76 research outputs found
An improvement to multifold euclidean geometry codes
This paper presents an improvement to the multifold Euclidean geometry codes introduced by Lin (1973).The improved multifold EG codes are proved to be maximal, and therefore they are more efficient than the multifold EG codes. Relationships between the improved multifold EG codes and other known majority-logic decodable codes are proved
The q-ary image of some qm-ary cyclic codes: permutation group and soft-decision decoding
Using a particular construction of generator matrices of
the q-ary image of qm-ary cyclic codes, it is proved that some of these codes are invariant under the action of particular permutation groups. The equivalence of such codes with some two-dimensional (2-D) Abelian codes and cyclic codes is deduced from this property. These permutations are also used in the area of the soft-decision decoding of some expanded Reed–Solomon (RS) codes to improve the performance of generalized minimum-distance decoding
Asymmetric Quantum LDPC Codes
Recently, quantum error-correcting codes were proposed that capitalize on the
fact that many physical error models lead to a significant asymmetry between
the probabilities for bit flip and phase flip errors. An example for a channel
which exhibits such asymmetry is the combined amplitude damping and dephasing
channel, where the probabilities of bit flips and phase flips can be related to
relaxation and dephasing time, respectively. We give systematic constructions
of asymmetric quantum stabilizer codes that exploit this asymmetry. Our
approach is based on a CSS construction that combines BCH and finite geometry
LDPC codes.Comment: 5 pages, 1 figure, 1 table, to appear in the Proceedings of the 2008
IEEE International Symposium on Information Theor
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
Mathematical structures for decoding projective geometry codes
Imperial Users onl
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
Some new results on majority-logic codes for correction of random errors
The main advantages of random error-correcting majority-logic
codes and majority-logic decoding in general are well known and
two-fold. Firstly, they offer a partial solution to a classical
coding theory problem, that of decoder complexity. Secondly, a
majority-logic decoder inherently corrects many more random error
patterns than the minimum distance of the code implies is possible.
The solution to the decoder complexity is only a partial one
because there are circumstances under which a majority-logic decoder
is too complex and expensive to implement. [Continues.
Recommended from our members
Contemporary Coding Theory
Coding Theory naturally lies at the intersection of a large number
of disciplines in pure and applied mathematics. A multitude of
methods and means has been designed to construct, analyze, and
decode the resulting codes for communication. This has suggested to
bring together researchers in a variety of disciplines within
Mathematics, Computer Science, and Electrical Engineering, in order
to cross-fertilize generation of new ideas and force global
advancement of the field. Areas to be covered are Network Coding,
Subspace Designs, General Algebraic Coding Theory, Distributed
Storage and Private Information Retrieval (PIR), as well as
Code-Based Cryptography
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