35 research outputs found
New Classes of Partial Geometries and Their Associated LDPC Codes
The use of partial geometries to construct parity-check matrices for LDPC
codes has resulted in the design of successful codes with a probability of
error close to the Shannon capacity at bit error rates down to . Such
considerations have motivated this further investigation. A new and simple
construction of a type of partial geometries with quasi-cyclic structure is
given and their properties are investigated. The trapping sets of the partial
geometry codes were considered previously using the geometric aspects of the
underlying structure to derive information on the size of allowable trapping
sets. This topic is further considered here. Finally, there is a natural
relationship between partial geometries and strongly regular graphs. The
eigenvalues of the adjacency matrices of such graphs are well known and it is
of interest to determine if any of the Tanner graphs derived from the partial
geometries are good expanders for certain parameter sets, since it can be
argued that codes with good geometric and expansion properties might perform
well under message-passing decoding.Comment: 34 pages with single column, 6 figure
Circulant Arrays on Cyclic Subgroups of Finite Fields: Rank Analysis and Construction of Quasi-Cyclic LDPC Codes
This paper consists of three parts. The first part presents a large class of
new binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose
parity-check matrices are constructed based on cyclic subgroups of finite
fields. Experimental results show that the codes constructed perform well over
the binary-input AWGN channel with iterative decoding using the sum-product
algorithm (SPA). The second part analyzes the ranks of the parity-check
matrices of codes constructed based on finite fields with characteristic of 2
and gives combinatorial expressions for these ranks. The third part identifies
a subclass of constructed QC-LDPC codes that have large minimum distances.
Decoding of codes in this subclass with the SPA converges very fast.Comment: 26 pages, 6 figures, submitted to IEEE Transaction on Communication
Decomposition Methods for Large Scale LP Decoding
When binary linear error-correcting codes are used over symmetric channels, a
relaxed version of the maximum likelihood decoding problem can be stated as a
linear program (LP). This LP decoder can be used to decode error-correcting
codes at bit-error-rates comparable to state-of-the-art belief propagation (BP)
decoders, but with significantly stronger theoretical guarantees. However, LP
decoding when implemented with standard LP solvers does not easily scale to the
block lengths of modern error correcting codes. In this paper we draw on
decomposition methods from optimization theory, specifically the Alternating
Directions Method of Multipliers (ADMM), to develop efficient distributed
algorithms for LP decoding.
The key enabling technical result is a "two-slice" characterization of the
geometry of the parity polytope, which is the convex hull of all codewords of a
single parity check code. This new characterization simplifies the
representation of points in the polytope. Using this simplification, we develop
an efficient algorithm for Euclidean norm projection onto the parity polytope.
This projection is required by ADMM and allows us to use LP decoding, with all
its theoretical guarantees, to decode large-scale error correcting codes
efficiently.
We present numerical results for LDPC codes of lengths more than 1000. The
waterfall region of LP decoding is seen to initiate at a slightly higher
signal-to-noise ratio than for sum-product BP, however an error floor is not
observed for LP decoding, which is not the case for BP. Our implementation of
LP decoding using ADMM executes as fast as our baseline sum-product BP decoder,
is fully parallelizable, and can be seen to implement a type of message-passing
with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the
49th Annual Allerton Conference, September 2011. This version to appear in
IEEE Transactions on Information Theor
Novel Code-Construction for (3, k) Regular Low Density Parity Check Codes
Communication system links that do not have the ability to retransmit generally rely
on forward error correction (FEC) techniques that make use of error correcting codes
(ECC) to detect and correct errors caused by the noise in the channel. There are
several ECC’s in the literature that are used for the purpose. Among them, the low
density parity check (LDPC) codes have become quite popular owing to the fact that
they exhibit performance that is closest to the Shannon’s limit.
This thesis proposes a novel code-construction method for constructing not only (3, k)
regular but also irregular LDPC codes. The choice of designing (3, k) regular LDPC
codes is made because it has low decoding complexity and has a Hamming distance,
at least, 4. In this work, the proposed code-construction consists of information submatrix
(Hinf) and an almost lower triangular parity sub-matrix (Hpar). The core design
of the proposed code-construction utilizes expanded deterministic base matrices in
three stages. Deterministic base matrix of parity part starts with triple diagonal matrix
while deterministic base matrix of information part utilizes matrix having all elements
of ones. The proposed matrix H is designed to generate various code rates (R) by
maintaining the number of rows in matrix H while only changing the number of
columns in matrix Hinf.
All the codes designed and presented in this thesis are having no rank-deficiency, no
pre-processing step of encoding, no singular nature in parity part (Hpar), no girth of
4-cycles and low encoding complexity of the order of (N + g2) where g2«N. The
proposed (3, k) regular codes are shown to achieve code performance below 1.44 dB
from Shannon limit at bit error rate (BER) of 10
−6
when the code rate greater than
R = 0.875. They have comparable BER and block error rate (BLER) performance
with other techniques such as (3, k) regular quasi-cyclic (QC) and (3, k) regular
random LDPC codes when code rates are at least R = 0.7. In addition, it is also shown
that the proposed (3, 42) regular LDPC code performs as close as 0.97 dB from
Shannon limit at BER 10
−6
with encoding complexity (1.0225 N), for R = 0.928 and
N = 14364 – a result that no other published techniques can reach
Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning
The paper introduces the application of information geometry to describe the
ground states of Ising models by utilizing parity-check matrices of cyclic and
quasi-cyclic codes on toric and spherical topologies. The approach establishes
a connection between machine learning and error-correcting coding. This
proposed approach has implications for the development of new embedding methods
based on trapping sets. Statistical physics and number geometry applied for
optimize error-correcting codes, leading to these embedding and sparse
factorization methods. The paper establishes a direct connection between DNN
architecture and error-correcting coding by demonstrating how state-of-the-art
architectures (ChordMixer, Mega, Mega-chunk, CDIL, ...) from the long-range
arena can be equivalent to of block and convolutional LDPC codes (Cage-graph,
Repeat Accumulate). QC codes correspond to certain types of chemical elements,
with the carbon element being represented by the mixed automorphism
Shu-Lin-Fossorier QC-LDPC code. The connections between Belief Propagation and
the Permanent, Bethe-Permanent, Nishimori Temperature, and Bethe-Hessian Matrix
are elaborated upon in detail. The Quantum Approximate Optimization Algorithm
(QAOA) used in the Sherrington-Kirkpatrick Ising model can be seen as analogous
to the back-propagation loss function landscape in training DNNs. This
similarity creates a comparable problem with TS pseudo-codeword, resembling the
belief propagation method. Additionally, the layer depth in QAOA correlates to
the number of decoding belief propagation iterations in the Wiberg decoding
tree. Overall, this work has the potential to advance multiple fields, from
Information Theory, DNN architecture design (sparse and structured prior graph
topology), efficient hardware design for Quantum and Classical DPU/TPU (graph,
quantize and shift register architect.) to Materials Science and beyond.Comment: 71 pages, 42 Figures, 1 Table, 1 Appendix. arXiv admin note: text
overlap with arXiv:2109.08184 by other author