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arXiv:0901.0536

Polar Codes: Characterization of Exponent, Bounds, and Constructions

Satish Babu Korada, Eren Şaşoğlu, Rudiger L. Urbanke

Published Jan 5, 2009• 30 citations
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Abstract

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 $2 \times 2$ matrix $\bigl[ 1 &0 1& 1 \bigr]$. It was shown by Arıkan and Telatar that this construction achieves an error exponent of $\frac12$, 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 $\frac12$. We first show that any $\ell \times \ell$ 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 $\frac12$. Further, we give a general construction based on BCH codes which for large $n$ achieves exponents arbitrarily close to 1 and which exceeds $\frac12$ for size 16.

DOI
10.48550/arxiv.0901.0536

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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.
Polar codes, introduced by Arıkan, are linear block codes that can achieve the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. Polar codes of block length N are constructed by channel polarization method, which consists of channel combining and splitting operations to obtain N polarized subchannels from N copies of binary-input discrete memoryless channels. As N grows, symmetric channel capacities of the polarized subchannels converge to either 0 or 1. Polar codes are also close cousins of Reed-Muller codes and start to differ from each other for N≥32. Encoding and decoding of polar or Reed-Muller codes can be performed by using a factor graph, obtained from the n-th Kronecker product G=F^(⊗n) of F=[(1&0@1&1)] with N=2^n. Such a factor graph contains n=log⁡N stages; hence, by changing the order of stages with respect to each other, n! different factor graphs can be obtained. In the literature, some decoders using multiple factor graphs instead of a single factor graph are suggested. Therefore, it is of interest whether i) the K×N generator matrix of the code chosen by K active bits at the input of the encoder, and ii) the sum of the capacities of the K active channels that connect each input bit to the output vector of the encoder are invariant under stage permutations. In this study, we give an alternative proof of the fact that the answer to the first question is positive. It is also shown that the sum of the capacities of the K active channels is not invariant under stage permutations. Belief Propagation decoding performances on single and multiple factor graph decoders of polar and Reed-Muller codes over binary erasure channels are evaluated and compared. For multiple factor graph decoders, practical choice of factor graph sets that gives the best performance with low complexity is examined.
Polar codes, introduced by Arıkan, achieve symmetric capacity of any discrete memoryless channels under low encoding and decoding complexity. Recently, non-binary polar codes have been investigated. In this paper, we calculate error probability of non-binary polar codes constructed on the basis of Reed-Solomon matrices by numerical simulations. It is confirmed that 4-ary polar codes have significantly better performance than binary polar codes on binary-input AWGN channel. We also discuss an interpretation of polar codes in terms of algebraic geometry codes, and further show that polar codes using Hermitian codes have asymptotically good performance.
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.
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.
Polar codes were introduced by Arikan in 2008 and are the first family of error-correcting codes achieving the symmetric capacity of an arbitrary binary-input discrete memoryless channel under low complexity encoding and using an efficient successive cancellation decoding strategy. Recently, non-binary polar codes have been studied, in which one can use different algebraic geometry codes to achieve better error decoding probability. In this paper, we study the performance of binary polar codes that are obtained from non-binary algebraic geometry codes using concatenation. For binary polar codes (i.e. binary kernels) of a given length nnn, we compare numerically the use of short algebraic geometry codes over large fields versus long algebraic geometry codes over small fields. We find that for each nnn there is an optimal choice. For binary kernels of size up to n≤1,800n \leq 1,800n≤1,800 a concatenated Reed-Solomon code outperforms other choices. For larger kernel sizes concatenated Hermitian codes or Suzuki codes will do better.
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.