Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 De-randomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 De-randomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Generalizing the high-noise decoding methods of [1, 19] to the class of algebraic-geometric codes, we design the first polynomialtime algorithms to decode algebraic-geometric codes significantly beyond the conventional error-correction bound. Applying our results to codes obtained from curves with many rational points, we construct arbitrarily long, constant-rate linear codes over a fixed field F q such that a codeword is efficiently, non-uniquely reconstructible after a majority of its letters have been arbitrarily corrupted. We also construct codes such that a codeword is uniquely and efficiently reconstructible after a majority of its letters have been corrupted by noise which is random in a specified sense. We summarize our results in terms of bounds on asymptotic parameters, giving a new characterization of decoding beyond the error-correction bound. 1 Introduction Error-correcting codes, originally designed to accommodate reliable transmission of information through unreliable ...

proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op woensdag 4 juni 2003 om 16.00 uur door

We consider monotone embeddings of a nite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l 2 , while, (in a sense to be made precise later), for almost every n-point metric space, every monotone map must be into a space of dimension n) (Lemma 3).

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We develop new coset bounds for algebraic geometric codes. The bounds have a natural interpretation as an adversary threshold for algebraic geometric secret sharing schemes and lead to improved bounds for the minimum distance of an AG code. Our bounds improve both floor bounds and order bounds and provide for the first time a connection between the two types of bounds.

A common theme in mathematics is the classification of mathematical objects by assigning numerical invariants to them. There are two ways in which such numerical invariants can appear in relation to computational complexity. On the one hand, mathematical invariants are used in the context of proving lower complexity bounds: they serve as obstructions to the existence of fast algorithms for solving certain problems. On the other hand, it is the computational complexity of actually computing such invariants that is of interest. The first part of this thesis is concerned with lower bounds for the problems of computing linear and bilinear maps. The invariants used, namely the mean square volume, singular values, and rigidity, belong to linear algebra. One of the main results is a tight lower bound of order Ω(n log n) for the problem of multiplying two polynomials, in the model of bounded coefficient circuits. This lower bound is extended to circuits for which a limited number of unbounded scalar multiplications (help gates) are allowed. The second part is concerned with the complexity of actually computing numerical invariants. The objects of study are two of the most prominent invariants in algebraic geometry and topology: the Euler characteristic and the Hilbert polynomial of complex projective varieties. These problems are studied within the framework of counting complexity classes. It is shown that the problem of computing the Euler characteristic of a complex projective variety is on essentially the same level of difficulty as the problem of counting the number of solutions of a system of polynomial equations. A similar result is proved for the Hilbert polynomials, when the input variety is assumed to be smooth and equidimensional. iv

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f. Die beste bekannte untere Schranke für die Schaltkreiskomplexität einer explizit angegebenen Funktion ist noch linear in der Zahl der Variablen. Es konnten jedoch superpolynomiale untere Schranken für monotone Schaltkreise bewiesen werden. Ein Schaltkreis heiÿt monoton, wenn er keine NICHT-Gatter (Negationen) aufweist. Die Approximationsmethode liefert superpolynomiale untere Schranken für die monotone Schaltkreiskomplexität verschiedener Funktionen. Auch ist bekannt, dass Negationen zur Berechnung so genannter Slice-Funktionen fast keinen Beitrag leisten können. Eine superpolynomiale untere Schranke für die monotone Komplexität einer Slice-Funktion impliziert eine superpolynomiale untere Schranke für ihre nicht-monotone Komplexität. Allerdings reichen die heute bekannten Methoden anscheinend nicht aus, um ausreichende untere Schranken für die Komplexität von Slice-Funktionen zu zeigen. Daher ist es gerechtfertigt,