arXiv:1905.13340 · IEEE Transactions on Information Theory

Log-Logarithmic Time Pruned Polar Coding

Hsin-Po Wang, Iwan Duursma

Published Dec 1, 2020• 9 citations

Abstract

A pruned variant of polar coding is proposed for binary erasure channel (BEC). Fix any BEC. For sufficiently small ε > 0, we construct a series of capacity achieving codes with block length N = ε-4.9, code rate R = Capacity - O(ε), block error probability P = ε, and encoding and decoding time complexity bC = O(log|log ε|) per information bit. The given per-bit complexity bC is log-logarithmic in N, in Capacity - R, and in P. Beyond BEC, there is a generalization: Fix a prime q and fix a symmetric, q-ary-input, discrete-output memoryless channel. For sufficiently small ε > 0, we construct a series of error correction codes with block length N = ε-constant, code rate R = Capacity - O(ε), block error probability P = ε, and encoding and decoding time complexity bC = O(log|log ε|) per information bit. Over general channels, this family of codes has the lowest per-bit time complexity among all capacity-achieving codes known to date.

DOI: 10.1109/tit.2020.3041523

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Polar Codes’ Simplicity, Random Codes’ Durability

Hsin-Po Wang, Iwan Duursma

arXiv:1912.08995IEEE Transactions on Information TheoryDec 1, 202017 citations

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:1304.4778IEEE Transactions on Information TheoryJul 25, 2014108 citations

Unified Scaling of Polar Codes: Error Exponent, Scaling Exponent, Moderate Deviations, and Error Floors

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arXiv:1905.13340 · IEEE Transactions on Information Theory

Log-Logarithmic Time Pruned Polar Coding

Hsin-Po Wang, Iwan Duursma

Published Dec 1, 2020• 9 citations

arXiv HTML PDF

Abstract

DOI: 10.1109/tit.2020.3041523

Related papers

Polar Codes’ Simplicity, Random Codes’ Durability

Hsin-Po Wang, Iwan Duursma

Over any discrete memoryless channel, we offer error correction codes such that: for one, their block error probabilities and code rates scale like random codes'; and for two, their encoding and decoding complexities scale like polar codes'. Quantitatively, for any constants π, ρ > 0 π +2ρπ) , code rate N-ρless than the Shannon capacity, and encoding and decoding complexity O(N log N}) per code block. The core theme is to incorporate polar coding (which limits the complexity to polar's realm) with large, random, dynamic kernels (which boosts the performance to random's realm). The putative codes are optimal in the following manner: Should π +2ρ>1 , no such codes exist over generic channels regardless of complexity.Over any discrete memoryless channel, we offer error correction codes such that: for one, their block error probabilities and code rates scale like random codes'; and for two, their encoding and decoding complexities scale like polar codes'. Quantitatively, for any constants π, ρ > 0 π +2ρπ) , code rate N-ρless than the Shannon capacity, and encoding and decoding complexity O(N log N}) per code block. The core theme is to incorporate polar coding (which limits the complexity to polar's realm) with large, random, dynamic kernels (which boosts the performance to random's realm). The putative codes are optimal in the following manner: Should π +2ρ>1 , no such codes exist over generic channels regardless of complexity.

arXiv:1912.08995IEEE Transactions on Information TheoryDec 1, 202017 citations

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 $\overline {\mu }=6$ .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:1304.4778IEEE Transactions on Information TheoryJul 25, 2014108 citations

Unified Scaling of Polar Codes: Error Exponent, Scaling Exponent, Moderate Deviations, and Error Floors

Marco Mondelli, Rudiger L. Urbanke, Hamed Hassani

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.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.

arXiv:1501.02444IEEE International Symposium on Information Theory (ISIT)Jun 1, 2015127 citations

