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Modular algorithms and interpolation
Kubát, David ; Stanovský, David (advisor) ; Žemlička, Jan (referee)
This thesis concerns with the polynomial interpolation problem and the rational function reconstruction problem (Cauchy interpolation, Padé approximation). It does so from the algebraical point of view. Moreover, it involves some applications of the generalized Chinese remainder theorem (Hermite interpolation, partial fraction decomposition). An important theoretical concept regarding the above mentioned problems is the Euclidean algorithm, which is studied in case of polynomial rings. The structure of the thesis is based on the book by von zur Gathen and Gerhard called Modern Computer Algebra. Its exercises are the main content of the thesis. They usually extend the theory involved. Powered by TCPDF (www.tcpdf.org)
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Quasi-monoidic codes
Snítilá, Jitka ; Žemlička, Jan (advisor) ; Šťovíček, Jan (referee)
This thesis focuses on the problem of the key size in McEliece cryptosystem and its solution using quasi- monoidic codes, especially quasi-monoidic Goppa codes. Required theory of quasi-monoidic Cauchy matrices and Goppa codes is introduced along with algebraic structures necessary for quasi-monoidic codes description. Suitable Abelian groups for this class of codes are specified. This thesis also presents efficient algorithms for constructing quasi-monoidic Cauchy matrices and quasi-monoidic Goppa codes. Reduction of the key size using this class of algebraic codes is presented as well. Powered by TCPDF (www.tcpdf.org)
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Classical structural attack on the Niederreiter cryptosystem based upon GRS codes
Hrubešová, Tereza ; Drápal, Aleš (advisor) ; Žemlička, Jan (referee)
The main purpose of this bachelor thesis is the description of the attack on the Niederreiter cryptosystem based on GRS codes. This attack was published by Sidelnikov and Shestakov in 1992. In the beginning the problem of group action, which is used in the attack, is introduced. A short preface into the coding theory follows, GRS codes are described and McEliece and Niederreiter cryptosystems are introduced, both as representatives of post-quantum cryptography. The following part of the thesis is dedicated to the attack itself. It is showed how one uses the group action, the process of the attack is also described in detail and its computing complexity is mentioned. Everything is illustrated by examples. Powered by TCPDF (www.tcpdf.org)
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Discrete Channel Capacity
Butora, Jan ; Holub, Štěpán (advisor) ; Žemlička, Jan (referee)
Title: Discrete Channel Capacity Author: Jan Butora Department: Department of Algebra Supervisor: doc. Mgr. Štěpán Holub, Ph.D. et Ph.D., Department of Algebra Abstract: This Bachelor thesis introduces and examines C.E. Shannon's discrete channel capacity theory, which was first published in 1948 as one of the founding studies in the field of mathematical information theory. In the first place, possible way of information measurement is presented and communication systems are described. Additionally, emphasis is given to discrete noiseless channel and the theorem on calculating the capacity of such channels is examined and proven. Shannon's proof is examined in detail as it contains several non-trivial results in finite differences. Finally, calculation of channel capacity using the theorem is shown in practice. Keywords: difference equations, generating function, channel capacity
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Position coding on line and plane
Těthalová, Gabriela ; Tůma, Jiří (advisor) ; Žemlička, Jan (referee)
The introduction into the position coding consists of two examples of position codes on line which use the positional number system of base 4 and De Bruijn sequence constructed with the use of LFSR. The main aim of this thesis is the description of two positional codes on plane based on the patents of the Swedish company Anoto. These positional codes use mixed radix number representation and also a specific type of sequences called quasi De Bruijn sequences. The second of the described position codes represents an application of the Chinese Remainder Theorem. The thesis focuses on the position code construction, description of decoding, and the proofs of detectability. A detailed comparison of the principles of the coding of the coordinate y of both discussed codes is provided in the final part of the thesis. Keywords: positional code, quasi De Bruijn sequence, mixed radix, Anoto Pattern Powered by TCPDF (www.tcpdf.org)
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p-Adic numbers
Dubiel, Richard ; Šťovíček, Jan (advisor) ; Žemlička, Jan (referee)
Univerzita Karlova v Praze Matematicko-fyzikální fakulta BAKALÁŘSKÁ PRÁCE Richard Dubiel p-adická čísla Katedra algebry Vedoucí bakalářské práce: Mgr. Jan Šťovíček, Ph.D. Studijní program: Matematika Studijní obor: obecná matematika Abstract: This thesis deals with construction of the field of p-adic numbers as a completion of rational numbers field and introduces several important properties of this field. It will introduce concepts of an absolute value, metric, ultrametric and completion of field with respect to absolute value. Then we introduce a p-adic absolute value - one that measures "how much" is a number divsible by a prime number p. Then we construct the completion of the field of rational numbers with respect to this absolute value - field of p-adic numbers. We show, how can one represent these numbers. At last, we introduce two of the most important results of the theory of p-adic numbers - Hensel lemma and Hasse-Minkowski theorem.
