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Preface These notes treat the problem of counting the number of rational points on a curve defined over a finite field. The notes are an extended version of an earlier set of notes Aritmetisk Algebraisk Geometri – Kurver by Johan P. Hansen [Han] on the same subject. In Chapter 1 we summarize the basic notions of algebraic geometry, especially rational… (More)

For any integral convex polytope in R 2 there is an explicit construction of an error-correcting code of length (q − 1) 2 over the finite field F q , obtained by evaluation of rational functions on a toric surface associated to the polytope. The dimension of the code is equal to the number of integral points in the given polytope and the minimum distance is… (More)

This note is meant to be an introduction to coho-mological methods and their use in the theory of error-correcting codes. In particular we consider evaluation codes on a complete intersection. The dimension of the code is determined by the Koszul complex for X ⊂ P 2 and a lower bound for the minimal distance is obtained through linkage. By way of example… (More)

We treat toric surfaces and their application to construction of error-correcting codes and determination of the parameters of the codes, surveying and expanding the results of [4]. For any integral convex polytope in R 2 there is an explicit construction of a unique error-correcting code of length (q − 1) 2 over the finite field Fq. The dimension of the… (More)

Foundation 1 Abstract We consider dynamic evaluation of algebraic functions (matrix mul-) is an algebraic problem, we consider serving on-line requests of the form " change input x i to value v " or " what is the value of output y i ? ". We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The… (More)

We study subsets of Grassmann varieties G(l, m) over a field F , such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study the linear spans of, and in case of positive characteristic, the number of Fq-rational points on such unions. Moreover we study a geometric duality of such unions, and give a combinatorial… (More)

— We study subsets of Grassmann varieties G(l, m) over a field F , such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study such sets in detail, and give applications to coding theory, in particular for Grassmann codes. For l = 2 much is known about such Schubert unions with a maximal number of Fq-rational points for a… (More)