On bit error rate performance of polar codes in finite regime

Ali Eslami, Hossein Pishro-Nik

Polar codes have been recently proposed as the first low complexity class of codes that can provably achieve the capacity of symmetric binary-input memoryless channels. Here, we study the bit error rate performance of finite-length polar codes under Belief Propagation (BP) decoding. We analyze the stopping sets of polar codes and the size of the minimal stopping set, called “stopping distance”. Stopping sets, as they contribute to the decoding failure, play an important role in bit error rate and error floor performance of the code. We show that the stopping distance for binary polar codes, if carefully designed, grows as O(√N) where N is the code-length. We provide bit error rate (BER) simulations for polar codes over binary erasure and gaussian channels, showing no sign of error floor down to the BERs of 10-11. Our simulations asserts that while finite-length polar codes do not perform as good as LDPC codes in terms of bit error rate, they show superior error floor performance. Motivated by good error floor performance, we introduce a modified version of BP decoding employing a guessing algorithm to improve the BER performance of polar codes. Our simulations for this guessing algorithm show two orders of magnitude improvement over simple BP decoding for the binary erasure channel (BEC), and up to 0.3 dB improvement for the gaussian channel at BERs of 10-6.Polar codes have been recently proposed as the first low complexity class of codes that can provably achieve the capacity of symmetric binary-input memoryless channels. Here, we study the bit error rate performance of finite-length polar codes under Belief Propagation (BP) decoding. We analyze the stopping sets of polar codes and the size of the minimal stopping set, called “stopping distance”. Stopping sets, as they contribute to the decoding failure, play an important role in bit error rate and error floor performance of the code. We show that the stopping distance for binary polar codes, if carefully designed, grows as O(√N) where N is the code-length. We provide bit error rate (BER) simulations for polar codes over binary erasure and gaussian channels, showing no sign of error floor down to the BERs of 10-11. Our simulations asserts that while finite-length polar codes do not perform as good as LDPC codes in terms of bit error rate, they show superior error floor performance. Motivated by good error floor performance, we introduce a modified version of BP decoding employing a guessing algorithm to improve the BER performance of polar codes. Our simulations for this guessing algorithm show two orders of magnitude improvement over simple BP decoding for the binary erasure channel (BEC), and up to 0.3 dB improvement for the gaussian channel at BERs of 10-6.

Allerton Conference on Communication, Control, and ComputingSep 1, 201053 citations

Polar Codes: Finite Length Implementation, Error Correlations and Multilevel Modulation

Mani Bastani Parizi

Shannon, in his seminal work, formalized the transmission of data over a communication channel and determined its fundamental limits. He characterized the relation between communication rate and error probability and showed that as long as the communication rate is below the capacity of the channel, error probability can be made as small as desirable by using appropriate coding over the communication channel and letting the codeword length approach infinity. He provided the formula for capacity of discrete memoryless channel. However, his proposed coding scheme was too complex to be practical in communication systems. Polar codes, recently introduced by Arikan, are the first practical codes that are known to achieve the capacity for a large class of channel and have low encoding and decoding complexity. The original polar codes of Arikan achieve a block error probability decaying exponentially in the square root of the block length as it goes to infinity. However, it is interesting to investigate their performance in finite length as this is the case in all practical communication schemes. In this dissertation, after a brief overview on polar codes, we introduce a practical framework for simulation of error correcting codes in general. We introduce the importance sampling concept to efficiently evaluate the performance of polar codes with finite bock length. Next, based on simulation results, we investigate the performance of different genie aided decoders to mitigate the poor performance of polar codes in low to moderate block length and propose single-error correction methods to improve the performance dramatically in expense of complexity of decoder. In this context, we also study the correlation between error events in a successive cancellation decoder. Finally, we investigate the performance of polar codes in non-binary channels. We compare the code construction of Sasoglu for Q-ary channels and classical multilevel codes. We construct multilevel polar codes for Q-ary channels and provide a thorough comparison of complexity and performance of two methods in finite length.Shannon, in his seminal work, formalized the transmission of data over a communication channel and determined its fundamental limits. He characterized the relation between communication rate and error probability and showed that as long as the communication rate is below the capacity of the channel, error probability can be made as small as desirable by using appropriate coding over the communication channel and letting the codeword length approach infinity. He provided the formula for capacity of discrete memoryless channel. However, his proposed coding scheme was too complex to be practical in communication systems. Polar codes, recently introduced by Arikan, are the first practical codes that are known to achieve the capacity for a large class of channel and have low encoding and decoding complexity. The original polar codes of Arikan achieve a block error probability decaying exponentially in the square root of the block length as it goes to infinity. However, it is interesting to investigate their performance in finite length as this is the case in all practical communication schemes. In this dissertation, after a brief overview on polar codes, we introduce a practical framework for simulation of error correcting codes in general. We introduce the importance sampling concept to efficiently evaluate the performance of polar codes with finite bock length. Next, based on simulation results, we investigate the performance of different genie aided decoders to mitigate the poor performance of polar codes in low to moderate block length and propose single-error correction methods to improve the performance dramatically in expense of complexity of decoder. In this context, we also study the correlation between error events in a successive cancellation decoder. Finally, we investigate the performance of polar codes in non-binary channels. We compare the code construction of Sasoglu for Q-ary channels and classical multilevel codes. We construct multilevel polar codes for Q-ary channels and provide a thorough comparison of complexity and performance of two methods in finite length.