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Banschewského funkce na komplementárních modulárních svazech
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
Title: Banaschewski function on countable complemented modular lattices Author: Samuel Mokriš Department: Department of Algebra Supervisor of the bachelor thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: A Banaschewski function on a bounded lattice L is an antitone self-map on L that picks a complement for each element of L. On any at most countable complemented modular lattice L, there exists a Banaschewski function with a Boolean range M. Moreover, such M is a maximal Boolean sublattice of L and is uniquely determined up to isomorphism. In the thesis we give a negative answer to the related question whether all maximal Boolean sublattices in an arbitrary countable complemented modular lattice are isomorphic and whether every max- imal Boolean sublattice in an arbitrary countable complemented modular lattice L is the range of some Banaschewski function on L. We also generalize the coun- terexample to greater cardinalities; for a given infinite cardinal κ we construct a complemented modular lattice L of cardinality κ and maximal Boolean sublat- tices B and E of L such that B is not the range of any Banaschewski function on L, that there exists a Banaschewski function on L of range E, and that B is not isomorphic to E. Keywords: complemented modular lattice, Banaschewski function, von...
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Division rings in coding theory
Svatoňová, Petra ; Žemlička, Jan (advisor) ; Holub, Štěpán (referee)
Wireless communication using multiple transmit and receive antennas is very effective when suitable space-time codes are used. Reliability of codes for com- munication can be assessed using various criteria. One of the basic is the Rank Criterion. In the presented work, we focus on this criterion and using division algebras we construct space-time block codes that satisfy it. First, we introduce the general construction. Then we apply it to create specific examples of division algebras and respective codes. At the end, we follow up with the Coding Gain Criterion - we present the Golden Code and using division algebras we outline general construction of perfect space-time codes.
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Structure of self-small groups and modules
Dvořák, Josef ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Title: Structure of self-small groups and modules Author: Josef Dvořák Department: Department of Algebra Supervisor: Mgr. Žemlička Jan, Ph.D. Supervisor's e-mail address: zemlicka@karlin.mff.cuni.cz Abstract: The thesis sums up the basic properties of self-small groups. Furthermore it thoroughly builds the theory od quotient categories by Serre classes, with focus on quotient category modulo the class B of boun- ded groups, which, as demonstrated, is equivalent to the quasicategory, i.e. category of abelian groups with Hom-sets being Q⊗Z HomA (A, B). This approach is developed into the theory of generalized quasi-categories. The dualities between quasi-caterogories od torsion-free and quotient-divisible categories of finite rank, resp. between categories of finite-rank self-small groups are studied and they are emloyed to the partial solution of Fuchs' problem no. 34. Keywords: self-small group, quotient divisible group, quasicategory, quo- tient category 1
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Tall rings
Penk, Tomáš ; Žemlička, Jan (advisor) ; Šťovíček, Jan (referee)
Perfect and max rings are known for over fifty years. Their theory is being steadily and intensively studied. The conditions defining them are mainly interesting while studying non-noetherian modules. In this work we summarize at first basic information about rings and modules with previous knowledge requiring just in elementary level. After summing up basic results in the theory of noetherian modules we will be prepaired for the definition of tall modules and tall rings. We show then that they are a generalization of prefect and max rings in a specific way. We bring out some examples of tall and non-tall rings with accenting commutative rings. Information which we obtain we try to generalize and use for searching some necessary and some sufficient conditions with the goal to be able to say about a commutative ring if it is tall or not. At the end we point out that in case of a commutative noetherian ring they are equivalent to each other and they give together to the concept tall ring an equivalent characterization.
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