118 research outputs found
Applications of finite geometries to designs and codes
This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures.
A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples.
We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs.
Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes
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
Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds
Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new
homological quantum error correcting codes. They are LDPC codes with linear
rate and distance . Their rate is evaluated via Euler
characteristic arguments and their distance using -systolic
geometry. This construction answers a queston of Z\'emor, who asked whether
homological codes with such parameters could exist at all.Comment: 21 page
Quantum stabilizer codes and beyond
The importance of quantum error correction in paving the way to build a
practical quantum computer is no longer in doubt. This dissertation makes a
threefold contribution to the mathematical theory of quantum error-correcting
codes. Firstly, it extends the framework of an important class of quantum codes
-- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes
to classical codes over quadratic extension fields, provides many new
constructions of quantum codes, and develops further the theory of optimal
quantum codes and punctured quantum codes. Secondly, it contributes to the
theory of operator quantum error correcting codes also called as subsystem
codes. These codes are expected to have efficient error recovery schemes than
stabilizer codes. This dissertation develops a framework for study and analysis
of subsystem codes using character theoretic methods. In particular, this work
establishes a close link between subsystem codes and classical codes showing
that the subsystem codes can be constructed from arbitrary classical codes.
Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum
codes and considers more realistic channels than the commonly studied
depolarizing channel. It gives systematic constructions of asymmetric quantum
stabilizer codes that exploit the asymmetry of errors in certain quantum
channels.Comment: Ph.D. Dissertation, Texas A&M University, 200
Asymmetric Quantum Codes: New Codes from Old
In this paper we extend to asymmetric quantum error-correcting codes (AQECC)
the construction methods, namely: puncturing, extending, expanding, direct sum
and the (u|u + v) construction. By applying these methods, several families of
asymmetric quantum codes can be constructed. Consequently, as an example of
application of quantum code expansion developed here, new families of
asymmetric quantum codes derived from generalized Reed-Muller (GRM) codes,
quadratic residue (QR), Bose-Chaudhuri-Hocquenghem (BCH), character codes and
affine-invariant codes are constructed.Comment: Accepted for publication Quantum Information Processin
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Coding Theory
Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances
Topology-Aware Exploration of Energy-Based Models Equilibrium: Toric QC-LDPC Codes and Hyperbolic MET QC-LDPC Codes
This paper presents a method for achieving equilibrium in the ISING
Hamiltonian when confronted with unevenly distributed charges on an irregular
grid. Employing (Multi-Edge) QC-LDPC codes and the Boltzmann machine, our
approach involves dimensionally expanding the system, substituting charges with
circulants, and representing distances through circulant shifts. This results
in a systematic mapping of the charge system onto a space, transforming the
irregular grid into a uniform configuration, applicable to Torical and Circular
Hyperboloid Topologies. The paper covers fundamental definitions and notations
related to QC-LDPC Codes, Multi-Edge QC-LDPC codes, and the Boltzmann machine.
It explores the marginalization problem in code on the graph probabilistic
models for evaluating the partition function, encompassing exact and
approximate estimation techniques. Rigorous proof is provided for the
attainability of equilibrium states for the Boltzmann machine under Torical and
Circular Hyperboloid, paving the way for the application of our methodology.
Practical applications of our approach are investigated in Finite Geometry
QC-LDPC Codes, specifically in Material Science. The paper further explores its
effectiveness in the realm of Natural Language Processing Transformer Deep
Neural Networks, examining Generalized Repeat Accumulate Codes,
Spatially-Coupled and Cage-Graph QC-LDPC Codes. The versatile and impactful
nature of our topology-aware hardware-efficient quasi-cycle codes equilibrium
method is showcased across diverse scientific domains without the use of
specific section delineations.Comment: 16 pages, 29 figures. arXiv admin note: text overlap with
arXiv:2307.1577
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