List Viterbi Decoding of PAC Codes

Rowshan, Mohammad; Viterbo, Emanuele

doi:10.36227/techrxiv.12619706.v1

“…Finally, Viterbi algorithm (VA) [32] with similar hardware resources as list decoding (except the 2L-value sorter, replaced by a 2-value comparator) provides a close performance to Fano and list decoders.…”

Section: Fano Decodingmentioning

confidence: 99%

PAC Codes: Sequential Decoding vs List Decoding

Rowshan

¹

,

Burg

²

,

Viterbo

³

2023

Preprint

Self Cite

1

0

View full text Add to dashboard Cite

In the Shannon lecture at the 2019 International Symposium on Information Theory (ISIT), Arıkan proposed to employ a one-to-one convolutional transform as a pre-coding step before the polar transform. The resulting codes of this concatenation are called polarization-adjusted convolutional (PAC) codes. In this scheme, a pair of polar mapper and demapper as pre- and postprocessing devices are deployed around a memoryless channel, which provides polarized information to an outer decoder leading to improved error correction performance of the outer code. In this paper, the list decoding and sequential decoding (including Fano decoding and stack decoding) are first adapted for use to decode PAC codes. Then, to reduce the complexity of sequential decoding of PAC/polar codes, we propose (i) an adaptive heuristic metric, (ii) tree search constraints for backtracking to avoid exploration of unlikely sub-paths, and (iii) tree search strategies consistent with the pattern of error occurrence in polar codes. These contribute to the reduction of the average decoding time complexity from 50% to 80%, trading with 0.05 to 0.3 dB degradation in error correction performance within FER=10^-3 range, respectively, relative to not applying the corresponding search strategies. Additionally, as an important ingredient in Fano decoding of PAC/polar codes, an efficient computation method for the intermediate LLRs and partial sums is provided. This method is effective in backtracking and avoids storing the intermediate information or restarting the decoding process. Eventually, all three decoding algorithms are compared in terms of performance, complexity, and resource requirements.

show abstract

“…However, if we allow the impulse response to vary with time, then essentially the same performance can be achieved with constraint length as low as

(which is the minimum possible, since

by assumption). This observation is of importance if trellis methods (such as list-Viterbi decoding, as suggested in [ 2 , 49 ]) are used to decode PAC codes. Indeed, reducing the constraint length from

to

reduces the number of states in the resulting trellis from 64 to 4, respectively.…”

Section: Pac Codes With Random Time-varying Convolutional Precodingmentioning

confidence: 99%

“…While these papers investigate various aspects of PAC codes, none of them considers list decoding thereof. Finally, we note the work of [ 48 , 49 ], which investigates both Fano decoding and list decoding of PAC codes. This work is apparently independent from and contemporaneous with our results herein.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

List Decoding of Arıkan’s PAC Codes

Yao

¹

,

Vardy

²

2021

Entropy

View full text Add to dashboard Cite

Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short blocklengths under standard successive cancellation decoding is far from optimal. A well-known way to improve the performance of polar codes at short blocklengths is CRC precoding followed by successive-cancellation list decoding. This approach, along with various refinements thereof, has largely remained the state of the art in polar coding since it was introduced in 2011. Recently, Arıkan presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. At short blocklengths, such codes offer a dramatic improvement in performance as compared to CRC-aided list decoding of conventional polar codes. PAC codes are based primarily upon the following main ideas: replacing CRC codes with convolutional precoding (under appropriate rate profiling) and replacing list decoding by sequential decoding. One of our primary goals in this paper is to answer the following question: is sequential decoding essential for the superior performance of PAC codes? We show that similar performance can be achieved using list decoding when the list size L is moderately large (say, L⩾128). List decoding has distinct advantages over sequential decoding in certain scenarios, such as low-SNR regimes or situations where the worst-case complexity/latency is the primary constraint. Another objective is to provide some insights into the remarkable performance of PAC codes. We first observe that both sequential decoding and list decoding of PAC codes closely match ML decoding thereof. We then estimate the number of low weight codewords in PAC codes, and use these estimates to approximate the union bound on their performance. These results indicate that PAC codes are superior to both polar codes and Reed–Muller codes. We also consider random time-varying convolutional precoding for PAC codes, and observe that this scheme achieves the same superior performance with constraint length as low as ν=2.

show abstract

“…The Viterbi algorithm is a maximum likelihood decoding method that examines the entire state space of the encoder at each step. We have studied Viterbi decoding of PAC codes in [32].…”

Section: B Convolutional Codes and Fano Decodingmentioning

confidence: 99%

Polarization-adjusted Convolutional (PAC) Codes: Sequential Decoding vs List Decoding

Rowshan¹,

Burg²,

Viterbo³

2021

Preprint

Self Cite

3

0

View full text Add to dashboard Cite

In the Shannon lecture at the 2019 International Symposium on Information Theory (ISIT), Arıkan proposed to employ a one-to-one convolutional transform as a pre-coding step before the polar transform. The resulting codes of this concatenation are called polarization-adjusted convolutional (PAC) codes. In this scheme, a pair of polar mapper and demapper as pre- and postprocessing devices are deployed around a memoryless channel, which provides polarized information to an outer decoder leading to improved error correction performance of the outer code. In this paper, the list decoding and sequential decoding (including Fano decoding and stack decoding) are first adapted for use to decode PAC codes. Then, to reduce the complexity of sequential decoding of PAC/polar codes, we propose (i) an adaptive heuristic metric, (ii) tree search constraints for backtracking to avoid exploration of unlikely sub-paths, and (iii) tree search strategies consistent with the pattern of error occurrence in polar codes. These contribute to the reduction of the average decoding time complexity from 50% to 80%, trading with 0.05 to 0.3 dB degradation in error correction performance within FER=10^-3 range, respectively, relative to not applying the corresponding search strategies. Additionally, as an important ingredient in Fano decoding of PAC/polar codes, an efficient computation method for the intermediate LLRs and partial sums is provided. This method is effective in backtracking and avoids storing the intermediate information or restarting the decoding process. Eventually, all three decoding algorithms are compared in terms of performance, complexity, and resource requirements.

show abstract

List Viterbi Decoding of PAC Codes

Cited by 7 publications

References 14 publications

PAC Codes: Sequential Decoding vs List Decoding

PAC Codes: Sequential Decoding vs List Decoding

List Decoding of Arıkan’s PAC Codes

Polarization-adjusted Convolutional (PAC) Codes: Sequential Decoding vs List Decoding

Contact Info

Product

Resources

About