195 research outputs found
Belief Propagation Decoding of Polar Codes on Permuted Factor Graphs
We show that the performance of iterative belief propagation (BP) decoding of
polar codes can be enhanced by decoding over different carefully chosen factor
graph realizations. With a genie-aided stopping condition, it can achieve the
successive cancellation list (SCL) decoding performance which has already been
shown to achieve the maximum likelihood (ML) bound provided that the list size
is sufficiently large. The proposed decoder is based on different realizations
of the polar code factor graph with randomly permuted stages during decoding.
Additionally, a different way of visualizing the polar code factor graph is
presented, facilitating the analysis of the underlying factor graph and the
comparison of different graph permutations. In our proposed decoder, a high
rate Cyclic Redundancy Check (CRC) code is concatenated with a polar code and
used as an iteration stopping criterion (i.e., genie) to even outperform the
SCL decoder of the plain polar code (without the CRC-aid). Although our
permuted factor graph-based decoder does not outperform the SCL-CRC decoder, it
achieves, to the best of our knowledge, the best performance of all iterative
polar decoders presented thus far.Comment: in IEEE Wireless Commun. and Networking Conf. (WCNC), April 201
Improved Logical Error Rate via List Decoding of Quantum Polar Codes
The successive cancellation list decoder (SCL) is an efficient decoder for
classical polar codes with low decoding error, approximating the maximum
likelihood decoder (MLD) for small list sizes. Here we adapt the SCL to the
task of decoding quantum polar codes and show that it inherits the high
performance and low complexity of the classical case, and can approximate the
quantum MLD for certain channels. We apply SCL decoding to a novel version of
quantum polar codes based on the polarization weight (PW) method, which
entirely avoids the need for small amounts of entanglement assistance apparent
in previous quantum polar code constructions. When used to find the precise
error pattern, the quantum SCL decoder (SCL-E) shows competitive performance
with surface codes of similar size and low-density parity check codes of
similar size and rate. The SCL decoder may instead be used to approximate the
probability of each equivalence class of errors, and then choose the most
likely class. We benchmark this class-oriented decoder (SCL-C) against the
SCL-E decoder and find a noticeable improvement in the logical error rate. This
improvement stems from the fact that the contributions from just the low-weight
errors give a reasonable approximation to the error class probabilities. Both
SCL-E and SCL-C maintain the complexity O(LN logN) of SCL for code size N and
list size L. We also show that the list decoder can be used to gain insight
into the weight distribution of the codes and how this impacts the effect of
degenerate errors.Comment: 14 pages, 7 figure
How to Achieve the Capacity of Asymmetric Channels
We survey coding techniques that enable reliable transmission at rates that
approach the capacity of an arbitrary discrete memoryless channel. In
particular, we take the point of view of modern coding theory and discuss how
recent advances in coding for symmetric channels help provide more efficient
solutions for the asymmetric case. We consider, in more detail, three basic
coding paradigms.
The first one is Gallager's scheme that consists of concatenating a linear
code with a non-linear mapping so that the input distribution can be
appropriately shaped. We explicitly show that both polar codes and spatially
coupled codes can be employed in this scenario. Furthermore, we derive a
scaling law between the gap to capacity, the cardinality of the input and
output alphabets, and the required size of the mapper.
The second one is an integrated scheme in which the code is used both for
source coding, in order to create codewords distributed according to the
capacity-achieving input distribution, and for channel coding, in order to
provide error protection. Such a technique has been recently introduced by
Honda and Yamamoto in the context of polar codes, and we show how to apply it
also to the design of sparse graph codes.
The third paradigm is based on an idea of B\"ocherer and Mathar, and
separates the two tasks of source coding and channel coding by a chaining
construction that binds together several codewords. We present conditions for
the source code and the channel code, and we describe how to combine any source
code with any channel code that fulfill those conditions, in order to provide
capacity-achieving schemes for asymmetric channels. In particular, we show that
polar codes, spatially coupled codes, and homophonic codes are suitable as
basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published
in IEEE Trans. Inform. Theor
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