A class of functions with low-valued Walsh spectrum

Let $l\equiv 3\pmod 4$, $l\ne 3$, be a prime, $N=l^2$, $f=\frac{l(l-1)}2$ the multiplicative order of a prime $p$ modulo $N$, and $q=p^f$. In this paper, we investigate the Walsh spectrum of the monomial functions $f(x)={\rm Tr}_{q/p}(x^{\frac{q-1}{l^2}})$ in index two case. In special, we explicitly present the value distribution of the Walsh transform of $f(x)$ if $1+l=4p^h$, where $h$ is a class number of $\Bbb Q(\sqrt{-l})$.