Explicit constructions of asymptotically good towers of function fields

Thesis (MSc)--Stellenbosch University, 2003ENGLISH ABSTRACT: A tower of global function fields :F = (FI, F2' ... ) is an infinite tower of separable extensions of algebraic function fields of one variable such that the constituent function fields have the same (finite) field of constants and the genus of these tend to infinity. A study can be made of the asymptotic behaviour of the ratio of the number of places of degree one over the genus of FJWq as i tends to infinity. A tower is called asymptotically good if this limit is a positive number. The well-known Drinfeld- Vladut bound provides a general upper bound for this limit. In practise, asymptotically good towers are rare. While the first examples were non-explicit, we focus on explicit towers of function fields, that is towers where equations recursively defining the extensions Fi+d F; are known. It is known that if the field of constants of the tower has square cardinality, it is possible to attain the Drinfeld- Vladut upper bound for this limit, even in the explicit case. If the field of constants does not have square cardinality, it is unknown how close the limit of the tower can come to this upper bound. In this thesis, we will develop the theory required to construct and analyse the asymptotic behaviour of explicit towers of function fields. Various towers will be exhibited, and general families of explicit formulae for which the splitting behaviour and growth of the genus can be computed in a tower will be discussed. When the necessary theory has been developed, we will focus on the case of towers over fields of non-square cardinality and the open problem of how good the asymptotic behaviour of the tower can be under these circumstances.AFRIKAANSE OPSOMMING: 'n Toring van globale funksieliggame F = (FI, F2' ... ) is 'n oneindige toring van skeibare uitbreidings van algebraïese funksieliggame van een veranderlike sodat die samestellende funksieliggame dieselfde (eindige) konstante liggaam het en die genus streef na oneindig. 'n Studie kan gemaak word van die asimptotiese gedrag van die verhouding van die aantal plekke van graad een gedeel deur die genus van Fi/F q soos i streef na oneindig. 'n Toring word asimptoties goed genoem as hierdie limiet 'n positiewe getal is. Die bekende Drinfeld- Vladut grens verskaf 'n algemene bogrens vir hierdie limiet. In praktyk is asimptoties goeie torings skaars. Terwyl die eerste voorbeelde nie eksplisiet was nie, fokus ons op eksplisiete torings, dit is torings waar die vergelykings wat rekursief die uitbreidings Fi+d F; bepaal bekend is. Dit is bekend dat as die kardinaliteit van die konstante liggaam van die toring 'n volkome vierkant is, dit moontlik is om die Drinfeld- Vladut bogrens vir die limiet te behaal, selfs in die eksplisiete geval. As die konstante liggaam nie 'n kwadratiese kardinaliteit het nie, is dit onbekend hoe naby die limiet van die toring aan hierdie bogrens kan kom. In hierdie tesis salons die teorie ontwikkel wat benodig word om eksplisiete torings van funksieliggame te konstrueer, en hulle asimptotiese gedrag te analiseer. Verskeie torings sal aangebied word en algemene families van eksplisiete formules waarvoor die splitsingsgedrag en groei van die genus in 'n toring bereken kan word, sal bespreek word. Wanneer die nodige teorie ontwikkel is, salons fokus op die geval van torings oor liggame waarvan die kardinaliteit nie 'n volkome vierkant is nie, en op die oop probleem aangaande hoe goed die asimptotiese gedrag van 'n toring onder hierdie omstandighede kan wees

Error Correcting Codes on Algebraic Surfaces

Error correcting codes are defined and important parameters for a code are explained. Parameters of new codes constructed on algebraic surfaces are studied. In particular, codes resulting from blowing up points in \proj^2 are briefly studied, then codes resulting from ruled surfaces are covered. Codes resulting from ruled surfaces over curves of genus 0 are completely analyzed, and some codes are discovered that are better than direct product Reed Solomon codes of similar length. Ruled surfaces over genus 1 curves are also studied, but not all classes are completely analyzed. However, in this case a family of codes are found that are comparable in performance to the direct product code of a Reed Solomon code and a Goppa code. Some further work is done on surfaces from higher genus curves, but there remains much work to be done in this direction to understand fully the resulting codes. Codes resulting from blowing points on surfaces are also studied, obtaining necessary parameters for constructing infinite families of such codes. Also included is a paper giving explicit formulas for curves with more \field{q}-rational points than were previously known for certain combinations of field size and genus. Some upper bounds are now known to be optimal from these examples.Comment: This is Chris Lomont's PhD thesis about error correcting codes from algebriac surface

Hash Families and Cover-Free Families with Cryptographic Applications

This thesis is focused on hash families and cover-free families and their application to problems in cryptography. We present new necessary conditions for generalized separating hash families, and provide new explicit constructions. We then consider three cryptographic applications of hash families and cover-free families. We provide a stronger de nition of anonymity in the context of shared symmetric key primitives and give a new scheme with improved anonymity properties. Second, we observe that nding the invalid signatures in a set of digital signatures that fails batch veri cation is a group testing problem, then apply and compare many group testing algorithms to solve this problem e ciently. In particular, we apply group testing algorithms based on cover-free families. Finally, we construct a one-time signature scheme based on cover-free families with short signatures

Knots, Trees, and Fields: Common Ground Between Physics and Mathematics

One main theme of this thesis is a connection between mathematical physics (in particular, the three-dimensional topological quantum field theory known as Chern-Simons theory) and three-dimensional topology. This connection arises because the partition function of Chern-Simons theory provides an invariant of three-manifolds, and the Wilson-loop observables in the theory define invariants of knots. In the first chapter, we review this connection, as well as more recent work that studies the classical limit of quantum Chern-Simons theory, leading to relations to another knot invariant known as the A-polynomial. (Roughly speaking, this invariant can be thought of as the moduli space of flat SL(2,C) connections on the knot complement.) In fact, the connection can be deepened: through an embedding into string theory, categorifications of polynomial knot invariants can be understood as spaces of BPS states. We go on to study these homological knot invariants, and interpret spectral sequences that relate them to one another in terms of perturbations of supersymmetric theories. Our point is more general than the application to knots; in general, when one perturbs any modulus of a supersymmetric theory and breaks a symmetry, one should expect a spectral sequence to relate the BPS states of the unperturbed and perturbed theories. We consider several diverse instances of this general lesson. In another chapter, we consider connections between supersymmetric quantum mechanics and the de Rham version of homotopy theory developed by Sullivan; this leads to a new interpretation of Sullivan's minimal models, and of Massey products as vacuum states which are entangled between different degrees of freedom in these models. We then turn to consider a discrete model of holography: a Gaussian lattice model defined on an infinite tree of uniform valence. Despite being discrete, the matching of bulk isometries and boundary conformal symmetries takes place as usual; the relevant group is PGL(2,Qp), and all of the formulas developed for holography in the context of scalar fields on fixed backgrounds have natural analogues in this setting. The key observation underlying this generalization is that the geometry underlying AdS3/CFT2 can be understood algebraically, and the base field can therefore be changed while maintaining much of the structure. Finally, we give some analysis of A-polynomials under change of base (to finite fields), bringing things full circle.</p

On the splitting of places in a tower of function fields meeting the Drinfeld-Vladut bound

A description of how places split in an asymptotically optimal tower of function fields studied by Garcia and Stichtenoth (1995) is provided and an exact count of the number of places of degree one is given. This information is useful in the setting up of generator matrices for algebraic-geometry codes constructed over this function field tower. These long codes have performance that asymptotically improves upon the Gilbert-Varshamov bound