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

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper we present a unified way to determine the values and their multiplicities of the exponential sums <emphasis><formula formulatype="display"><tex>$$\sum _{x\in\BBF _{q}}\zeta _{p}^{{\rm Tr}\left(af(x)+bx\right)}\left(a,b\in \BBF _{q},q=p^{m},p\ge 3\right)$$</tex> </formula></emphasis> for all perfect nonlinear functions <emphasis><formula formulatype="inline"><tex>$f$</tex> </formula></emphasis> which is a Dembowski–Ostrom polynomial or <emphasis><formula formulatype="inline"><tex>$p\!=\!3$</tex></formula></emphasis>, <emphasis><formula formulatype="inline"><tex>$f\!=\!x^{{ 3^{k} + 1}\over { 2}}$</tex></formula></emphasis> where <emphasis><formula formulatype="inline"><tex>$k$</tex></formula></emphasis> is odd and <emphasis><formula formulatype="inline"><tex>$(k,m)\!=\!1.\break$</tex></formula></emphasis>As applications, we determine 1) the correlation distribution of the <emphasis><formula formulatype="inline"><tex>$m$</tex></formula></emphasis>-sequence <emphasis><formula formulatype="inline"><tex>$\left \{a_{\lambda }= {\rm Tr}(\gamma ^{\lambda })\right \}({\lambda =0,1,\ldots })$</tex></formula></emphasis> and the sequence <emphasis><formula formulatype="inline"><tex>$\left \{b_{\lambda }= {\rm Tr}\left (f(\gamma ^{\lambda })\right)\right \}({\lambda =0,1,\ldots })$</tex></formula></emphasis> over <emphasis><formula formulatype="inline"><tex>$\BBF _{p}$</tex></formula></emphasis> where <emphasis><formula formulatype="inline"><tex>$\gamma $</tex></formula></emphasis> is a primitive element of <emphasis><formula formulatype="inline"><tex>$\BBF _{q}$</tex> </formula></emphasis> and 2) the weight distributions of the linear codes over <emphasis><formula formulatype="inline"><tex>$\BBF _{p}$</tex></formula></emphasis> defined by <emphasis><formula formulatype="inline"><tex>$f$</tex></formula></emphasis>. </para>