19 research outputs found
Spatially-Coupled MacKay-Neal Codes and Hsu-Anastasopoulos Codes
Kudekar et al. recently proved that for transmission over the binary erasure
channel (BEC), spatial coupling of LDPC codes increases the BP threshold of the
coupled ensemble to the MAP threshold of the underlying LDPC codes. One major
drawback of the capacity-achieving spatially-coupled LDPC codes is that one
needs to increase the column and row weight of parity-check matrices of the
underlying LDPC codes.
It is proved, that Hsu-Anastasopoulos (HA) codes and MacKay-Neal (MN) codes
achieve the capacity of memoryless binary-input symmetric-output channels under
MAP decoding with bounded column and row weight of the parity-check matrices.
The HA codes and the MN codes are dual codes each other.
The aim of this paper is to present an empirical evidence that
spatially-coupled MN (resp. HA) codes with bounded column and row weight
achieve the capacity of the BEC. To this end, we introduce a spatial coupling
scheme of MN (resp. HA) codes. By density evolution analysis, we will show that
the resulting spatially-coupled MN (resp. HA) codes have the BP threshold close
to the Shannon limit.Comment: Corrected typos in degree distributions \nu and \mu of MN and HA
code
Spatially-Coupled Precoded Rateless Codes
Raptor codes are rateless codes that achieve the capacity on the binary
erasure channels. However the maximum degree of optimal output degree
distribution is unbounded. This leads to a computational complexity problem
both at encoders and decoders. Aref and Urbanke investigated the potential
advantage of universal achieving-capacity property of proposed
spatially-coupled (SC) low-density generator matrix (LDGM) codes. However the
decoding error probability of SC-LDGM codes is bounded away from 0. In this
paper, we investigate SC-LDGM codes concatenated with SC low-density
parity-check codes. The proposed codes can be regarded as SC Hsu-Anastasopoulos
rateless codes. We derive a lower bound of the asymptotic overhead from
stability analysis for successful decoding by density evolution. The numerical
calculation reveals that the lower bound is tight. We observe that with a
sufficiently large number of information bits, the asymptotic overhead and the
decoding error rate approach 0 with bounded maximum degree
Spatially-Coupled Precoded Rateless Codes with Bounded Degree Achieve the Capacity of BEC under BP decoding
Raptor codes are known as precoded rateless codes that achieve the capacity
of BEC. However the maximum degree of Raptor codes needs to be unbounded to
achieve the capacity. In this paper, we prove that spatially-coupled precoded
rateless codes achieve the capacity with bounded degree under BP decoding
Spatially-Coupled MacKay-Neal Codes with No Bit Nodes of Degree Two Achieve the Capacity of BEC
Obata et al. proved that spatially-coupled (SC) MacKay-Neal (MN) codes
achieve the capacity of BEC. However, the SC-MN codes codes have many variable
nodes of degree two and have higher error floors. In this paper, we prove that
SC-MN codes with no variable nodes of degree two achieve the capacity of BEC
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