9 research outputs found
Spectral Shape of Check-Hybrid GLDPC Codes
This paper analyzes the asymptotic exponent of both the weight spectrum and
the stopping set size spectrum for a class of generalized low-density
parity-check (GLDPC) codes. Specifically, all variable nodes (VNs) are assumed
to have the same degree (regular VN set), while the check node (CN) set is
assumed to be composed of a mixture of different linear block codes (hybrid CN
set). A simple expression for the exponent (which is also referred to as the
growth rate or the spectral shape) is developed. This expression is consistent
with previous results, including the case where the normalized weight or
stopping set size tends to zero. Furthermore, it is shown how certain symmetry
properties of the local weight distribution at the CNs induce a symmetry in the
overall weight spectral shape function.Comment: 6 pages, 3 figures. Presented at the IEEE ICC 2010, Cape Town, South
Africa. A minor typo in equation (9) has been correcte
Rate-Equivocation Optimal Spatially Coupled LDPC Codes for the BEC Wiretap Channel
We consider transmission over a wiretap channel where both the main channel
and the wiretapper's channel are Binary Erasure Channels (BEC). We use
convolutional LDPC ensembles based on the coset encoding scheme. More
precisely, we consider regular two edge type convolutional LDPC ensembles. We
show that such a construction achieves the whole rate-equivocation region of
the BEC wiretap channel.
Convolutional LDPC ensemble were introduced by Felstr\"om and Zigangirov and
are known to have excellent thresholds. Recently, Kudekar, Richardson, and
Urbanke proved that the phenomenon of "Spatial Coupling" converts MAP threshold
into BP threshold for transmission over the BEC.
The phenomenon of spatial coupling has been observed to hold for general
binary memoryless symmetric channels. Hence, we conjecture that our
construction is a universal rate-equivocation achieving construction when the
main channel and wiretapper's channel are binary memoryless symmetric channels,
and the wiretapper's channel is degraded with respect to the main channel.Comment: Working pape
Stability of Iterative Decoding of Multi-Edge Type Doubly-Generalized LDPC Codes Over the BEC
Using the EXIT chart approach, a necessary and sufficient condition is
developed for the local stability of iterative decoding of multi-edge type
(MET) doubly-generalized low-density parity-check (D-GLDPC) code ensembles. In
such code ensembles, the use of arbitrary linear block codes as component codes
is combined with the further design of local Tanner graph connectivity through
the use of multiple edge types. The stability condition for these code
ensembles is shown to be succinctly described in terms of the value of the
spectral radius of an appropriately defined polynomial matrix.Comment: 6 pages, 3 figures. Presented at Globecom 2011, Houston, T
Performance Analysis and Design of Two Edge Type LDPC Codes for the BEC Wiretap Channel
We consider transmission over a wiretap channel where both the main channel
and the wiretapper's channel are Binary Erasure Channels (BEC). We propose a
code construction method using two edge type LDPC codes based on the coset
encoding scheme. Using a standard LDPC ensemble with a given threshold over the
BEC, we give a construction for a two edge type LDPC ensemble with the same
threshold. If the given standard LDPC ensemble has degree two variable nodes,
our construction gives rise to degree one variable nodes in the code used over
the main channel. This results in zero threshold over the main channel. In
order to circumvent this problem, we numerically optimize the degree
distribution of the two edge type LDPC ensemble. We find that the resulting
ensembles are able to perform close to the boundary of the rate-equivocation
region of the wiretap channel.
There are two performance criteria for a coding scheme used over a wiretap
channel: reliability and secrecy. The reliability measure corresponds to the
probability of decoding error for the intended receiver. This can be easily
measured using density evolution recursion. However, it is more challenging to
characterize secrecy, corresponding to the equivocation of the message for the
wiretapper. M\'easson, Montanari, and Urbanke have shown how the equivocation
can be measured for a broad range of standard LDPC ensembles for transmission
over the BEC under the point-to-point setup. By generalizing the method of
M\'easson, Montanari, and Urbanke to two edge type LDPC ensembles, we show how
the equivocation for the wiretapper can be computed. We find that relatively
simple constructions give very good secrecy performance and are close to the
secrecy capacity. However finding explicit sequences of two edge type LDPC
ensembles which achieve secrecy capacity is a more difficult problem. We pose
it as an interesting open problem.Comment: submitted to IEEE Transactions on Information Theory. Updated versio
Spectral Shape of Doubly-Generalized LDPC Codes: Efficient and Exact Evaluation
This paper analyzes the asymptotic exponent of the weight spectrum for
irregular doubly-generalized LDPC (D-GLDPC) codes. In the process, an efficient
numerical technique for its evaluation is presented, involving the solution of
a 4 x 4 system of polynomial equations. The expression is consistent with
previous results, including the case where the normalized weight or stopping
set size tends to zero. The spectral shape is shown to admit a particularly
simple form in the special case where all variable nodes are repetition codes
of the same degree, a case which includes Tanner codes; for this case it is
also shown how certain symmetry properties of the local weight distribution at
the CNs induce a symmetry in the overall weight spectral shape function.
Finally, using these new results, weight and stopping set size spectral shapes
are evaluated for some example generalized and doubly-generalized LDPC code
ensembles.Comment: 17 pages, 6 figures. To appear in IEEE Transactions on Information
Theor