2,360 research outputs found
Capacity-Achieving Codes with Bounded Graphical Complexity on Noisy Channels
We introduce a new family of concatenated codes with an outer low-density
parity-check (LDPC) code and an inner low-density generator matrix (LDGM) code,
and prove that these codes can achieve capacity under any memoryless
binary-input output-symmetric (MBIOS) channel using maximum-likelihood (ML)
decoding with bounded graphical complexity, i.e., the number of edges per
information bit in their graphical representation is bounded. In particular, we
also show that these codes can achieve capacity on the binary erasure channel
(BEC) under belief propagation (BP) decoding with bounded decoding complexity
per information bit per iteration for all erasure probabilities in (0, 1). By
deriving and analyzing the average weight distribution (AWD) and the
corresponding asymptotic growth rate of these codes with a rate-1 inner LDGM
code, we also show that these codes achieve the Gilbert-Varshamov bound with
asymptotically high probability. This result can be attributed to the presence
of the inner rate-1 LDGM code, which is demonstrated to help eliminate high
weight codewords in the LDPC code while maintaining a vanishingly small amount
of low weight codewords.Comment: 17 pages, 2 figures. This paper is to be presented in the 43rd Annual
Allerton Conference on Communication, Control and Computing, Monticello, IL,
USA, Sept. 28-30, 200
On Universal Properties of Capacity-Approaching LDPC Ensembles
This paper is focused on the derivation of some universal properties of
capacity-approaching low-density parity-check (LDPC) code ensembles whose
transmission takes place over memoryless binary-input output-symmetric (MBIOS)
channels. Properties of the degree distributions, graphical complexity and the
number of fundamental cycles in the bipartite graphs are considered via the
derivation of information-theoretic bounds. These bounds are expressed in terms
of the target block/ bit error probability and the gap (in rate) to capacity.
Most of the bounds are general for any decoding algorithm, and some others are
proved under belief propagation (BP) decoding. Proving these bounds under a
certain decoding algorithm, validates them automatically also under any
sub-optimal decoding algorithm. A proper modification of these bounds makes
them universal for the set of all MBIOS channels which exhibit a given
capacity. Bounds on the degree distributions and graphical complexity apply to
finite-length LDPC codes and to the asymptotic case of an infinite block
length. The bounds are compared with capacity-approaching LDPC code ensembles
under BP decoding, and they are shown to be informative and are easy to
calculate. Finally, some interesting open problems are considered.Comment: Published in the IEEE Trans. on Information Theory, vol. 55, no. 7,
pp. 2956 - 2990, July 200
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
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