141 research outputs found

    Explicit Riemann-Roch spaces in the Hilbert class field

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    Let K\mathbf K be a finite field, XX and YY two curves over K\mathbf K, and Y→XY\rightarrow X an unramified abelian cover with Galois group GG. Let DD be a divisor on XX and EE its pullback on YY. Under mild conditions the linear space associated with EE is a free K[G]{\mathbf K}[G]-module. We study the algorithmic aspects and applications of these modules

    Bases and applications of Riemann-Roch Spaces of Function Fields with Many Rational Places

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    Algebraic geometry codes are generalizations of Reed-Solomon codes, which are implemented in nearly all digital communication devices. In ground-breaking work, Tsfasman, Vladut, and Zink showed the existence of a sequence of algebraic geometry codes that exceed the Gilbert-Varshamov bound, which was previously thought unbeatable. More recently, it has been shown that multipoint algebraic geometry codes can outperform comparable one-point algebraic geometry codes. In both cases, it is desirable that these function fields have many rational places. The prototypical example of such a function field is the Hermitian function field which is maximal. In 2003, Geil produced a new family of function fields which contain the Hermitian function field as a special case. This family, known as the norm-trace function field, has the advantage that codes from it may be defined over alphabets of larger size. The main topic of this dissertation is function fields arising from linearized polynomials; these are a generalization of Geil\u27s norm-trace function field. We also consider applications of this function field to error-correcting codes and small-bias sets. Additionally, we study certain Riemann-Roch spaces and codes arising from the Suzuki function field. The Weierstrass semigroup of a place on a function field is an object of classical interest and is tied to the dimension of associated Riemann-Roch spaces. In this dissertation, we derive Weierstrass semigroups of m-tuples of places on the norm-trace function fields. In addition, we also discuss Weierstrass semigroups from finite graphs

    AG Codes from Polyhedral Divisors

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    A description of complete normal varieties with lower dimensional torus action has been given by Altmann, Hausen, and Suess, generalizing the theory of toric varieties. Considering the case where the acting torus T has codimension one, we describe T-invariant Weil and Cartier divisors and provide formulae for calculating global sections, intersection numbers, and Euler characteristics. As an application, we use divisors on these so-called T-varieties to define new evaluation codes called T-codes. We find estimates on their minimum distance using intersection theory. This generalizes the theory of toric codes and combines it with AG codes on curves. As the simplest application of our general techniques we look at codes on ruled surfaces coming from decomposable vector bundles. Already this construction gives codes that are better than the related product code. Further examples show that we can improve these codes by constructing more sophisticated T-varieties. These results suggest to look further for good codes on T-varieties.Comment: 30 pages, 9 figures; v2: replaced fansy cycles with fansy divisor

    Counting curves over finite fields

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    This is a survey on recent results on counting of curves over finite fields. It reviews various results on the maximum number of points on a curve of genus g over a finite field of cardinality q, but the main emphasis is on results on the Euler characteristic of the cohomology of local systems on moduli spaces of curves of low genus and its implications for modular forms.Comment: 25 pages, to appear in Finite Fields and their Application
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