3-DESIGNS DERIVED FROM PLANE ALGEBRAIC CURVES

3-designs Derived from Plane Algebraic Curves

In this paper, we develop a simple method for computing the stabilizer subgroup of a subgroup of D(g) = {α ∈ Fq | there is a β ∈ Fq such that β = g(α)} in PSL2(Fq), where q is a large odd prime power, n is a positive integer dividing q − 1, and g(x) ∈ Fq [x]. As an application, we construct new infinite families of 3-designs (cf. Examples 3.4 and 3.5).

Plane Algebraic Curves

We go over some of the basics of plane algebraic curves, which are planar curves described as the set of solutions of a polynomial in two variables. We study many basic notions, such as projective space, parametrization, and the intersection of two curves. We end with the group law on the cubic and search for torsion points.

Real Plane Algebraic Curves

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi–definite nondefinite or definite. We present a discussion about isolated points. By means of the operator, these points can ...

Topology of Plane Algebraic Curves

Fast decoding of codes from algebraic plane curves

Improvement to an earlier decoding algorithm for codes from algebraic geometry is presented. For codes from an arbitrary regular plane curve we correct up to d*/2 m2/8 + m /4 9/8 errors, where d* is the designed distance of the code and m is the degree of the curve. The complexity of finding the error locator is O(n7/3), where n is the length of the code. For codes from Hermitian curves the com...