11 research outputs found
Reed-Muller codes polarize
Reed-Muller (RM) codes and polar codes are generated by the same matrix
but using different subset of rows. RM codes select simply rows having largest
weights. Polar codes select instead rows having the largest conditional mutual
information proceeding top to down in ; while this is a more elaborate and
channel-dependent rule, the top-to-down ordering has the advantage of making
the conditional mutual information polarize, giving directly a
capacity-achieving code on any binary memoryless symmetric channel (BMSC). RM
codes are yet to be proved to have such property.
In this paper, we reconnect RM codes to polarization theory. It is shown that
proceeding in the RM code ordering, i.e., not top-to-down but from the lightest
to the heaviest rows in , the conditional mutual information again
polarizes. We further demonstrate that it does so faster than for polar codes.
This implies that contains another code, different than the polar code
and called here the twin code, that is provably capacity-achieving on any BMSC.
This proves a necessary condition for RM codes to achieve capacity on BMSCs. It
further gives a sufficient condition if the rows with the largest conditional
mutual information correspond to the heaviest rows, i.e., if the twin code is
the RM code. We show here that the two codes bare similarity with each other
and give further evidence that they are likely the same
Multi-Factor Pruning for Recursive Projection-Aggregation Decoding of RM Codes
The recently introduced recursive projection aggregation (RPA) decoding
method for Reed-Muller (RM) codes can achieve near-maximum likelihood (ML)
decoding performance. However, its high computational complexity makes its
implementation challenging for time- and resource-critical applications. In
this work, we present a complexity reduction technique called multi-factor
pruning that reduces the computational complexity of RPA significantly. Our
simulation results show that the proposed pruning approach with appropriately
selected factors can reduce the complexity of RPA by up to for
while keeping the comparable error-correcting performance
Pipelined Architecture for Soft-decision Iterative Projection Aggregation Decoding for RM Codes
The recently proposed recursive projection-aggregation (RPA) decoding
algorithm for Reed-Muller codes has received significant attention as it
provides near-ML decoding performance at reasonable complexity for short codes.
However, its complicated structure makes it unsuitable for hardware
implementation. Iterative projection-aggregation (IPA) decoding is a modified
version of RPA decoding that simplifies the hardware implementation. In this
work, we present a flexible hardware architecture for the IPA decoder that can
be configured from fully-sequential to fully-parallel, thus making it suitable
for a wide range of applications with different constraints and resource
budgets. Our simulation and implementation results show that the IPA decoder
has 41% lower area consumption, 44% lower latency, four times higher
throughput, but currently seven times higher power consumption for a code with
block length of 128 and information length of 29 compared to a state-of-the-art
polar successive cancellation list (SCL) decoder with comparable decoding
performance
Recursive projection-aggregation decoding of Reed-Muller codes
We propose a new class of efficient decoding algorithms for Reed-Muller (RM)
codes over binary-input memoryless channels. The algorithms are based on
projecting the code on its cosets, recursively decoding the projected codes
(which are lower-order RM codes), and aggregating the reconstructions (e.g.,
using majority votes). We further provide extensions of the algorithms using
list-decoding.
We run our algorithm for AWGN channels and Binary Symmetric Channels at the
short code length () regime for a wide range of code rates.
Simulation results show that in both low code rate and high code rate regimes,
the new algorithm outperforms the widely used decoder for polar codes (SCL+CRC)
with the same parameters. The performance of the new algorithm for RM codes in
those regimes is in fact close to that of the maximal likelihood decoder.
Finally, the new decoder naturally allows for parallel implementations
Descodificação através de Machine Learning
In recent years, machine learning has become one of the most rapidly expanding
technologies in a variety of technological fields. In general, it allows a computer
to learn from data without being expressly designed for a particular purpose. This
thesis investigates the application of decoding methods inspired by machine learning
to linear block codes, such as Reed-Muller (RM) codes.Recentemente, o Machine Learning tornou-se uma das tecnologias em mais rápida
expansão numa variedade de campos tecnológicos. Em geral, permite que um
computador aprenda com os dados sem ser expressamente concebido para um fim
específico. Esta dissertação investiga a aplicação de métodos de descodificação
inspirados no Machine Learning a códigos de blocos lineares, tais como os códigos
de Reed-Muller
A proof that Reed-Muller codes achieve Shannon capacity on symmetric channels
Reed-Muller codes were introduced in 1954, with a simple explicit
construction based on polynomial evaluations, and have long been conjectured to
achieve Shannon capacity on symmetric channels. Major progress was made towards
a proof over the last decades; using combinatorial weight enumerator bounds, a
breakthrough on the erasure channel from sharp thresholds, hypercontractivity
arguments, and polarization theory. Another major progress recently established
that the bit error probability vanishes slowly below capacity. However, when
channels allow for errors, the results of Bourgain-Kalai do not apply for
converting a vanishing bit to a vanishing block error probability, neither do
the known weight enumerator bounds. The conjecture that RM codes achieve
Shannon capacity on symmetric channels, with high probability of recovering the
codewords, has thus remained open.
This paper closes the conjecture's proof. It uses a new recursive boosting
framework, which aggregates the decoding of codeword restrictions on
`subspace-sunflowers', handling their dependencies via an Boolean Fourier
analysis, and using a list-decoding argument with a weight enumerator bound
from Sberlo-Shpilka. The proof does not require a vanishing bit error
probability for the base case, but only a non-trivial probability, obtained
here for general symmetric codes. This gives in particular a shortened and
tightened argument for the vanishing bit error probability result of
Reeves-Pfister, and with prior works, it implies the strong wire-tap secrecy of
RM codes on pure-state classical-quantum channels
Partially Coupled Codes for TB-based Transmission
In this thesis, we mainly investigate the design of partially coupled codes for transport block (TB) based transmission protocol adopted in 4G/5G mobile network standards. In this protocol, an information sequence in a TB is segmented into multiple code blocks (CBs) and each CB is protected by a channel codeword independently. It is inefficient in terms of transmit power and spectrum efficiency because any erroneous CB in a TB leads to the retransmission of the whole TB. An important research problem related to this TB-based transmission is how to improve the TB error rate (TBER) performance so that the number of retransmissions reduces.
To tackle this challenge, we present a class of spatial coupling techniques called partial coupling in the TB encoding operation, which has two subclasses: partial information coupled (PIC) and partial parity coupling (PPC). To be specific, the coupling is performed such that a fraction of the information/parity sequence of one component code at the current CB is used as the input of the component encoder at the next CB, leading to improved TBER performance. One of the appealing features of partial coupling (both PIC and PPC) is that the coupling can be applied to any component codes without changing their encoding and decoding architectures, making them compatible with the TB-based transmission protocol.
The main body of this thesis consists of two parts. In the first part, we apply both PIC and PPC to turbo codes. We investigate various coupling designs and analysis the performance of the partially coupled turbo codes over the binary erasure channel via density evolution (DE). Both simulation results and DE analysis show that such a class of codes can approach channel capacity with a large blocklength. In the second part, we construct PIC-polar codes. We show that PIC can effectively improve the error performance of finite-length polar codes by utilizing the channel polarization phenomenon. The DE-based performance analysis is also conducted. For both turbo codes and polar codes, we have shown that the partially coupled codes have significant performance gain over their uncoupled counterpart, demonstrating the effectiveness of the partial coupling