Let $p$ be a prime. An $[n, k]$-linear code $\mathcal{C}$ over the finite field $\mathbb{F}_p$ is a $k$-dimensional linear subspace of $\mathbb{F}_p^n$. Moreover, if $(c_0, c_1, \cdots, c_{n-1})\in \mathcal{C}$ implies $(c_{n-1}, c_0, \cdots, c_{n-2}) \in \mathcal{C}$ then $\mathcal{C}$ is called a cyclic code. For a cyclic code $\mathcal{C}$ with length $n$ over $\mathbb{F}_p$, let $A_i$ be the number of codewords in $\mathcal{C}$ with Hamming weight $i$. The sequence $(1, A_1, A_2, \cdots, A_n)$ is called the weight distribution of $\mathcal{C}$. The weight distribution of a code is an important research object in coding theory. If $\mathcal{C}$ is cyclic, the weight of each codeword can be expressed by exponential sums, so the weight distribution of $\mathcal{C}$ can be determined if the corresponding exponential sums (or their certain combinations) can be calculated explicitly (see [1-8]).

The value distribution of the exponential sum $S(\alpha, \beta)=\sum\limits_{x\in\mathbb{F}_{p^m}}\chi(\alpha x^{d_1}+\beta x^{d_2})$ and the weight distribution of the cyclic code

$ C=\left\{c(\alpha, \beta)=\left({\rm{Tr}}_1^m(\alpha x^{d_1}+\beta x^{d_2})\right)_{x\in\mathbb{F}_{p^m}^*}|(\alpha, \beta)\in\mathbb{F}_{p^m}^2\right\} $

were extensively studied, where $\chi(\cdot)=\zeta_p^{{\rm{Tr}}_1^m(\cdot)}$ is the canonical additive character on the finite field $\mathbb{F}_{p^m}, {\rm{Tr}}_1^m(\cdot)$ is the trace mapping from $\mathbb{F}_{p^m}$ to $\mathbb{F}_p$, and $\zeta_p=\exp({2\pi\sqrt{-1}}/p)$ is a primitive $p$-th root of unity. For $d_1=p^k+1, d_2=2$, the exponential sum $S(\alpha, \beta)$ and the associated cyclic code $\mathcal{C}$ were studied in [2]. For $d_1=(p^k+1)/2, d_2=1$, the value distribution of $S(\alpha, \beta)$ and the weight distribution of $\mathcal{C}$ were derived in [3]. When $d_1=p^k+1, d_2=p^{3k}+1$, the weight distribution of $\mathcal{C}$ was determined in [5] for ${\rm{for}}\frac{m}{{\gcd \left( {m,k} \right)}}$ odd, and in [6, 7] for ${\rm{for}}\frac{m}{{\gcd \left( {m,k} \right)}}$ even.

In this paper, we will study the exponential sum

$ S(\alpha, \beta)=\sum\limits_{x\in\mathbb{F}_{p^m}}\chi(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2}), $

and determine the weight distibution of the cyclic code $\mathcal{C}=\left\{c=c(\alpha, \beta)\mid(\alpha, \beta)\in\mathbb{F}_{p^m}\right\}, $ where

$ c(\alpha, \beta)=\left({\rm{Tr}}_1^m(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2})\right)_{x\in\mathbb{F}_{p^m}^*}, $

$m/\gcd(m, k)$ is odd and $k/\gcd(m, k)$ is even.

This paper is presented as follows. In Section 2, we introduce some definitions and auxiliary results that will be needed later in this paper. In Section 3, we determine the value distribution of $S(\alpha, \beta)$ and the weight distribution of the cyclic code $\mathcal{C}$.

The following notations are fixed throughout this paper.

(a) Let $m$ and $k$ be positive integers such that $s=m/e$ is odd, $k/e$ is even and $m\geq3$, where $e=\gcd(m, k)$. Let $p$ be an odd prime, $q=p^m$, $q_0=p^e$, $q_0^*=(-1)^\frac{q_0-1}{2}q_0$.

(b) Let $\mathbb{F}_{p^i}$ be the finite field with $p^i$ elements, and $\mathbb{F}_{p^i}^\ast=\mathbb{F}_{p^i}\backslash\{0\}$.

