Please use this identifier to cite or link to this item: http://hdl.handle.net/10603/26209
Title: Efficient codes with class errors of SK-Metric and polynomial power-product composition for codes
Researcher: Gaur, Ankita
Guide(s): Sharma, Bhu Dev
Keywords: Efficient Code
Metrix
Patterns
Random Errors
Upload Date: 30-Sep-2014
University: Jaypee Institute of Information Technology
Completed Date: 31/05/2014
Abstract: Partitions of sets are an important combinatorial tool. The concept of metric is another basic tool across all branches of mathematics and has wide applications including those in statistics. The thesis is on coding theory where metric plays key role in error control. One metric, widely used is Hamming metric and has been used rather in ad-hoc manner. The only systematic approach to find all possible metrics through partitions of sets over Zq was undertaken by Sharma and Kaushik, who defined partitions which induced metrics over Zq . Using this new type of partitions of Zq , Sharma and Kaushik explored all possible metrics, including Hamming and Lee matrices. Such a class of metrics was not available before. In this thesis, schemes of limited error patters that provide upper bounds on paritychecks for ‘random limited error patterns are considered. Considering SK-distances and examining the sufficient condition for the existence of a parity check matrix for a given number of parity-checks, also obtained an upper bound on the number of parity check digits. Another important link is the channel. Having used SK-partitions for metric in coding, there is need to study of channels also matching these partitions. Introducing a very general way of the q-nary channel, generalizing BSC, results have been obtained on probabilities of errors for different kinds of errors, also bounds on probability of error for code correcting designed additive class errors that address to the reliability aspect of block coding. With broadening the concept of metrics through SK-partitions, there arises a wider search of perfect codes. In the thesis this area is explored and extended. Bounds on the number of parity check digits for codes correcting errors of different SK-weights are derived. These bounds with equality give condition for perfect codes. Existence of perfect codes correcting error in t positions are reported. Constructing larger codes by suitably composing lower order codes is of interest. Defining a new product of polynomials called ‘ordered power product’ (OPP) a new product of two cyclic codes is developed that introduces a method of getting a cyclic code. The technique is further applied for another important class that of quasi-cyclic codes. This dissertation contains more than 100 relevant references.
URI: http://hdl.handle.net/10603/26209
Appears in Departments:Department of Mathematics

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01 title.pdfAttached File26.91 kBAdobe PDFView/Open
02 contents.pdf47.08 kBAdobe PDFView/Open
03 certificate and declaration.pdf7.56 kBAdobe PDFView/Open
04 acknowledgement.pdf6.76 kBAdobe PDFView/Open
05 abstract.pdf8.89 kBAdobe PDFView/Open
06 list of tables and abbreviations.pdf26.84 kBAdobe PDFView/Open
07 chapter1.pdf137.04 kBAdobe PDFView/Open
08 chapter2.pdf100.64 kBAdobe PDFView/Open
09 chapter3.pdf72.03 kBAdobe PDFView/Open
10 chapter4.pdf119.2 kBAdobe PDFView/Open
11 chapter5.pdf82.82 kBAdobe PDFView/Open
12 chapter 6.pdf210.98 kBAdobe PDFView/Open
13 chapter7.pdf32.85 kBAdobe PDFView/Open
14 references.pdf30.11 kBAdobe PDFView/Open
15 list of publication.pdf11.22 kBAdobe PDFView/Open
16 synopsis.pdf87.8 kBAdobe PDFView/Open
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