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arXiv:1304.5220

Scaling Exponent of List Decoders with Applications to Polar Codes

Marco Mondelli, Hamed Hassani, Rudiger L. Urbanke

Published Apr 18, 2013• 3 citations
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Abstract

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 $\rm MAP$ 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 block length grows large. Then, it is shown that under $\rm MAP$ 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.

DOI
10.48550/arxiv.1304.5220

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Scaling exponent of list decoders with applications to polar codes

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Finite-Length Scaling for Polar Codes

Hamed Hassani, Kasra Alishahi, Rudiger L. Urbanke

Consider a binary-input memoryless output-symmetric channel\(W\). Such a channel has a capacity, call it\(I(W)\), and for any\(R<I(W)\)and strictly positive constant\(P_{\rm e}\)we know that we can construct a coding scheme that allows transmission at rate\(R\)with an error probability not exceeding\(P_{\rm e}\). Assume now that we let the rate\(R\)tend to\(I(W)\)and we ask how we have to scale the blocklength\(N\)in order to keep the error probability fixed to\(P_{\rm e}\). We refer to this as the finite-length scaling behavior. This question was addressed by Strassen as well as Polyanskiy, Poor, and Verdu, and the result is that\(N\)must grow at least as the square of the reciprocal of\(I(W)-R\). Polar codes are optimal in the sense that they achieve capacity. In this paper, we are asking to what degree they are also optimal in terms of their finite-length behavior. Since the exact scaling behavior depends on the choice of the channel, our objective is to provide scaling laws that hold universally for all binary-input memoryless output-symmetric channels. Our approach is based on analyzing the dynamics of the un-polarized channels. More precisely, we provide bounds on (the exponent of) the number of subchannels whose Bhattacharyya constant falls in a fixed interval\([a,b]\). Mathematically, this can be stated as bounding the sequence\(\bigl \{{1}/{n} \log {\rm {Pr}}(Z_{n} \!\in \! [a,b])\bigr \}_{n \in \mathbb {N}}\), where\(Z_{n}\)is the Bhattacharyya process. We then use these bounds to derive tradeoffs between the rate and the block-length. The main results of this paper can be summarized as follows. Consider the sum of Bhattacharyya parameters of subchannels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given value\(P_{\rm e}>0\), then the required block-length\(N\)scales in terms of the rate\(R < I(W)\)as\(N \geq {\alpha }/{(I(W)-R)^{\underline {\mu }}}\), where\(\alpha \)is a positive constant that depends on\(P_{\rm e}\)and\(I(W)\). We show that\(\underline {\mu } = 3.579\)is a valid choice, and we conjecture that indeed the value of\(\underline {\mu }\)can be improved to\(\underline {\mu }=3.627\), the parameter for the binary erasure channel. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate like\(N \leq {\beta }/{(I(W)-R)^{\overline {\mu }}}\), where\(\beta \)is a constant that depends on\(P_{\rm e}\)and\(I(W)\), and\(\overline {\mu }=6\).

arXiv:1304.4778•IEEE Transactions on Information Theory•Jul 25, 2014•108 citations
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.
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.
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.
Successive cancellation list decoding of polar codes provides very good performance for short to moderate block lengths. However, the list size required to approach the performance of maximum-likelihood decoding is still not well understood theoretically. This work identifies information-theoretic quantities that are closely related to this required list size. It also provides a natural approximation for these quantities that can be computed efficiently even for very long codes. Simulation results are provided for the binary erasure channel as well as the binary-input additive white Gaussian noise channel.
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.
This work identifies information-theoretic quantities that are closely related to the required list size on average for successive cancellation list (SCL) decoding to implement maximum-likelihood decoding over general binary memoryless symmetric (BMS) channels. It also provides upper and lower bounds for these quantities that can be computed efficiently for very long codes. For the binary erasure channel (BEC), we provide a simple method to estimate the mean accurately via density evolution. The analysis shows how to modify, e.g., Reed-Muller codes, to improve the performance when practical list sizes, e.g.,L∈[8,1024]L\in {[{8, 1024}]}L∈[8,1024], are adopted. Exemplary constructions with block lengthsN∈{128,512}N\in \{128,512\}N∈{128,512}outperform polar codes of 5G over the binary-input additive white Gaussian noise channel. It is further shown that there is a concentration around the mean of the logarithm of the required list size for sufficiently large block lengths, over discrete-output BMS channels. We provide the probability mass functions (p.m.f.s) of this logarithm, over the BEC, for a sequence of the modified RM codes with an increasing block length via simulations, which illustrate that the p.m.f.s concentrate around the estimated mean.
Consider a binary-input memoryless output-symmetric channelWWW. Such a channel has a capacity, call itI(W)I(W)I(W), and for anyR<I(W)R<I(W)R<I(W)and strictly positive constantPeP_{\rm e}Pe​we know that we can construct a coding scheme that allows transmission at rateRRRwith an error probability not exceedingPeP_{\rm e}Pe​. Assume now that we let the rateRRRtend toI(W)I(W)I(W)and we ask how we have to scale the blocklengthNNNin order to keep the error probability fixed toPeP_{\rm e}Pe​. We refer to this as the finite-length scaling behavior. This question was addressed by Strassen as well as Polyanskiy, Poor, and Verdu, and the result is thatNNNmust grow at least as the square of the reciprocal ofI(W)−RI(W)-RI(W)−R. Polar codes are optimal in the sense that they achieve capacity. In this paper, we are asking to what degree they are also optimal in terms of their finite-length behavior. Since the exact scaling behavior depends on the choice of the channel, our objective is to provide scaling laws that hold universally for all binary-input memoryless output-symmetric channels. Our approach is based on analyzing the dynamics of the un-polarized channels. More precisely, we provide bounds on (the exponent of) the number of subchannels whose Bhattacharyya constant falls in a fixed interval[a,b][a,b][a,b]. Mathematically, this can be stated as bounding the sequence{1/nlog⁡Pr(Zn ⁣∈ ⁣[a,b])}n∈N\bigl \{{1}/{n} \log {\rm {Pr}}(Z_{n} \!\in \! [a,b])\bigr \}_{n \in \mathbb {N}}{1/nlogPr(Zn​∈[a,b])}n∈N​, whereZnZ_{n}Zn​is the Bhattacharyya process. We then use these bounds to derive tradeoffs between the rate and the block-length. The main results of this paper can be summarized as follows. Consider the sum of Bhattacharyya parameters of subchannels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given valuePe>0P_{\rm e}>0Pe​>0, then the required block-lengthNNNscales in terms of the rateR<I(W)R < I(W)R<I(W)asN≥α/(I(W)−R)μ‾N \geq {\alpha }/{(I(W)-R)^{\underline {\mu }}}N≥α/(I(W)−R)μ​, whereα\alpha αis a positive constant that depends onPeP_{\rm e}Pe​andI(W)I(W)I(W). We show thatμ‾=3.579\underline {\mu } = 3.579μ​=3.579is a valid choice, and we conjecture that indeed the value ofμ‾\underline {\mu }μ​can be improved toμ‾=3.627\underline {\mu }=3.627μ​=3.627, the parameter for the binary erasure channel. Also, we show that with the same requirement on the sum of Bhattacharyya parameters, the block-length scales in terms of the rate likeN≤β/(I(W)−R)μ‾N \leq {\beta }/{(I(W)-R)^{\overline {\mu }}}N≤β/(I(W)−R)μ​, whereβ\beta βis a constant that depends onPeP_{\rm e}Pe​andI(W)I(W)I(W), andμ‾=6\overline {\mu }=6μ​=6.