Explicit evaluation of some exponential sums
Let \(m\) be a positive integer and let \(r\) be a prime such that 2 is a primitive root modulo \(r^m\). Set \(q=2^{\varphi (r^m)}\), \(\varphi\) the Euler function, and let \(\chi\) be the canonical additive character of \(\mathbb F_q\). For \(a,b\in \mathbb F_q\), the exponential sum \[ S(a,b)=\sum_{x\in\mathbb F_q^*} \chi (ax^{(q-1)/r^m}+bx) \] is evaluated. The answer is in terms of the Hamming weights of subvectors of a certain coordinate vector of \(ab^{-(q-1)/r^m}\). This is applied to the weight distribution of the dual of a binary cyclic code.
scientific article; zbMATH DE number 5651168