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arXiv:1701.01628

A Lower Bound on the Probability of Error of Polar Codes over BMS Channels

Boaz Shuval, Ido Tal

Published Jan 6, 2017• 0 citations
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Abstract

Polar codes are a family of capacity-achieving codes that have explicit and low-complexity construction, encoding, and decoding algorithms. Decoding of polar codes is based on the successive-cancellation decoder, which decodes in a bit- wise manner. A decoding error occurs when at least one bit is erroneously decoded. The various codeword bits are correlated, yet performance analysis of polar codes ignores this dependence: the upper bound is based on the union bound, and the lower bound is based on the worst-performing bit. Improvement of the lower bound is afforded by considering error probabilities of two bits simultaneously. These are difficult to compute explicitly due to the large alphabet size inherent to polar codes. In this research we propose a method to lower-bound the error probabilities of bit pairs. We develop several transformations on pairs of synthetic channels that make the resultant synthetic channels amenable to alphabet reduction. Our method yields lower bounds that significantly improve upon currently known lower bounds for polar codes under successive-cancellation decoding.

DOI
10.48550/arxiv.1701.01628

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Polar codes are a family of capacity-achieving codes that have explicit and low-complexity construction, encoding, and decoding algorithms. Decoding of polar codes is based on the successive-cancellation decoder, which decodes in a bit- wise manner. A decoding error occurs when at least one bit is erroneously decoded. The various codeword bits are correlated, yet performance analysis of polar codes ignores this dependence: the upper bound is based on the union bound, and the lower bound is based on the worst-performing bit. Improvement of the lower bound is afforded by considering error probabilities of two bits simultaneously. These are difficult to compute explicitly due to the large alphabet size inherent to polar codes. In this research we propose a method to lower-bound the error probabilities of bit pairs. We develop several transformations on pairs of synthetic channels that make the resultant synthetic channels amenable to alphabet reduction. Our method yields lower bounds that significantly improve upon currently known lower bounds for polar codes under successive-cancellation decoding.
Polar codes are a family of capacity-achieving codes that have explicit and low-complexity construction, encoding, and decoding algorithms. Decoding of polar codes is based on the successive-cancellation decoder, which decodes in a bit-wise manner. A decoding error occurs when at least one bit is erroneously decoded. The various codeword bits are correlated, yet performance analysis of polar codes ignores this dependence: the upper bound is based on the union bound, and the lower bound is based on the worst-performing bit. Improvement of the lower bound is afforded by considering error probabilities of two bits simultaneously. These are difficult to compute explicitly due to the large alphabet size inherent to polar codes. In this paper, we propose a method to lower-bound the error probabilities of bit pairs. We develop several transformations on pairs of synthetic channels that make the resultant synthetic channels amenable to alphabet reduction. Our method yields lower bounds that significantly improve upon currently known lower bounds for polar codes under successive-cancellation decoding.
The min-sum approximation is widely used in the decoding of polar codes. Although it is a numerical approximation, hardly any penalties are incurred in practice. We give a theoretical justification for this. We consider the common case of a binary-input, memoryless, and symmetric channel, decoded using successive cancellation and the min-sum approximation. Under mild assumptions, we show the following. For the finite length case, we show how to exactly calculate the error probabilities of all synthetic (bit) channels in timeO(N1.585)\mathcal{O}\left(N^{1.585}\right)O(N1.585), whereNNNis the codeword length. This implies a code construction algorithm with the above complexity. For the asymptotic case, we develop two rate thresholds, denotedRL=RL(λ)R_{\mathrm{L}}=R_{\mathrm{L}}(\lambda)RL​=RL​(λ)andRU=RU(λ)R_{\mathrm{U}}=R_{\mathrm{U}}(\lambda)RU​=RU​(λ), whereλ(⋅)\lambda(\cdot)λ(⋅)is the labeler of the channel outputs (essentially, a quantizer). For any0<β<120<\beta<\frac{1}{2}0<β<21​and any code rateR<RLR<R_{\mathrm{L}}R<RL​, there exists a family of polar codes with growing lengths such that their rates are at leastRRRand their error probabilities are at most2−Nβ2^{-N^{\beta}}2−Nβ. That is, strong polarization continues to hold under the min-sum approximation. Conversely, for code rates exceedingRUR_{\mathrm{U}}RU​, the error probability approaches 1 as the code-length increases, irrespective of which bits are frozen. We show that0<RL≤RU≤C0<R_{\mathrm{L}} \leq R_{\mathrm{U}} \leq C0<RL​≤RU​≤C, whereCCCis the channel capacity. The last inequality is often strict, in which case the ramification of using the min-sum approximation is that we can no longer achieve capacity.
A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size.
Most existing works of polar codes focus on the analysis of block error probability. However, in many scenarios, bit error probability is also important for evaluating the performance of channel codes. In this paper, we establish a new framework to analyze the bit error probability of polar codes. Specifically, by revisiting the error event of bit-channel, we first introduce the conditional bit error probability as a metric to evaluate the reliability of bit-channel for both systematic and non-systematic polar codes. Guided by the concept of polar subcode, we then derive an upper bound on the conditional bit error probability of each bit-channel, and accordingly, an upper bound on the bit error probability of polar codes. Based on these, two types of construction metrics aiming at minimizing the bit error probability of polar codes are proposed, which are of linear computational complexity and explicit forms. Simulation results show that the polar codes constructed by the proposed methods can outperform those constructed by the conventional methods.
Polar codes were recently introduced by Arıkan. They achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels under a low complexity successive cancellation decoding strategy. The original polar code construction is closely related to the recursive construction of Reed-Muller codes and is based on the 2×22 \times 22×2 matrix $\bigl[ 1 &0 1& 1 \bigr]$. It was shown by Arıkan and Telatar that this construction achieves an error exponent of 12\frac1221​, i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the length. It was already mentioned by Arıkan that in principle larger matrices can be used to construct polar codes. A fundamental question then is to see whether there exist matrices with exponent exceeding 12\frac1221​. We first show that any ℓ×ℓ\ell \times \ellℓ×ℓ matrix none of whose column permutations is upper triangular polarizes symmetric channels. We then characterize the exponent of a given square matrix and derive upper and lower bounds on achievable exponents. Using these bounds we show that there are no matrices of size less than 15 with exponents exceeding 12\frac1221​. Further, we give a general construction based on BCH codes which for large nnn achieves exponents arbitrarily close to 1 and which exceeds 12\frac1221​ for size 16.