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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
- Applicable Algebra in Engineering, Communication…
- 1990

We construct a series of algebraic geometric codes using a class of curves which have many rational points. We obtain codes of lengthq 2 over $$\mathbb{F}$$ q , whereq = 2q 0 2 andq 0 = 2 n , such that dimension + minimal distance ≧q 2 + 1 − q 0 (q − 1). The codes are ideals in the group algebra $$\mathbb{F}$$ q [S], whereS is a Sylow-2-subgroup of orderq 2… (More)

- Johan P. Hansen
- Applicable Algebra in Engineering, Communication…
- 2002

For any integral convex polytope in ℝ 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
- Applicable Algebra in Engineering, Communication…
- 2003

This note is meant to be an introduction to cohomological 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⊂ℙ2 and a lower bound for the minimal distance is obtained through linkage. By way of example our… (More)

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, . . . , xn) = (y1, . . . , ym) is an algebraic problem, we consider serving on-line requests of the form “change input xi to value v” or “what is the value of output yi?”. We present… (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)

Abstract. 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 there is an explicit construction of a unique error-correcting code of length (q − 1) over the finite field Fq. The dimension of… (More)

- Peter P Groenewegen, Reinhard Busse, +4 authors Willemijn Schäfer
- European journal of public health
- 2011

- Willemijn Schäfer, Peter P Groenewegen, Johan Hansen, Nick Black
- Quality in primary care
- 2011

BACKGROUND
All European health systems face several common challenges related to increases in lifestyle and chronic diseases, a decreasing future workforce, inequalities in health and the consequences of societal changes. Primary care, which has the potential to help meet these challenges, would benefit from the contribution of health services research… (More)

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