692 research outputs found
Local Optimality Certificates for LP Decoding of Tanner Codes
We present a new combinatorial characterization for local optimality of a
codeword in an irregular Tanner code. The main novelty in this characterization
is that it is based on a linear combination of subtrees in the computation
trees. These subtrees may have any degree in the local code nodes and may have
any height (even greater than the girth). We expect this new characterization
to lead to improvements in bounds for successful decoding.
We prove that local optimality in this new characterization implies
ML-optimality and LP-optimality, as one would expect. Finally, we show that is
possible to compute efficiently a certificate for the local optimality of a
codeword given an LLR vector
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
Raptor Codes in the Low SNR Regime
In this paper, we revisit the design of Raptor codes for binary input
additive white Gaussian noise (BIAWGN) channels, where we are interested in
very low signal to noise ratios (SNRs). A linear programming degree
distribution optimization problem is defined for Raptor codes in the low SNR
regime through several approximations. We also provide an exact expression for
the polynomial representation of the degree distribution with infinite maximum
degree in the low SNR regime, which enables us to calculate the exact value of
the fractions of output nodes of small degrees. A more practical degree
distribution design is also proposed for Raptor codes in the low SNR regime,
where we include the rate efficiency and the decoding complexity in the
optimization problem, and an upper bound on the maximum rate efficiency is
derived for given design parameters. Simulation results show that the Raptor
code with the designed degree distributions can approach rate efficiencies
larger than 0.95 in the low SNR regime.Comment: Submitted to the IEEE Transactions on Communications. arXiv admin
note: text overlap with arXiv:1510.0772
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
Efficient Linear Programming Decoding of HDPC Codes
We propose several improvements for Linear Programming (LP) decoding
algorithms for High Density Parity Check (HDPC) codes. First, we use the
automorphism groups of a code to create parity check matrix diversity and to
generate valid cuts from redundant parity checks. Second, we propose an
efficient mixed integer decoder utilizing the branch and bound method. We
further enhance the proposed decoders by removing inactive constraints and by
adapting the parity check matrix prior to decoding according to the channel
observations. Based on simulation results the proposed decoders achieve near-ML
performance with reasonable complexity.Comment: Submitted to the IEEE Transactions on Communications, November 200
Bilayer Low-Density Parity-Check Codes for Decode-and-Forward in Relay Channels
This paper describes an efficient implementation of binning for the relay
channel using low-density parity-check (LDPC) codes. We devise bilayer LDPC
codes to approach the theoretically promised rate of the decode-and-forward
relaying strategy by incorporating relay-generated information bits in
specially designed bilayer graphical code structures. While conventional LDPC
codes are sensitively tuned to operate efficiently at a certain channel
parameter, the proposed bilayer LDPC codes are capable of working at two
different channel parameters and two different rates: that at the relay and at
the destination. To analyze the performance of bilayer LDPC codes, bilayer
density evolution is devised as an extension of the standard density evolution
algorithm. Based on bilayer density evolution, a design methodology is
developed for the bilayer codes in which the degree distribution is iteratively
improved using linear programming. Further, in order to approach the
theoretical decode-and-forward rate for a wide range of channel parameters,
this paper proposes two different forms bilayer codes, the bilayer-expurgated
and bilayer-lengthened codes. It is demonstrated that a properly designed
bilayer LDPC code can achieve an asymptotic infinite-length threshold within
0.24 dB gap to the Shannon limits of two different channels simultaneously for
a wide range of channel parameters. By practical code construction,
finite-length bilayer codes are shown to be able to approach within a 0.6 dB
gap to the theoretical decode-and-forward rate of the relay channel at a block
length of and a bit-error probability (BER) of . Finally, it is
demonstrated that a generalized version of the proposed bilayer code
construction is applicable to relay networks with multiple relays.Comment: Submitted to IEEE Trans. Info. Theor
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