additive combinatorics. We also investigate polar codes for q-ary input channels using non-canonical kernels to construct the generator matrix and present applications of our results to constructing polar codes with significantly improved error probability compared to the canonical construction.

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Simplified Successive Cancellation Decoding of Polar Codes With Medium-Dimensional Binary Kernels

Huang

Zhang

et al. 2018

IEEE Access

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A method for efficiently successive cancellation (SC) decoding of polar codes with high-dimensional linear binary kernels (HDLBK) is presented and analyzed. We devise a lexpressions method which can obtain simplified recursive formulas of SC decoder in likelihood ratio form for arbitrary linear binary kernels to reduce the complexity of corresponding SC decoder. By considering the bit-channel transition probabilities W (⋅) G (⋅ 0) and W (⋅) G (⋅ 1) separately, a W -expressions method is proposed to further reduce the complexity of HDLBK based SC decoder. For a m × m binary kernel, the complexity of straightforward SC decoder is O(2 m N log N ). With W -expressions, we reduce the complexity of straightforward SC decoder to O(m 2 N log N ) when m ≤ 16. Simulation results show that 16×16 kernel polar codes offer significant advantages in terms of error performances compared with 2 × 2 kernel polar codes under SC and list SC decoders.

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On the scaling exponent of binary polarization kernels

Cited by 58 publications

References 4 publications

Efficient decoding of polar codes with some 16×16 kernels

Efficient decoding of polar codes with some 16×16 kernels

Entropies of Weighted Sums in Cyclic Groups and an Application to Polar Codes

Simplified Successive Cancellation Decoding of Polar Codes With Medium-Dimensional Binary Kernels

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