(c) Let ${\rm{Tr}}_i^j:\mathbb{F}_{p^j}\rightarrow\mathbb{F}_{p^i}$ be the trace mapping defined by ${\rm{Tr}}_i^j(x)=\sum\limits_{l=0}^{j/i-1}x^{p^{il}}$ for $i|j$. For $x\in\mathbb{F}_q$, define $\chi(x)=\zeta_p^{{\rm{Tr}}_1^m(x)}$ to be the canonical additive character of $\mathbb{F}_q$, where $\zeta_p=\exp({2\pi\sqrt{-1}}/p)$ is a $p$-th root of unity.

From now on, we assume that $\lambda$ is a fixed nonsquare in $\mathbb{F}_{q_0}$. Note that $s$ is odd and $k/e$ is even. It is easy to get that $\lambda$ is also a nonsquare in $\mathbb{F}_q$ and $\lambda^{(p^{3k}+1)/2}=\lambda^{(p^k+1)/2}=\lambda$. Since the union of the images of maps $x\mapsto x^2$ and $x\mapsto \lambda x^2$ covers each element of $\mathbb{F}_q$ exactly two times, then we have

$ S(\alpha ,\beta )\quad = \quad \sum\limits_{x \in {\mathbb{F}_q}} \chi \left( {\alpha {x^{\frac{{{p^k} + 1}}{2}}} + \beta {x^{\frac{{{p^{3k}} + 1}}{2}}}} \right) = \quad \frac{1}{2}\left( {T(\alpha ,\beta ) + T(\lambda \alpha ,\lambda \beta )} \right), $

(2.1)

where

$ T(\alpha, \beta)=\sum\limits_{x\in \mathbb{F}_q}\zeta_p^{{\rm{Tr}}_1^m(\alpha x^{p^k+1}+\beta x^{p^{3k}+1})}. $

(2.2)

The exponential sum $T(\alpha, \beta)$ have been extensively studied in [4-6]. This is an important tool we will use.

Definition 2.1  [9] The quadratic character of $\mathbb{F}_{q_0}$ is defined as

$ \eta_0(x)=\left\{ \begin{array}{ll} 1, \ \ \ \ \ \ \ \ \ \ \ &\mbox{if $x$ is a nonzero square in $\mathbb{F}_{q_0}$}, \\ -1, \ \ \ \ \ \ \ \ \ \ \ &\mbox{if $x$ is a nonsquare in $\mathbb{F}_{q_0}$}, \\ 0, \ \ \ &\mbox{if $x=0$}. \end{array} \right. $

Definition 2.2  [9] A quadratic form in $s$ indeterminates over $\mathbb{F}_{q_0}$ is a homogeneous polynomial in $\mathbb{F}_{q_0}[x_1, x_2, \cdots, x_s]$ of degree $2$ and can be uniquely expressed as

$ f(x_1, x_2, \cdots, x_s)=\sum\limits_{i, j=1}^s h_{ij}x_ix_j \ \ {\rm with}~~\ h_{ij}=h_{ji}\in\mathbb{F}_{q_0}. $

The $s\times s$ symmetric matrix $H$ whose $(i, j)$ entry is $h_{ij}$ is called the coefficient matrix of $f$. Let $r$ be the rank of $H$. Then, there exists $M\in GL_s(\mathbb{F}_{q_0})$ such that $H'=MHM^T=\mathrm {diag}(a_1, \cdots, a_r, 0, \cdots, 0)$ is a diagonal matrix where $a_i\in\mathbb{F}_{q_0}^\ast(1\leq i\leq r)$. Let $X=(x_1, x_2, \cdots, x_s)$, making a nonsingular linear substitution $X=YM$ with $Y=(y_1, y_2, \cdots, y_s)\in\mathbb{F}_{q_0}^s$, then we have

$ f(X)=XHX^T=YMHM^TY^T=\sum\limits_{i=1}^ra_iy_i^2. $

(2.3)

Let $\Delta=a_1a_2\cdots a_r$ (we assume $\Delta=1$ when $r=0$), and $\eta_0$ be the quadratic (multiplicative) character of $\mathbb{F}_{q_0}$. Then $\eta_0(\Delta)$ is an invariant of $H$ under the conjugate action of $M\in GL_s(\mathbb{F}_{q_0})$.

