Hyper-Kloosterman Sums and their Applications to the Coding Theory

The dual code of the Melas code is called the Kloosterman code. The weights of its codewords can be expressed by the Kloosterman sums, and are uniformly distributed with respect to the Sato-Tate measure. In this paper, the hyper-Kloosterman code C _m (q), a generalization of the Kloosterman code is defined, and the uniform distribution property is deduced using the hyper-Kloosterman sums when m is even and p− 1|m. Finally we discuss doubly-evenness for the weights in the binary case. It shows that we can construct infinitely many doubly-even codes in a non-trivial way. Received: October 30, 2000; revised version: February 28, 2001