The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ l $\end{document}</tex-math></inline-formula> be a prime with <inline-formula><tex-math id="M2">\begin{document}$ l\equiv 3\pmod 4 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ l\ne 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N = l^m $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula> a positive integer, <inline-formula><tex-math id="M6">\begin{document}$ f = \phi(N)/2 $\end{document}</tex-math></inline-formula> the multiplicative order of a prime <inline-formula><tex-math id="M7">\begin{document}$ p $\end{document}</tex-math></inline-formula> modulo <inline-formula><tex-math id="M8">\begin{document}$ N $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ q = p^f $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M10">\begin{document}$ \phi(\cdot) $\end{document}</tex-math></inline-formula> is the Euler-function. Let <inline-formula><tex-math id="M11">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> be a primitive element of a finite field <inline-formula><tex-math id="M12">\begin{document}$ \Bbb F_{q} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ C_0^{(N,q)} = \langle \alpha^N\rangle $\end{document}</tex-math></inline-formula> a cyclic subgroup of the multiplicative group <inline-formula><tex-math id="M14">\begin{document}$ \Bbb F_q^* $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M15">\begin{document}$ C_i^{(N,q)} = \alpha^i\langle \alpha^N\rangle $\end{document}</tex-math></inline-formula> the cosets, <inline-formula><tex-math id="M16">\begin{document}$ i = 0,\ldots, N-1 $\end{document}</tex-math></inline-formula>. In this paper, we use Gaussian sums to obtain the explicit values of <inline-formula><tex-math id="M17">\begin{document}$ \eta_i^{(N, q)} = \sum_{x \in C_i^{(N,q)}}\psi(x) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M18">\begin{document}$ i = 0,1,\cdots, N-1 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M19">\begin{document}$ \psi $\end{document}</tex-math></inline-formula> is the canonical additive character of <inline-formula><tex-math id="M20">\begin{document}$ \Bbb F_{q} $\end{document}</tex-math></inline-formula>. Moreover, we also compute the explicit values of <inline-formula><tex-math id="M21">\begin{document}$ \eta_i^{(2N, q)} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M22">\begin{document}$ i = 0,1,\cdots, 2N-1 $\end{document}</tex-math></inline-formula>, if <inline-formula><tex-math id="M23">\begin{document}$ q $\end{document}</tex-math></inline-formula> is a power of an odd prime <inline-formula><tex-math id="M24">\begin{document}$ p $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>As an application, we investigate the weight distribution of a <inline-formula><tex-math id="M25">\begin{document}$ p $\end{document}</tex-math></inline-formula>-ary linear code: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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}\}, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where its defining set <inline-formula><tex-math id="M26">\begin{document}$ D $\end{document}</tex-math></inline-formula> is given by <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ D = \{x\in \Bbb{F}_{q}^{*}: \operatorname{Tr}_{q/p}(x^{\frac{q-1}{l^{m}}}) = 0\} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>and <inline-formula><tex-math id="M27">\begin{document}$ \operatorname{Tr}_{q/p} $\end{document}</tex-math></inline-formula> denotes the trace function from <inline-formula><tex-math id="M28">\begin{document}$ \Bbb F_{q} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M29">\begin{document}$ \Bbb F_p $\end{document}</tex-math></inline-formula>.