If we regard $\mathbb{F}_q$ as an $\mathbb{F}_{q_0}$-linear space of dimension $s$, then

$ Q_{\alpha, \beta}(x):={\rm{Tr}}_e^m(\alpha x^{p^k+1}+\beta x^{p^{3k}+1}) $

is a quadratic form over $\mathbb{F}_{q_0}$. Let $H_{\alpha, \beta}$ be the coefficient matrix of $Q_{\alpha, \beta}(x)$, $r_{\alpha, \beta}$ be the rank of $H_{\alpha, \beta}$, we have

$ T(\alpha, \beta)=\sum\limits_{x\in \mathbb{F}_q}\zeta_p^{{\rm{Tr}}_1^e\left(Q_{\alpha, \beta}(x)\right)}=\sum\limits_{x\in \mathbb{F}_q}\zeta_p^{{\rm{Tr}}_1^e(XH_{\alpha, \beta}X^T)} $

(2.4)

and

$ T(\lambda\alpha, \lambda\beta)=\sum\limits_{x\in \mathbb{F}_q}\zeta_p^{{\rm{Tr}}_1^e(XH_{\lambda\alpha, \lambda\beta}X^T)} =\sum\limits_{x\in \mathbb{F}_q}\zeta_p^{{\rm{Tr}}_1^e(\lambda XH_{\alpha, \beta}X^T)}, $

(2.5)

where $H_{\lambda\alpha, \lambda\beta}=\lambda H_{\alpha, \beta}$ and $r_{\lambda\alpha, \lambda\beta}=r_{\alpha, \beta}$.

Now we give the following lemmas, which will be used in the next section.

Lemma 2.1  (see Theorems 5.15 and 5.33 of [9]) For $a\in\mathbb{F}_{q_0}^\ast$, let $\eta_0$ be the quadratic (multiplicative) character of $\mathbb{F}_{q_0}$. Then we have

$ \sum\limits_{x\in \mathbb{F}_{q_0}}\zeta_p^{{\rm{Tr}}_1^e(ax^2)}=\left\{ \begin{array}{ll} \eta_0(a)(-1)^{e-1}q_0^\frac{1}{2}, \ \ \ \ \ \ \ \ \ \ \ &\mbox{if $p\equiv1\pmod 4$}, \\ \eta_0(a)(-1)^{e-1}\left(\sqrt{-1}\right)^eq_0^\frac{1}{2}, \ \ \ &\mbox{if $p\equiv3\pmod4$}. \end{array} \right. $

From Lemma 2.1 and (2.3), it is easy to get the following lemmas.

Lemma 2.2  With the notations as above, we have

$ T(\alpha, \beta)\quad=\quad\sum\limits_{x\in \mathbb{F}_q}\zeta_p^{{\rm{Tr}}_1^e\left(Q_{\alpha, \beta}(x)\right)} \\ \quad\quad\quad\quad=\quad\left\{ \begin{array}{ll} \eta_0(\Delta)(-1)^{(e-1)r_{\alpha, \beta}}q_0^{s-\frac{r_{\alpha, \beta}}{2}}, \ \ \ \ \ \ \ \ \ \ \ &\mbox{if $p\equiv1\pmod 4$}, \\ \eta_0(\Delta)(-1)^{(e-1)r_{\alpha, \beta}}\left(\sqrt{-1}\right)^{e\cdot r_{\alpha, \beta}}q_0^{s-\frac{r_{\alpha, \beta}}{2}}, \ \ \ &\mbox{if $p\equiv3\pmod4$}. \end{array} \right. $

Lemma 2.3  (see [4]) For $(\alpha, \beta)\in\mathbb{F}_q^2\backslash\{(0, 0)\}$, we have $r_{\alpha, \beta}=s-i$, $0\leq i\leq2$.

Combining (2.3), (2.4) and (2.5), by repeatedly using Lemma 2.1 wo obtain the following conclusion.

