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IEEE International Symposium on Information Theory (ISIT)

An empirical scaling law for polar codes

Satish Babu Korada, Andrea Montanari A, Emre Telatar, Rudiger L. Urbanke

Published Jun 1, 2010• 62 citations
Publisher

Abstract

Using scaling laws, we obtain estimates of the block error probability of polar codes under successive cancellation decoding. For the binary erasure channel we present an upper and a lower bound for the scaling parameter. Numerically these two bounds match. We also present a scaling law for general binary discrete memoryless channels.
DOI
10.1109/isit.2010.5513579

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IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2017•7 citations
We consider the problem of determining the trade-off between the rate and the block-length of polar codes for a given block error probability when we use the successive cancellation decoder. We take the sum of the Bhattacharyya parameters as a proxy for the block error probability, and show that there exists a universal parameter μμμ such that for any binary memoryless symmetric channel WWW with capacity I(W)I(W)I(W), reliable communication requires rates that satisfy R<I(W)−αN−1μR< I(W)-αN^{-\frac{1}μ}R<I(W)−αN−μ1​, where ααα is a positive constant and NNN is the block-length. We provide lower bounds on μμμ, namely μ≥3.553μ\geq 3.553μ≥3.553, and we conjecture that indeed μ=3.627μ=3.627μ=3.627, the parameter for the binary erasure channel.
We consider the problem of determining the tradeoff between the rate and the block-length of polar codes for a given block error probability when we use the successive cancellation decoder. We take the sum of the Bhattacharyya parameters as a proxy for the block error probability, and show that there exists a universal parameter μ such that for any binary memoryless symmetric channel W with capacity I(W), reliable communication requires rates that satisfy R-1/μ, where α is a positive constant and N is the block-length. We provide lower bounds on μ, namely μ ≥ 3.553, and we conjecture that indeed μ = 3.627, the parameter for the binary erasure channel.
Consider the transmission of a polar code of block length N and rate R over a binary memoryless symmetric channel W and let Pebe the block error probability under successive cancellation decoding. In this paper, we develop new bounds that characterize the relationship of the parameters R, N, Pe, and the quality of the channel W quantified by its capacity I(W) and its Bhattacharyya parameter Z(W). In previous work, two main regimes were studied. In the error exponent regime, the channel W and the rate R-√N. In the scaling exponent approach, the channel W and the error probability Pe are fixed and it was proved that the gap to capacity I(W) - R scales as N-1/μ. Here, μ is called scaling exponent and this scaling exponent depends on the channel W. A heuristic computation for the binary erasure channel (BEC) gives μ = 3.627 and it was shown that, for any channel W, 3.579 ≤ μ ≤ 5.702. Our contributions are as follows. First, we provide the tighter upper bound μ <;≤ 4.714 valid for any W. With the same technique, we obtain the upper bound μ ≤ 3.639 for the case of the BEC; this upper bound approaches very closely the heuristically derived value for the scaling exponent of the erasure channel. Second, we develop a trade-off between the gap to capacity I(W)- R and the error probability Pe as the functions of the block length N. In other words, we neither fix the gap to capacity (error exponent regime) nor the error probability (scaling exponent regime), but we do consider a moderate deviations regime in which we study how fast both quantities, as the functions of the block length N, simultaneously go to 0. Third, we prove that polar codes are not affected by error floors. To do so, we fix a polar code of block length N and rate R. Then, we vary the channel W and study the impact of this variation on the error probability. We show that the error probability Pe scales as the Bhattacharyya parameter Z(W) raised to a power that scales roughly like VN. This agrees with the scaling in the error exponent regime.
Motivated by the significant performance gains which polar codes experience when they are decoded with successive cancellation list decoders, we study how the scaling exponent changes as a function of the list size L. In particular, we fix the block error probability Peand we analyze the tradeoff between the blocklength N and the back-off from capacity C-R using scaling laws. By means of a Divide and Intersect procedure, we provide a lower bound on the error probability under MAP decoding with list size L for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the blocklength grows large. We show that, although list decoding can significantly improve the involved constants, the scaling exponent itself, i.e., the speed at which capacity is approached, stays unaffected. This result applies in particular to polar codes, since their minimum distance tends to infinity as N increases. Some considerations are also pointed out for the genie-aided successive cancellation decoder when transmission takes place over the binary erasure channel.
Motivated by the significant performance gains which polar codes experience under successive cancellation list decoding, their scaling exponent is studied as a function of the list size. In particular, the error probability is fixed and the trade-off between block length and back-off from capacity is analyzed. A lower bound is provided on the error probability under MAP\rm MAPMAP decoding with list size LLL for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the block length grows large. Then, it is shown that under MAP\rm MAPMAP decoding, although the introduction of a list can significantly improve the involved constants, the scaling exponent itself, i.e., the speed at which capacity is approached, stays unaffected for any finite list size. In particular, this result applies to polar codes, since their minimum distance tends to infinity as the block length increases. A similar result is proved for genie-aided successive cancellation decoding when transmission takes place over the binary erasure channel, namely, the scaling exponent remains constant for any fixed number of helps from the genie. Note that since genie-aided successive cancellation decoding might be strictly worse than successive cancellation list decoding, the problem of establishing the scaling exponent of the latter remains open.