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arXiv:2501.13092 · IEEE International Symposium on Information Theory (ISIT)

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

Nir Chisnevski, Ido Tal, Shlomo Shitz Shamai

Published Jun 22, 2025• 1 citation
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Abstract

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.

DOI
10.1109/isit63088.2025.11195226

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Finite-length scaling of polar codes

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Consider a polar code designed for some binary memoryless symmetric channel. We develop a lower bound on the probability of error of this polar code under successive-cancellation decoding. The bound exploits the correlation between the various codeword bits and improves upon existing lower bounds.

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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 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.

IEEE Transactions on Information Theory•Oct 12, 2018•7 citations

Universal Bounds on the Scaling Behavior of Polar Codes

Ali Goli, Hamed Hassani, Rudiger L. Urbanke

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 $μ$ such that for any binary memoryless symmetric channel $W$ with capacity $I(W)$, reliable communication requires rates that satisfy $R< I(W)-αN^{-\frac{1}μ}$, where $α$ is a positive constant and $N$ is the block-length. We provide lower bounds on $μ$, namely $μ\geq 3.553$, and we conjecture that indeed $μ=3.627$, the parameter for the binary erasure channel.

arXiv:1205.2876•May 13, 2012•3 citations
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.
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.
Consider a binary-input memoryless output-symmetric channel WWW. Such a channel has a capacity, call it I(W)I(W)I(W), and for any R<I(W)R<I(W)R<I(W) and strictly positive constant PeP_{\rm e}Pe​ we know that we can construct a coding scheme that allows transmission at rate RRR with an error probability not exceeding PeP_{\rm e}Pe​. Assume now that we let the rate RRR tend to I(W)I(W)I(W) and we ask how we have to "scale" the blocklength NNN in order to keep the error probability fixed to PeP_{\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 that NNN must grow at least as the square of the reciprocal of I(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. Our approach is based on analyzing the dynamics of the un-polarized channels. The main results of this paper can be summarized as follows. Consider the sum of Bhattacharyya parameters of sub-channels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given value Pe>0P_{\rm e}>0Pe​>0, then the required block-length NNN scales in terms of the rate R<I(W)R < I(W)R<I(W) as N≥α(I(W)−R)μ‾N \geq \frac{\alpha}{(I(W)-R)^{\underline{\mu}}}N≥(I(W)−R)μ​α​, where α\alphaα is a positive constant that depends on PeP_{\rm e}Pe​ and I(W)I(W)I(W), and μ‾=3.579\underline{\mu} = 3.579μ​=3.579. 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≤β(I(W)−R)μ‾N \leq \frac{\beta}{(I(W)-R)^{\overline{\mu}}}N≤(I(W)−R)μ​β​, where β\betaβ is a constant that depends on PeP_{\rm e}Pe​ and I(W)I(W)I(W), and μ‾=6\overline{\mu}=6μ​=6.
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.
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.