1996 Project Report Weight Distributions

Weight distributions of certain generalized codes (Led by Giuliana Davidoff).

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.