297 research outputs found
Construction of Polar Codes with Sublinear Complexity
Consider the problem of constructing a polar code of block length for the
transmission over a given channel . Typically this requires to compute the
reliability of all the synthetic channels and then to include those that
are sufficiently reliable. However, we know from [1], [2] that there is a
partial order among the synthetic channels. Hence, it is natural to ask whether
we can exploit it to reduce the computational burden of the construction
problem.
We show that, if we take advantage of the partial order [1], [2], we can
construct a polar code by computing the reliability of roughly a fraction
of the synthetic channels. In particular, we prove that
is a lower bound on the number of synthetic channels to be
considered and such a bound is tight up to a multiplicative factor . This set of roughly synthetic channels is universal, in
the sense that it allows one to construct polar codes for any , and it can
be identified by solving a maximum matching problem on a bipartite graph.
Our proof technique consists of reducing the construction problem to the
problem of computing the maximum cardinality of an antichain for a suitable
partially ordered set. As such, this method is general and it can be used to
further improve the complexity of the construction problem in case a new
partial order on the synthetic channels of polar codes is discovered.Comment: 9 pages, 3 figures, presented at ISIT'17 and submitted to IEEE Trans.
Inform. Theor
Partitioned List Decoding of Polar Codes: Analysis and Improvement of Finite Length Performance
Polar codes represent one of the major recent breakthroughs in coding theory
and, because of their attractive features, they have been selected for the
incoming 5G standard. As such, a lot of attention has been devoted to the
development of decoding algorithms with good error performance and efficient
hardware implementation. One of the leading candidates in this regard is
represented by successive-cancellation list (SCL) decoding. However, its
hardware implementation requires a large amount of memory. Recently, a
partitioned SCL (PSCL) decoder has been proposed to significantly reduce the
memory consumption. In this paper, we examine the paradigm of PSCL decoding
from both theoretical and practical standpoints: (i) by changing the
construction of the code, we are able to improve the performance at no
additional computational, latency or memory cost, (ii) we present an optimal
scheme to allocate cyclic redundancy checks (CRCs), and (iii) we provide an
upper bound on the list size that allows MAP performance.Comment: 2017 IEEE Global Communications Conference (GLOBECOM
Sublinear Latency for Simplified Successive Cancellation Decoding of Polar Codes
This work analyzes the latency of the simplified successive cancellation
(SSC) decoding scheme for polar codes proposed by Alamdar-Yazdi and
Kschischang. It is shown that, unlike conventional successive cancellation
decoding, where latency is linear in the block length, the latency of SSC
decoding is sublinear. More specifically, the latency of SSC decoding is
, where is the block length and is the scaling
exponent of the channel, which captures the speed of convergence of the rate to
capacity. Numerical results demonstrate the tightness of the bound and show
that most of the latency reduction arises from the parallel decoding of
subcodes of rate or .Comment: 20 pages, 6 figures, presented in part at ISIT 2020 and accepted in
IEEE Transactions on Wireless Communication
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
Rate-Flexible Fast Polar Decoders
Polar codes have gained extensive attention during the past few years and
recently they have been selected for the next generation of wireless
communications standards (5G). Successive-cancellation-based (SC-based)
decoders, such as SC list (SCL) and SC flip (SCF), provide a reasonable error
performance for polar codes at the cost of low decoding speed. Fast SC-based
decoders, such as Fast-SSC, Fast-SSCL, and Fast-SSCF, identify the special
constituent codes in a polar code graph off-line, produce a list of operations,
store the list in memory, and feed the list to the decoder to decode the
constituent codes in order efficiently, thus increasing the decoding speed.
However, the list of operations is dependent on the code rate and as the rate
changes, a new list is produced, making fast SC-based decoders not
rate-flexible. In this paper, we propose a completely rate-flexible fast
SC-based decoder by creating the list of operations directly in hardware, with
low implementation complexity. We further propose a hardware architecture
implementing the proposed method and show that the area occupation of the
rate-flexible fast SC-based decoder in this paper is only of the total
area of the memory-based base-line decoder when 5G code rates are supported
Parallelism versus Latency in Simplified Successive-Cancellation Decoding of Polar Codes
This paper characterizes the latency of the simplified
successive-cancellation (SSC) decoding scheme for polar codes under hardware
resource constraints. In particular, when the number of processing elements
that can perform SSC decoding operations in parallel is limited, as is the case
in practice, the latency of SSC decoding is
, where is
the block length of the code and is the scaling exponent of the channel.
Three direct consequences of this bound are presented. First, in a
fully-parallel implementation where , the latency of SSC
decoding is , which is sublinear in the block
length. This recovers a result from our earlier work. Second, in a fully-serial
implementation where , the latency of SSC decoding scales as
. The multiplicative constant is also
calculated: we show that the latency of SSC decoding when is given by
. Third, in a semi-parallel
implementation, the smallest that gives the same latency as that of the
fully-parallel implementation is . The tightness of our bound on
SSC decoding latency and the applicability of the foregoing results is
validated through extensive simulations
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