13 research outputs found
Covering -Symbol Metric Codes and the Generalized Singleton Bound
Symbol-pair codes were proposed for the application in high density storage
systems, where it is not possible to read individual symbols. Yaakobi, Bruck
and Siegel proved that the minimum pair-distance of binary linear cyclic codes
satisfies and introduced -symbol metric
codes in 2016. In this paper covering codes in -symbol metrics are
considered. Some examples are given to show that the Delsarte bound and the
Norse bound for covering codes in the Hamming metric are not true for covering
codes in the pair metric. We give the redundancy bound on covering radius of
linear codes in the -symbol metric and give some optimal codes attaining
this bound. Then we prove that there is no perfect linear symbol-pair code with
the minimum pair distance and there is no perfect -symbol metric code if
. Moreover a lot of cyclic and algebraic-geometric codes
are proved non-perfect in the -symbol metric. The covering radius of the
Reed-Solomon code in the -symbol metric is determined. As an application the
generalized Singleton bound on the sizes of list-decodable -symbol metric
codes is also presented. Then an upper bound on lengths of general MDS
symbol-pair codes is proved.Comment: 21 page
New bounds for -Symbol Distances of Matrix Product Codes
Matrix product codes are generalizations of some well-known constructions of
codes, such as Reed-Muller codes, -construction, etc. Recently, a
bound for the symbol-pair distance of a matrix product code was given in
\cite{LEL}, and new families of MDS symbol-pair codes were constructed by using
this bound. In this paper, we generalize this bound to the -symbol distance
of a matrix product code and determine all minimum -symbol distances of
Reed-Muller codes. We also give a bound for the minimum -symbol distance of
codes obtained from the -construction, and use this bound to
construct some -linear -symbol almost MDS codes with arbitrary
length. All the minimum -symbol distances of -linear codes and
-linear codes for are determined. Some examples are
presented to illustrate these results
Two classes of reducible cyclic codes with large minimum symbol-pair distances
The high-density data storage technology aims to design high-capacity storage
at a relatively low cost. In order to achieve this goal, symbol-pair codes were
proposed by Cassuto and Blaum \cite{CB10,CB11} to handle channels that output
pairs of overlapping symbols. Such a channel is called symbol-pair read
channel, which introduce new concept called symbol-pair weight and minimum
symbol-pair distance. In this paper, we consider the parameters of two classes
of reducible cyclic codes under the symbol-pair metric. Based on the theory of
cyclotomic numbers and Gaussian period over finite fields, we show the possible
symbol-pair weights of these codes. Their minimum symbol-pair distances are
twice the minimum Hamming distances under some conditions. Moreover, we obtain
some three symbol-pair weight codes and determine their symbol-pair weight
distribution. A class of MDS symbol-pair codes is also established. Among other
results, we determine the values of some generalized cyclotomic numbers
Turbo codes: convergence phenomena & non-binary constructions
The introduction of turbo codes in 1993 provided a code structure that could approach Shannon limit performance whilst remaining practically decodeable. Much subsequent work has focused on this remarkable structure, attempting to explain its performance and to extend or modify it. This thesis builds on this research providing insights into the convergence behaviour of the iterative decoder for turbo codes and examining the potential of turbo codes constructed from non-binary component codes.
The first chapter of this thesis gives a brief history of coding theory, providing context for the work. Chapter two explains in detail both the turbo encoding and decoding structures considered. Chapter three presents new work on convergence phenomena observed in the iterative decoding process. These results emphasise the dynamic nature of the decoder and allow for both a stopping criteria and ARQ scheme to be proposed. Chapters four and five present the work on non-binary turbo codes. First the problem of choosing good component codes is discussed and an achievability bound on the dominant parameter affecting their performance is derived. Searches for good component codes over a number of small rings are then conducted, and simulation results presented. The new results, and suggestions for further work are summarised in the conclusion of Chapter six
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