6 research outputs found
Threshold Analysis of Non-Binary Spatially-Coupled LDPC Codes with Windowed Decoding
In this paper we study the iterative decoding threshold performance of
non-binary spatially-coupled low-density parity-check (NB-SC-LDPC) code
ensembles for both the binary erasure channel (BEC) and the binary-input
additive white Gaussian noise channel (BIAWGNC), with particular emphasis on
windowed decoding (WD). We consider both (2,4)-regular and (3,6)-regular
NB-SC-LDPC code ensembles constructed using protographs and compute their
thresholds using protograph versions of NB density evolution and NB extrinsic
information transfer analysis. For these code ensembles, we show that WD of
NB-SC-LDPC codes, which provides a significant decrease in latency and
complexity compared to decoding across the entire parity-check matrix, results
in a negligible decrease in the near-capacity performance for a sufficiently
large window size W on both the BEC and the BIAWGNC. Also, we show that
NB-SC-LDPC code ensembles exhibit gains in the WD threshold compared to the
corresponding block code ensembles decoded across the entire parity-check
matrix, and that the gains increase as the finite field size q increases.
Moreover, from the viewpoint of decoding complexity, we see that (3,6)-regular
NB-SC-LDPC codes are particularly attractive due to the fact that they achieve
near-capacity thresholds even for small q and W.Comment: 6 pages, 8 figures; submitted to 2014 IEEE International Symposium on
Information Theor
Proving Threshold Saturation for Nonbinary SC-LDPC Codes on the Binary Erasure Channel
We analyze nonbinary spatially-coupled low-density parity-check (SC-LDPC)
codes built on the general linear group for transmission over the binary
erasure channel. We prove threshold saturation of the belief propagation
decoding to the potential threshold, by generalizing the proof technique based
on potential functions recently introduced by Yedla et al.. The existence of
the potential function is also discussed for a vector sparse system in the
general case, and some existence conditions are developed. We finally give
density evolution and simulation results for several nonbinary SC-LDPC code
ensembles.Comment: in Proc. 2014 XXXIth URSI General Assembly and Scientific Symposium,
URSI GASS, Beijing, China, August 16-23, 2014. Invited pape
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
From Polar to Reed-Muller Codes:Unified Scaling, Non-standard Channels, and a Proven Conjecture
The year 2016, in which I am writing these words, marks the centenary of Claude Shannon, the father of information theory. In his landmark 1948 paper "A Mathematical Theory of Communication", Shannon established the largest rate at which reliable communication is possible, and he referred to it as the channel capacity. Since then, researchers have focused on the design of practical coding schemes that could approach such a limit. The road to channel capacity has been almost 70 years long and, after many ideas, occasional detours, and some rediscoveries, it has culminated in the description of low-complexity and provably capacity-achieving coding schemes, namely, polar codes and iterative codes based on sparse graphs. However, next-generation communication systems require an unprecedented performance improvement and the number of transmission settings relevant in applications is rapidly increasing. Hence, although Shannon's limit seems finally close at hand, new challenges are just around the corner. In this thesis, we trace a road that goes from polar to Reed-Muller codes and, by doing so, we investigate three main topics: unified scaling, non-standard channels, and capacity via symmetry. First, we consider unified scaling. A coding scheme is capacity-achieving when, for any rate smaller than capacity, the error probability tends to 0 as the block length becomes increasingly larger. However, the practitioner is often interested in more specific questions such as, "How much do we need to increase the block length in order to halve the gap between rate and capacity?". We focus our analysis on polar codes and develop a unified framework to rigorously analyze the scaling of the main parameters, i.e., block length, rate, error probability, and channel quality. Furthermore, in light of the recent success of a list decoding algorithm for polar codes, we provide scaling results on the performance of list decoders. Next, we deal with non-standard channels. When we say that a coding scheme achieves capacity, we typically consider binary memoryless symmetric channels. However, practical transmission scenarios often involve more complicated settings. For example, the downlink of a cellular system is modeled as a broadcast channel, and the communication on fiber links is inherently asymmetric. We propose provably optimal low-complexity solutions for these settings. In particular, we present a polar coding scheme that achieves the best known rate region for the broadcast channel, and we describe three paradigms to achieve the capacity of asymmetric channels. To do so, we develop general coding "primitives", such as the chaining construction that has already proved to be useful in a variety of communication problems. Finally, we show how to achieve capacity via symmetry. In the early days of coding theory, a popular paradigm consisted in exploiting the structure of algebraic codes to devise practical decoding algorithms. However, proving the optimality of such coding schemes remained an elusive goal. In particular, the conjecture that Reed-Muller codes achieve capacity dates back to the 1960s. We solve this open problem by showing that Reed-Muller codes and, in general, codes with sufficient symmetry are capacity-achieving over erasure channels under optimal MAP decoding. As the proof does not rely on the precise structure of the codes, we are able to show that symmetry alone guarantees optimal performance