How to Break Gilbert-Varshamov Bound

Modular curves provide the first known examples of
codes which are better than random ones. However, for
an explicit construction of a code one needs a
nonsingular model of the modular curve that initially
is defined via a very singular planar model given by
a modular equation. In this work we analyze the
structure of its affine singularities in terms of
class numbers of binary quadratic forms. In
principle this allows to describe the space of
regular differentials vanishing at a point needed for
the Goppa construction.

Autorentext

Dr.Orhun Kara received his PhD from Bilkent University in Turkeyin 2003.Currently he has been working as a chief researcher inTÜBITAK UEKAE.Professor Alexander Klyachko got his PhD in Saratov StateUniversity (Russia) in 1975. His mathematical interests includeLiegroups,Representation theory, Algebraic Geometry, Modular formsand Coding Theory.

Klappentext

Modular curves provide the first known examples ofcodes which are better than random ones. However, foran explicit construction of a code one needs anonsingular model of the modular curve that initiallyis defined via a very singular planar model given bya modular equation. In this work we analyze thestructure of its affine singularities in terms ofclass numbers of binary quadratic forms. Inprinciple this allows to describe the space ofregular differentials vanishing at a point needed forthe Goppa construction.