3,934 research outputs found
Codes Over Rings from Curves of Higher Genus
We construct certain error-correcting codes over finite rings and estimate their parameters. These codes are constructed using plane curves and the estimates for their parameters rely on constructing “lifts” of these curves and then estimating the size of certain exponential sums.
THE purpose of this paper is to construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools; notably, an estimate for the dimension of trace codes over rings (generalizing work of van der Vlugt over fields and some results on lifts of affin curves from field of characteristic p to Witt vectors of length two. This work partly generalizes our previous work on elliptic curves, although there are some differences which we will point out below
On the maximum number of rational points on singular curves over finite fields
We give a construction of singular curves with many rational points over
finite fields. This construction enables us to prove some results on the
maximum number of rational points on an absolutely irreducible projective
algebraic curve defined over Fq of geometric genus g and arithmetic genus
Two-point coordinate rings for GK-curves
Giulietti and Korchm\'aros presented new curves with the maximal number of
points over a field of size q^6. Garcia, G\"uneri, and Stichtenoth extended the
construction to curves that are maximal over fields of size q^2n, for odd n >=
3. The generalized GK-curves have affine equations x^q+x = y^{q+1} and
y^{q^2}-y^q = z^r, for r=(q^n+1)/(q+1). We give a new proof for the maximality
of the generalized GK-curves and we outline methods to efficiently obtain their
two-point coordinate ring.Comment: 16 page
Superspecial rank of supersingular abelian varieties and Jacobians
An abelian variety defined over an algebraically closed field k of positive
characteristic is supersingular if it is isogenous to a product of
supersingular elliptic curves and is superspecial if it is isomorphic to a
product of supersingular elliptic curves. In this paper, the superspecial
condition is generalized by defining the superspecial rank of an abelian
variety, which is an invariant of its p-torsion. The main results in this paper
are about the superspecial rank of supersingular abelian varieties and
Jacobians of curves. For example, it turns out that the superspecial rank
determines information about the decomposition of a supersingular abelian
variety up to isomorphism; namely it is a bound for the maximal number of
supersingular elliptic curves appearing in such a decomposition.Comment: V2: New coauthor, major rewrit
Exponential Sums Along p-adic Curves
Let K be a p-adic field, R the valuation ring of K, and P the maximal ideal
of R. Let be a non-singular closed curve, and Y_{m} its
image in R/P^{m} times R/P^{m}, i.e. the reduction modulo P^{m} of Y. We denote
by Psi an standard additive character on K. In this paper we discuss the
estimation of exponential sums of type S_{m}(z,Psi,Y,g):= sum\limits_{x in
Y_{m}} Psi(zg(x)), with z in K, and g a polynomial function on Y. We show that
if the p-adic absolute value of z is big enough then the complex absolute value
of S_{m}(z,Psi,Y,g) is O(q^{m(1-beta(f,g))}), for a positive constant beta(f,g)
satisfying 0<beta(f,g)<1.Comment: 9 pages. Accepted in Finite Fields and Their Application
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