The quadratic extension extractor for (hyper)elliptic curves in odd characteristic

We propose a simple and efficient deterministic extractor for the (hyper)elliptic curve C, defined over Fq2, where q is some power of an odd prime. Our extractor, for a given point P on C, outputs the first Fq-coefficient of the abscissa of the point P. We show that if a point P is chosen uniformly at random in C, the element extracted from the point P is indistinguishable from a uniformly random variable in Fq

The shift bound for cyclic, Reed-Muller and geometric Goppa codes

We give a generalization of the shift bound on the minimum distance for cyclic codes which applies to Reed—Muller and algebraic-geometric codes. The number of errors one can correct by majority coset decoding is up to hdlf the shift bound

Multiplicative structures on the minimal resolution of determinantal rings

It is shown that the minimal resolution of a determinantal ring has the structure of an associative differential graded algebra

On the Klein quartic, the Fano plane and curves representing designs

No abstract

On the existence of error-correcting pairs

Algebraic-geometric codes have a t-error-correcting pair which corrects errors up to half the designed minimum distance. A generalization of the Roos bound is given from cyclic to linear codes. An MDS code of minimum distance 5 has a 2-error-correcting pair if and only if it is an extended-generalized-Reed–Solomon code

Deformations of hypersurfaces with a one-dimensional singular locus

No abstract