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.
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.
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], are adopted. Exemplary constructions with block lengthsN∈{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.
Successive-cancellation list (SCL) decoding is a widely used and studied decoding algorithm for polar codes. For short blocklengths, empirical evidence shows that SCL decoding with moderate list sizes (say,L≤32) closely matches the performance of maximum-likelihood (ML) decoding. Hashemi et al. proved that on the binary erasure channel (BEC), SCL decoding actually coincides with ML decoding for list sizesL≥2γ, whereγis a new parameter we call the mixing factor. Loosely speaking, the mixing factor counts the number of information bits mixed-in among the frozen bits; more preciselyγ=∣{i∈Fc:i≤max{F}}∣, whereF⊂[n]denotes the set of frozen indices. Herein, we extend the aforementioned result of Hashemi et al. from the BEC to arbitrary binary-input memoryless symmetric channels. Our proof is based on capturing all2γdecoding paths that correspond to theγinformation bits appearing before the last frozen bit, and then finding the most-likely extension for each of these paths efficiently using a nearest coset decoding algorithm introduced herein. Furthermore, we present a hybrid successive-cancellation list (H -SCL) decoding algorithm, which is a hybrid between conventional SCL decoding and nearest coset decoding. We believe that the hybrid algorithm can outperform the conventional SCL decoder with lower decoding complexity.
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 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 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.
Successive cancellation list (SCL) decoding has been widely adopted for polar codes, which allows near maximum likelihood performance with sufficiently large list size. In this work, we show that, if the list size is 2γ, where γ is the fundamental quantity called mixing factor, then a modification to SCL decoding can implement Forney’s generalized decoding rule. Hence, it provides an efficient means to discard unreliable decisions. The performance achieved by short polar codes under the proposed generalized SCL decoding is analyzed via Monte Carlo simulations.
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.
A product code with single parity-check component codes can be described via the tools of a multi-kernel polar code, where the rows of the generator matrix are chosen according to the constraints imposed by the product code construction. Following this observation, successive cancellation decoding of such codes is introduced. In particular, the error probability of single parity-check product codes over binary memoryless symmetric channels under successive cancellation decoding is characterized. A bridge with the analysis of product codes introduced by Elias is also established for the binary erasure channel. Successive cancellation list decoding of single parity-check product codes is then described. For the provided example, simulations over the binary input additive white Gaussian channel show that successive cancellation list decoding outperforms belief propagation decoding applied to the code graph. Finally, the performance of the concatenation of a product code with a high-rate outer code is investigated via distance spectrum analysis. Examples of concatenations performing within 0.7 dB from the random coding union bound are provided.