Lemma 2.4  With the notations introduced above, we have

$ S(\alpha, \beta)=\frac{1}{2}\left(1+(\eta_0(\lambda))^{r_{\alpha, \beta}}\right)T(\alpha, \beta)=\frac{1}{2}\left(1+(-1)^{r_{\alpha, \beta}}\right)T(\alpha, \beta). $

In order to determine the frequency of each value of $S(\alpha, \beta)$ for $\alpha, \beta\in\mathbb{F}_q$, we also need some preliminary identities of $S(\alpha, \beta)$.

Lemma 2.5  Let $s$ be odd and $k/e$ be even. Then the following identities hold.

(ⅰ) $\sum\limits_{\alpha, \beta\in\mathbb{F}_q}S(\alpha, \beta)=p^{2m}$;

(ⅱ) $\sum\limits_{\alpha, \beta\in\mathbb{F}_q}S(\alpha, \beta)^2=p^{3m}$.

Proof  (ⅰ) We observe that

$ \sum\limits_{\alpha, \beta\in\mathbb{F}_q}S(\alpha, \beta) \quad=\quad\sum\limits_{\alpha, \beta\in\mathbb{F}_q}\sum\limits_{x\in\mathbb{F}_q}\chi\big(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2}\big) \\ \quad\quad\quad\quad\quad\quad=\quad\sum\limits_{x\in\mathbb{F}_q}\sum\limits_{\alpha\in\mathbb{F}_q}\chi\big(\alpha x^\frac{p^k+1}{2}\big)\times\sum\limits_{\beta \in\mathbb{F}_q}\chi\big(\beta x^\frac{p^{3k}+1}{2}\big)\\ \quad\quad\quad\quad\quad\quad=\quad p^{2m}. $

(ⅱ) We can calculate

$ \sum\limits_{\alpha, \beta\in\mathbb{F}_q}S(\alpha, \beta)^2 \quad=\quad\sum\limits_{\alpha, \beta\in\mathbb{F}_q}\sum\limits_{x, y\in\mathbb{F}_q}\chi\big(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2}\big)\times\chi\big(\alpha y^\frac{p^k+1}{2}+\beta y^\frac{p^{3k}+1}{2}\big) \\ \quad\quad\quad\quad\quad\quad=\quad\sum\limits_{x, y\in\mathbb{F}_q}\sum\limits_{\alpha\in\mathbb{F}_q}\chi\left(\alpha\big(x^\frac{p^k+1}{2}+y^\frac{p^k+1}{2}\big)\right)\times\sum\limits_{\beta \in\mathbb{F}_q}\chi\left(\beta\big(x^\frac{p^{3k}+1}{2}+y^\frac{p^{3k}+1}{2}\big)\right)\\ \quad\quad\quad\quad\quad\quad=\quad p^{2m}\cdot M, $

where

$ M\quad=\quad\#\left\{(x, y)\in\mathbb{F}_q^2\big|x^\frac{p^k+1}{2}+y^\frac{p^k+1}{2}=0, x^\frac{p^{3k}+1}{2}+y^\frac{p^{3k}+1}{2}=0\right\} \\ \quad\quad=\quad\#\left\{(x, y)\in\mathbb{F}_q^2\big|x^\frac{p^k+1}{2}+y^\frac{p^k+1}{2}=0\right\} \\ \quad\quad=\quad\#\left\{(x, y)\in\mathbb{F}_q^2\big|y=-x\right\} \\ \quad\quad=\quad p^m. $

Here the third equality follows from $\gcd\big((p^k+1)/2, p^m-1\big)=1$.

Hence, the result follows.

Now we give the value distribution of $S(\alpha, \beta)$ and the weight distribution of the cyclic code $\mathcal{C}$.

Theorem 3.1  The value distribution of the multiset

$ \left\{S(\alpha, \beta)=\sum\limits_{x\in \mathbb{F}_q}\chi\left(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2}\right)\bigg|\alpha, \beta\in\mathbb{F}_q\right\} $

is described as shown in Table Ⅰ.

