Adv. Math. Commun. | 2021
The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes
Abstract
Let \\begin{document}$ l $\\end{document} be a prime with \\begin{document}$ l\\equiv 3\\pmod 4 $\\end{document} and \\begin{document}$ l\\ne 3 $\\end{document} , \\begin{document}$ N = l^m $\\end{document} for \\begin{document}$ m $\\end{document} a positive integer, \\begin{document}$ f = \\phi(N)/2 $\\end{document} the multiplicative order of a prime \\begin{document}$ p $\\end{document} modulo \\begin{document}$ N $\\end{document} , and \\begin{document}$ q = p^f $\\end{document} , where \\begin{document}$ \\phi(\\cdot) $\\end{document} is the Euler-function. Let \\begin{document}$ \\alpha $\\end{document} be a primitive element of a finite field \\begin{document}$ \\Bbb F_{q} $\\end{document} , \\begin{document}$ C_0^{(N,q)} = \\langle \\alpha^N\\rangle $\\end{document} a cyclic subgroup of the multiplicative group \\begin{document}$ \\Bbb F_q^* $\\end{document} , and \\begin{document}$ C_i^{(N,q)} = \\alpha^i\\langle \\alpha^N\\rangle $\\end{document} the cosets, \\begin{document}$ i = 0,\\ldots, N-1 $\\end{document} . In this paper, we use Gaussian sums to obtain the explicit values of \\begin{document}$ \\eta_i^{(N, q)} = \\sum_{x \\in C_i^{(N,q)}}\\psi(x) $\\end{document} , \\begin{document}$ i = 0,1,\\cdots, N-1 $\\end{document} , where \\begin{document}$ \\psi $\\end{document} is the canonical additive character of \\begin{document}$ \\Bbb F_{q} $\\end{document} . Moreover, we also compute the explicit values of \\begin{document}$ \\eta_i^{(2N, q)} $\\end{document} , \\begin{document}$ i = 0,1,\\cdots, 2N-1 $\\end{document} , if \\begin{document}$ q $\\end{document} is a power of an odd prime \\begin{document}$ p $\\end{document} . As an application, we investigate the weight distribution of a \\begin{document}$ p $\\end{document} -ary linear code: \\begin{document}$ \\mathcal{C}_{D} = \\{C = ( \\operatorname{Tr}_{q/p}(c x_1), \\operatorname{Tr}_{q/p}(cx_2),\\ldots, \\operatorname{Tr}_{q/p}(cx_n)):c\\in \\Bbb{F}_{q}\\}, $\\end{document} where its defining set \\begin{document}$ D $\\end{document} is given by \\begin{document}$ D = \\{x\\in \\Bbb{F}_{q}^{*}: \\operatorname{Tr}_{q/p}(x^{\\frac{q-1}{l^{m}}}) = 0\\} $\\end{document} and \\begin{document}$ \\operatorname{Tr}_{q/p} $\\end{document} denotes the trace function from \\begin{document}$ \\Bbb F_{q} $\\end{document} to \\begin{document}$ \\Bbb F_p $\\end{document} .