On the scaling of polar codes: II. The behavior of un-polarized channels

Hassani,; Alishahi,; Urbanke, R.

doi:10.1109/isit.2010.5513585

Cited by 19 publications

(13 citation statements)

References 4 publications

(13 reference statements)

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“…Currently no rigorous statements regarding this convergence for the general case are known. But "calculations" suggest that, for a fixed desired error probability, the required blocklength scales like 1/δ µ , where δ is the additive gap to capacity and where µ depends on the channel and has a value around 4, [51], [52]. Note that random block codes under MAP decoding have a similar scaling behavior but with µ = 2.…”

Section: A Historical Perspectivementioning

confidence: 99%

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

Kudekar

Richardson

Urbanke

2013

IEEE Trans. Inform. Theory

357

318

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Abstract-We investigate spatially coupled code ensembles. For transmission over the binary erasure channel, it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a-priori threshold of the underlying component ensemble. This explains why convolutional LDPC ensembles, originally introduced by Felström and Zigangirov, perform so well over this channel.We show that the equivalent result holds true for transmission over general binary-input memoryless output-symmetric channels. More precisely, given a desired error probability and a gap to capacity, we can construct a spatially coupled ensemble which fulfills these constraints universally on this class of channels under belief propagation decoding. In fact, most codes in that ensemble have that property. The quantifier universal refers to the single ensemble/code which is good for all channels but we assume that the channel is known at the receiver.The key technical result is a proof that under belief propagation decoding spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble.We conclude by discussing some interesting open problems.

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Section: A Historical Perspectivementioning

confidence: 99%

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

Kudekar

Richardson

Urbanke

2013

IEEE Trans. Inform. Theory

357

318

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“…200021 − 121903 of the Swiss National Science Foundation. This paper was presented in part in [12], [13]. S. H. Hassani is with the Department of Computer Science, ETHZ, 8092 Zurich, Switzerland (e-mail: hamed@inf.ethz.ch).…”

Section: Introductionmentioning

confidence: 99%

Finite-Length Scaling for Polar Codes

Hassani

Alishahi

Urbanke

2014

IEEE Trans. Inform. Theory

Self Cite

135

163

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Consider a binary-input memoryless outputsymmetric channel W . Such a channel has a capacity, call it I(W ), and for any R < I(W ) and strictly positive constant Pe we know that we can construct a coding scheme that allows transmission at rate R with an error probability not exceeding Pe. 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 Pe. We refer to this as the "finitelength 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 binaryinput 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 sub-channels whose Bhattacharyya constant falls in a fixed interval [a, b]. Mathematically, this can be stated as bounding the sequence 1 n log Pr (Zn ∈ [a, b]) n∈N , where Zn is the Bhattacharyya process. We then use these bounds to derive trade-offs 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 sub-channels chosen (by the polar coding scheme) to transmit information. If we require this sum to be smaller than a given value Pe > 0, then the required block-length N scales in terms of the rate R < I(W ) as N ≥ α (I(W )−R) µ , where α is a positive constant that depends on Pe and I(W ). We show that µ = 3.579 is a valid choice, and we conjecture that indeed the value of µ can be improved to µ = 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 ≤ β (I(W )−R) µ , where β is a constant that depends on Pe and I(W ), and µ = 6.

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“…Proof: See Appendix E. According to [15] and [16], the quality of a subchannel W (j) N depends heavily on the first few least significant bits of the binary expansion of j − 1. Now, recalling the process of locating node B in Fig.…”

Section: B Sc Decoding Performance After U N −K1+1mentioning

confidence: 99%

A Split-Reduced Successive Cancellation List Decoder for Polar Codes

Zhang

Wang

et al. 2016

IEEE J. Select. Areas Commun.

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Abstract-This paper focuses on low complexity successive cancellation list (SCL) decoding of polar codes. In particular, using the fact that splitting may be unnecessary when the reliability of decoding the unfrozen bit is sufficiently high, a novel splitting rule is proposed. Based on this rule, it is conjectured that, if the correct path survives at some stage, it tends to survive till termination without splitting with high probability. On the other hand, the incorrect paths are more likely to split at the following stages. Motivated by these observations, a simple counter that counts the successive number of stages without splitting is introduced for each decoding path to facilitate the identification of correct and incorrect path. Specifically, any path with counter value larger than a predefined threshold ω is deemed to be the correct path, which will survive at the decoding stage, while other paths with counter value smaller than the threshold will be pruned, thereby reducing the decoding complexity. Furthermore, it is proved that there exists a unique unfrozen bit uN−K 1 +1, after which the successive cancellation decoder achieves the same error performance as the maximum likelihood decoder if all the prior unfrozen bits are correctly decoded, which enables further complexity reduction. Simulation results demonstrate that the proposed low complexity SCL decoder attains performance similar to that of the conventional SCL decoder, while achieving substantial complexity reduction.

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On the scaling of polar codes: II. The behavior of un-polarized channels

Cited by 19 publications

References 4 publications

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation

Finite-Length Scaling for Polar Codes

A Split-Reduced Successive Cancellation List Decoder for Polar Codes

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