Proof  It is clear that $S(\alpha, \beta)=p^m$ if $(\alpha, \beta)=(0, 0)$. For $(\alpha, \beta)\in\mathbb{F}_q^2\backslash\{(0, 0)\}$, by Lemmas 2.2, 2.3 and 2.4, we have

$ S(\alpha, \beta)\in\left\{0, \pm p^\frac{m+e}{2}\right\}. $

To determined the distribution of these values, we define

$ n_i=\#\left\{(\alpha, \beta)\in\mathbb{F}_q^2\backslash\{(0, 0)\}\big|(\alpha, \beta)=(-1)^ip^\frac{m+e}{2}\right\}, $

where $i=0, 1$. By Lemma 2.5, we immediately have

$ \left\{ \begin{array}{lll} (n_0-n_1)p^\frac{m+e}{2}+p^m=p^{2m}, \\ (n_0+n_1)p^{m+e}+p^{2m}=p^{3m}. \end{array} \right. $

Solving the system of equations, we get the result.

Theorem 3.2  Let $p$ be an odd prime, $m$ and $k$ be two positive integers with $e=\gcd(m, k)$, $m\geq3$. If $m/e$ is odd and $k/e$ is even, then the weight distribution of the code

$ \mathcal{C}=\left\{c(\alpha, \beta)=\left({\rm{Tr}}_1^m(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2})\right)_{i=0}^{p^m-2}\right\} $

is described as shown in Table Ⅱ.

Proof  The Hamming weight of the codeword $c=c(\alpha, \beta)$ in $\mathcal{C}$ is given by

$ w_H(c)\quad=\quad\#\left\{x\in\mathbb{F}_q^*\big{|}{\rm{Tr}}_1^m\big(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2}\big)\neq0\right\} \nonumber \\ \quad\quad\quad\quad=\quad q-1-\#\left\{x\in\mathbb{F}_q^*\big{|}{\rm{Tr}}_1^m\big(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2}\big)=0\right\} \nonumber \\ \quad\quad\quad\quad=\quad q-1-\frac{1}{p}\sum\limits_{x\in \mathbb{F}_q^*}\sum\limits_{a\in \mathbb{F}_p}\zeta_p^{a{\rm{Tr}}_1^m\big(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2}\big)} \nonumber \\ \quad\quad\quad\quad=\quad q-1-\frac{q-1}{p}-\frac{1}{p}\sum\limits_{a\in \mathbb{F}_p^\ast}\sum\limits _{x\in \mathbb{F}_q^*}\zeta_p^{a{\rm{Tr}}_1^m\big(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2}\big)} \nonumber \\ \quad\quad\quad\quad=\quad q-1-\frac{q-1}{p}+\frac{p-1}{p}-\frac{1}{p}\sum\limits_{a\in \mathbb{F}_p^*}\sum\limits _{x\in \mathbb{F}_q}\zeta_p^{{\rm{Tr}}_1^m\big(\alpha ax^\frac{p^k+1}{2}+\beta ax^\frac{p^{3k}+1}{2}\big)} \nonumber \\ \quad\quad\quad\quad=\quad p^{m-1}(p-1)-\frac{1}{p}\sum\limits_{a\in \mathbb{F}_p^*}\sum\limits _{x\in \mathbb{F}_q}\zeta_p^{{\rm{Tr}}_1^m\big(\alpha (ax)^\frac{p^k+1}{2}+\beta (ax)^\frac{p^{3k}+1}{2}\big)} \nonumber \\ \quad\quad\quad\quad=\quad p^{m-1}(p-1)-\frac{1}{p}\sum\limits_{a\in \mathbb{F}_p^*}\sum\limits _{x\in \mathbb{F}_q}\zeta_p^{{\rm{Tr}}_1^m\big(\alpha x^\frac{p^k+1}{2}+\beta x^\frac{p^{3k}+1}{2}\big)} \nonumber \\ \quad\quad\quad\quad=\quad p^{m-1}(p-1)-\frac{p-1}{p}S(\alpha, \beta), $

where for the sixth equality we use the fact that $a^\frac{p^k+1}{2}=a^\frac{p^{3k}+1}{2}=a$ for any $a\in\mathbb{F}_p$($k/e$ is even). By Theorem 3.1, we get the weight distribution of the code $\mathcal{C}$.