Jan 1, 20124 citations

An Analytical Study of the Min-Sum Approximation for Polar Codes

Nir Chisnevski, Ido Tal, Shlomo Shitz Shamai

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 time $\mathcal{O}\left(N^{1.585}\right)$ , where $N$ is the codeword length. This implies a code construction algorithm with the above complexity. For the asymptotic case, we develop two rate thresholds, denoted $R_{\mathrm{L}}=R_{\mathrm{L}}(\lambda)$ and $R_{\mathrm{U}}=R_{\mathrm{U}}(\lambda)$ , where $\lambda(\cdot)$ is the labeler of the channel outputs (essentially, a quantizer). For any $0<\beta<\frac{1}{2}$ and any code rate $R<R_{\mathrm{L}}$ , there exists a family of polar codes with growing lengths such that their rates are at least $R$ and their error probabilities are at most $2^{-N^{\beta}}$ . That is, strong polarization continues to hold under the min-sum approximation. Conversely, for code rates exceeding $R_{\mathrm{U}}$ , the error probability approaches 1 as the code-length increases, irrespective of which bits are frozen. We show that $0<R_{\mathrm{L}} \leq R_{\mathrm{U}} \leq C$ , where $C$ is 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.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 time$\mathcal{O}\left(N^{1.585}\right)$, where$N$is the codeword length. This implies a code construction algorithm with the above complexity. For the asymptotic case, we develop two rate thresholds, denoted$R_{\mathrm{L}}=R_{\mathrm{L}}(\lambda)$and$R_{\mathrm{U}}=R_{\mathrm{U}}(\lambda)$, where$\lambda(\cdot)$is the labeler of the channel outputs (essentially, a quantizer). For any$0<\beta<\frac{1}{2}$and any code rate$R<R_{\mathrm{L}}$, there exists a family of polar codes with growing lengths such that their rates are at least$R$and their error probabilities are at most$2^{-N^{\beta}}$. That is, strong polarization continues to hold under the min-sum approximation. Conversely, for code rates exceeding$R_{\mathrm{U}}$, the error probability approaches 1 as the code-length increases, irrespective of which bits are frozen. We show that$0<R_{\mathrm{L}} \leq R_{\mathrm{U}} \leq C$, where$C$is 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.

arXiv:2501.13092IEEE International Symposium on Information Theory (ISIT)Jun 22, 20251 citation

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)

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

\mathcal{O}\left(N^{1.585}\right)

, where

N

is the codeword length. This implies a code construction algorithm with the above complexity. For the asymptotic case, we develop two rate thresholds, denoted

R_{\mathrm{L}}=R_{\mathrm{L}}(\lambda)

and

R_{\mathrm{U}}=R_{\mathrm{U}}(\lambda)

, where

\lambda(\cdot)

is the labeler of the channel outputs (essentially, a quantizer). For any

0<\beta<\frac{1}{2}

and any code rate

R<R_{\mathrm{L}}

, there exists a family of polar codes with growing lengths such that their rates are at least

R

and their error probabilities are at most

2^{-N^{\beta}}

. That is, strong polarization continues to hold under the min-sum approximation. Conversely, for code rates exceeding

R_{\mathrm{U}}

, the error probability approaches 1 as the code-length increases, irrespective of which bits are frozen. We show that

0<R_{\mathrm{L}} \leq R_{\mathrm{U}} \leq C

, where

C

is 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.