The summer project can be described as follows: Let k be a field with
a prime number p of elements. The Kloosterman sum for a in k is
defined to be the exponential sum $ S(a,p) = \sum_{x \in k^*} exp(2
\pi i (ax + x^{-1})/p) $. Various generalizations of this sum have
been defined, including the so-called higher-order Kloosterman sums
which depend on positive integers and also those for finite extension
fields.
The exponents in these expressions appear in the study of weight
distributions and frequencies of certain cyclic codes. Extensive work
has been done in the cases p = 2,3 by Rene Schoof, Marcel van der
Vlugt and Gerard van der Geer.
The group used various generalized sums to learn more about the
properties of the related codes in associated higher genus settings.
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