69 research outputs found
Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications
Coding; Communications; Engineering; Networks; Information Theory; Algorithm
Improved Spectral Bound for Quasi-Cyclic Codes
Spectral bounds form a powerful tool to estimate the minimum distances of
quasi-cyclic codes. They generalize the defining set bounds of cyclic codes to
those of quasi-cyclic codes. Based on the eigenvalues of quasi-cyclic codes and
the corresponding eigenspaces, we provide an improved spectral bound for
quasi-cyclic codes. Numerical results verify that the improved bound
outperforms the Jensen bound in almost all cases. Based on the improved bound,
we propose a general construction of quasi-cyclic codes with excellent designed
minimum distances. For the quasi-cyclic codes produced by this general
construction, the improved spectral bound is always sharper than the Jensen
bound
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
Codes and Designs Related to Lifted MRD Codes
Lifted maximum rank distance (MRD) codes, which are constant dimension codes,
are considered. It is shown that a lifted MRD code can be represented in such a
way that it forms a block design known as a transversal design. A slightly
different representation of this design makes it similar to a analog of a
transversal design. The structure of these designs is used to obtain upper
bounds on the sizes of constant dimension codes which contain a lifted MRD
code. Codes which attain these bounds are constructed. These codes are the
largest known codes for the given parameters. These transversal designs can be
also used to derive a new family of linear codes in the Hamming space. Bounds
on the minimum distance and the dimension of such codes are given.Comment: Submitted to IEEE Transactions on Information Theory. The material in
this paper was presented in part in the 2011 IEEE International Symposium on
Information Theory, Saint Petersburg, Russia, August 201
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