Value Distributions of Exponential Sums From Perfect Nonlinear Functions and Their Applications
Keqin Feng, Jinquan Luo · IEEE Transactions on Information Theory · 2007
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$.