9 research outputs found
Quasi-Cyclic Asymptotically Regular LDPC Codes
Families of "asymptotically regular" LDPC block code ensembles can be formed
by terminating (J,K)-regular protograph-based LDPC convolutional codes. By
varying the termination length, we obtain a large selection of LDPC block code
ensembles with varying code rates, minimum distance that grows linearly with
block length, and capacity approaching iterative decoding thresholds, despite
the fact that the terminated ensembles are almost regular. In this paper, we
investigate the properties of the quasi-cyclic (QC) members of such an
ensemble. We show that an upper bound on the minimum Hamming distance of
members of the QC sub-ensemble can be improved by careful choice of the
component protographs used in the code construction. Further, we show that the
upper bound on the minimum distance can be improved by using arrays of
circulants in a graph cover of the protograph.Comment: To be presented at the 2010 IEEE Information Theory Workshop, Dublin,
Irelan
Asymptotically Good LDPC Convolutional Codes Based on Protographs
LDPC convolutional codes have been shown to be capable of achieving the same
capacity-approaching performance as LDPC block codes with iterative
message-passing decoding. In this paper, asymptotic methods are used to
calculate a lower bound on the free distance for several ensembles of
asymptotically good protograph-based LDPC convolutional codes. Further, we show
that the free distance to constraint length ratio of the LDPC convolutional
codes exceeds the minimum distance to block length ratio of corresponding LDPC
block codes.Comment: Proceedings of the 2008 IEEE International Symposium on Information
Theory, Toronto, ON, Canada, July 6 - 11, 200
Rate-compatible protograph LDPC code families with linear minimum distance
Digital communication coding methods are shown, which generate certain types of low-density parity-check (LDPC) codes built from protographs. A first method creates protographs having the linear minimum distance property and comprising at least one variable node with degree less than 3. A second method creates families of protographs of different rates, all having the linear minimum distance property, and structurally identical for all rates except for a rate-dependent designation of certain variable nodes as transmitted or non-transmitted. A third method creates families of protographs of different rates, all having the linear minimum distance property, and structurally identical for all rates except for a rate-dependent designation of the status of certain variable nodes as non-transmitted or set to zero. LDPC codes built from the protographs created by these methods can simultaneously have low error floors and low iterative decoding thresholds, and families of such codes of different rates can be decoded efficiently using a common decoding architecture
Spatially Coupled LDPC Codes Constructed from Protographs
In this paper, we construct protograph-based spatially coupled low-density
parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or
uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L,
we obtain a flexible family of code ensembles with varying rates and frame
lengths that can share the same encoding and decoding architecture for
arbitrary L. We demonstrate that the resulting codes combine the best features
of optimized irregular and regular codes in one design: capacity approaching
iterative belief propagation (BP) decoding thresholds and linear growth of
minimum distance with block length. In particular, we show that, for
sufficiently large L, the BP thresholds on both the binary erasure channel
(BEC) and the binary-input additive white Gaussian noise channel (AWGNC)
saturate to a particular value significantly better than the BP decoding
threshold and numerically indistinguishable from the optimal maximum
a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all
variable nodes in the coupled chain have degree greater than two,
asymptotically the error probability converges at least doubly exponentially
with decoding iterations and we obtain sequences of asymptotically good LDPC
codes with fast convergence rates and BP thresholds close to the Shannon limit.
Further, the gap to capacity decreases as the density of the graph increases,
opening up a new way to construct capacity achieving codes on memoryless
binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor
Topologically Driven Methods for Construction Of Multi-Edge Type (Multigraph with nodes puncturing) Quasi-Cyclic Low-density Parity-check Codes for Wireless Channel, WDM Long-Haul and Archival Holographic Memory
In this Phd thesis discusses modern methods for constructing MET QC-LDPC
codes with a given error correction ("waterfall, error-floor") and complexity
(parallelism level according circulant size plus scheduler orthogonality of
checks) profiles: 1. weight enumerators optimization, protograph construction
using Density Evolution, MI (P/Exit-chart) and it approximation: Gaussian
Approximation, Reciprocal-channel approximation and etc; 2. Covariance
evolution and it approximation; 3. Lifting methods for QC codes
construction:PEG, Guest-and-Test, Hill-Climbing with girth, EMD, ACE
optimization; 4. Upper and lower bounds on code distance estimation and its
parallel implementation using CPU/GPU; 5. Brouwer-Zimmerman and Number Geometry
code distance estimation methods; 6. Importance Sampling for error-floor
estimation; 7. Length and rate adaption methods for QC codes based on cyclic
group decomposition; 8. Methods for interaction screening which allow to
improve performance (decorrelate variables) under BP and it's approximation. We
proposed several state-of-the-art methods: Simulated Annealing lifting for MET
QC-LDPC codes construction; fast EMD and code distance estimation; floor scale
modular lifting for lenght adaption; fast finite-length covariance evolution
rate penalty from threshold for code construction and it hardware friendly
compression for fast decoder's LLRs unbiasing due SNR's estimation error. We
found topology reason's of efficient of such methods using topology thickening
(homotopy of continuous and discrete curvature) under matched metric space
which allow to generalize this idea to a class of nonlinear codes for Signal
Processing and Machine Learning. Using the proposed algorithms several
generations of WDM Long-Haul error-correction codes were built. It was applied
for "5G eMBB" 3GPP TS38.212 and other applications like Flash storage,
Compressed sensing measurement matrix.Comment: Phd Thesis, 176 pages, in Russian, 62 pictures, 13 tables, 5 appendix
including links to binary and source code
Enumerators for protograph ensembles of LDPC codes
This paper considers the problem of finding average
enumerators for the class of protograph ensembles, which are
related in a certain way to quasi-cyclic codes. Our methods,
which are necessarily different from those used to compute
enumerators for classical irregular ensembles, can be applied to both codeword and stopping set weight enumerators. The method divides codewords into types based on their partial weight enumerator. For each type, an exponent can be computed for the average number of codewords of that type. Maximizing over types of fixed average weight gives the average enumerator which
we seek. Although this maximization step is in general difficult because of non-unique local maxima, we can compute it for simple cases. We show that certain ensembles exist which have a linearly growing minimum distance with high probability, while others have at most sublinearly growing minimum distance with high probability