Value Distributions of Exponential Sums From Perfect Nonlinear Functions and Their Applications

In this paper we present a unified way to determine the values and their multiplicities of the exponential sums $$\sum _{x\in\BBF _{q}}\zeta _{p}^{{\rm Tr}\left(af(x)+bx\right)}\left(a,b\in \BBF _{q},q=p^{m},p\ge 3\right)$$ for all perfect nonlinear functions $f$ which is a Dembowski–Ostrom polynomial or $p\!=\!3$, $f\!=\!x^{{ 3^{k} + 1}\over { 2}}$ where $k$ is odd and $(k,m)\!=\!1.\break$As applications, we determine 1) the correlation distribution of the $m$-sequence $\left \{a_{\lambda }= {\rm Tr}(\gamma ^{\lambda })\right \}({\lambda =0,1,\ldots })$ and the sequence $\left \{b_{\lambda }= {\rm Tr}\left (f(\gamma ^{\lambda })\right)\right \}({\lambda =0,1,\ldots })$ over $\BBF _{p}$ where $\gamma $ is a primitive element of $\BBF _{q}$ and 2) the weight distributions of the linear codes over $\BBF _{p}$ defined by $f$.