63 research outputs found
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
Distance bounds for algebraic geometric codes
Various methods have been used to obtain improvements of the Goppa lower
bound for the minimum distance of an algebraic geometric code. The main methods
divide into two categories and all but a few of the known bounds are special
cases of either the Lundell-McCullough floor bound or the Beelen order bound.
The exceptions are recent improvements of the floor bound by
Guneri-Stichtenoth-Taskin, and Duursma-Park, and of the order bound by
Duursma-Park and Duursma-Kirov. In this paper we provide short proofs for all
floor bounds and most order bounds in the setting of the van Lint and Wilson AB
method. Moreover, we formulate unifying theorems for order bounds and formulate
the DP and DK order bounds as natural but different generalizations of the
Feng-Rao bound for one-point codes.Comment: 29 page
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