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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)

- Johan P. Hansen, Henning Stichtenoth
- Appl. Algebra Eng. Commun. Comput.
- 1990

- Johan P. Hansen
- Appl. Algebra Eng. Commun. Comput.
- 2002

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)

- Johan P. Hansen
- Appl. Algebra Eng. Commun. Comput.
- 2003

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)

- Johan P. Hansen, Trygve Johnsen, Kristian Ranestad
- Finite Fields and Their Applications
- 2007

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)

- JOHAN P. HANSEN
- 2014

Secret Sharing Schemes with a large number of players from Tori Secret Sharing Schemes The methods of toric varieties Toric surfaces, fans, Cartier divisors and cohomology

- Johan P. Hansen
- IEEE Trans. Information Theory
- 1987

— 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)