13 research outputs found

    Covering bb-Symbol Metric Codes and the Generalized Singleton Bound

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    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 d2β‰₯⌈3dH/2βŒ‰d_2 \geq \lceil 3d_H/2 \rceil and introduced bb-symbol metric codes in 2016. In this paper covering codes in bb-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 bb-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 77 and there is no perfect bb-symbol metric code if bβ‰₯n+12b\geq \frac{n+1}{2}. Moreover a lot of cyclic and algebraic-geometric codes are proved non-perfect in the bb-symbol metric. The covering radius of the Reed-Solomon code in the bb-symbol metric is determined. As an application the generalized Singleton bound on the sizes of list-decodable bb-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 bb-Symbol Distances of Matrix Product Codes

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    Matrix product codes are generalizations of some well-known constructions of codes, such as Reed-Muller codes, [u+v,uβˆ’v][u+v,u-v]-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 bb-symbol distance of a matrix product code and determine all minimum bb-symbol distances of Reed-Muller codes. We also give a bound for the minimum bb-symbol distance of codes obtained from the [u+v,uβˆ’v][u+v,u-v]-construction, and use this bound to construct some [2n,2nβˆ’2]q[2n,2n-2]_q-linear bb-symbol almost MDS codes with arbitrary length. All the minimum bb-symbol distances of [n,nβˆ’1]q[n,n-1]_q-linear codes and [n,nβˆ’2]q[n,n-2]_q-linear codes for 1≀b≀n1\leq b\leq n are determined. Some examples are presented to illustrate these results

    Two classes of reducible cyclic codes with large minimum symbol-pair distances

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    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

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    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

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    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
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