10 research outputs found
PAC Code Rate-Profile Design Using Search-Constrained Optimization Algorithms
In this paper, we introduce a novel rate-profile design based on
search-constrained optimization techniques to assess the performance of
polarization-adjusted convolutional (PAC) codes under Fano (sequential)
decoding. The results demonstrate that the resulting PAC code offers much
reduced computational complexity compared to a construction based on a
conventional genetic algorithm without a performance loss in error-correction
performance. As the fitness function of our algorithm, we propose an adaptive
successive cancellation list decoding algorithm to determine the weight
distribution of the rate profiles. The simulation results indicate that, for a
PAC(256, 128) code, only 8% of the population requires that their fitness
function be evaluated with a large list size. This represents an improvement of
almost 92% over a conventional evolutionary algorithm. For a PAC(64, 32) code,
this improvement is about 99%. We also plotted the performance of the high-rate
PAC(128, 105) and PAC(64, 51) codes, and the results show that they exhibit
superior performance compared to other algorithms
Polarization-Adjusted Convolutional (PAC) Codes as a Concatenation of Inner Cyclic and Outer Polar- and Reed-Muller-like Codes
Polarization-adjusted convolutional (PAC) codes are a new family of linear
block codes that can perform close to the theoretical bounds in the short
block-length regime. These codes combine polar coding and convolutional coding.
In this study, we show that PAC codes are equivalent to a new class of codes
consisting of inner cyclic codes and outer polar- and Reed-Muller-like codes.
We leverage the properties of cyclic codes to establish that PAC codes
outperform polar- and Reed-Muller-like codes in terms of minimum distance
On the Weight Spectrum Improvement of Pre-transformed Reed-Muller Codes and Polar Codes
Pre-transformation with an upper-triangular matrix (including cyclic
redundancy check (CRC), parity-check (PC) and polarization-adjusted
convolutional (PAC) codes) improves the weight spectrum of Reed-Muller (RM)
codes and polar codes significantly. However, a theoretical analysis to
quantify the improvement is missing. In this paper, we provide asymptotic
analysis on the number of low-weight codewords of the original and
pre-transformed RM codes respectively, and prove that pre-transformation
significantly reduces low-weight codewords, even in the order sense. For polar
codes, we prove that the average number of minimum-weight codewords does not
increase after pre-transformation. Both results confirm the advantages of
pre-transformation
On the Weight Distribution of Weights Less than in Polar Codes
The number of low-weight codewords is critical to the performance of
error-correcting codes. In 1970, Kasami and Tokura characterized the codewords
of Reed-Muller (RM) codes whose weights are less than , where
represents the minimum weight. In this paper, we extend their
results to decreasing polar codes. We present the closed-form expressions for
the number of codewords in decreasing polar codes with weights less than
. Moreover, the proposed enumeration algorithm runs in polynomial
time with respect to the code length
Error Coefficient-reduced Polar/PAC Codes
Polar codes are normally designed based on the reliability of the
sub-channels in the polarized vector channel. There are various methods with
diverse complexity and accuracy to evaluate the reliability of the
sub-channels. However, designing polar codes solely based on the sub-channel
reliability may result in poor Hamming distance properties. In this work, we
propose a different approach to design the information set for polar codes and
PAC codes where the objective is to reduce the number of codewords with minimum
weight (a.k.a. error coefficient) of a code designed for maximum reliability.
This approach is based on the coset-wise characterization of the rows of polar
transform involved in the formation of the minimum-weight
codewords. Our analysis capitalizes on the properties of the polar transform
based on its row and column indices. The numerical results show that the
designed codes outperform PAC codes and CRC-Polar codes at the practical block
error rate of . Furthermore, a by-product of the combinatorial
properties analyzed in this paper is an alternative enumeration method of the
minimum-weight codewords.Comment: 19 pages, 10 figures, 4 tables, 2 listing
PAC Codes: Sequential Decoding vs List Decoding
In the Shannon lecture at the 2019 International Symposium on Information
Theory (ISIT), Ar{\i}kan 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.Comment: 14 pages, 12 figures, 1 table, 6 algorithm