Two classes of two-weight linear codes over finite fields

1.   Introduction

Let $\Bbb F_q$ be a finite field with $q$ elements, where $q$ is a power of a prime $2$. Let $\operatorname{Tr}_{q/2}$ be the trace function from $\Bbb F_{q}$ onto $\Bbb F_{2}$. An $[n, k, d]$ binary linear code $\mathcal{C}$ is a $k$-dimensional subspace of $\Bbb F_2^n$ with minimum Hamming distance $d$. Let $D = \{x_{1}, x_{2}, \ldots, x_{n}\}\subseteq \Bbb F_{q}$. A binary linear code of length $n$ over $\Bbb F_{2}$ is defined by

If the set $D$ is well chosen, the code $\mathcal{C}_{D}$ may have good parameters. Let $A_i$ be the number of codewords in $\mathcal C_D$ with Hamming weight $i$. The weight enumerator of $\mathcal C_D$ is defined by

The sequence $(1, A_1, A_2, \ldots, A_l)$ is called the weight distribution of $\mathcal C_D$. It is said to be a $t$-weight code if the number of nonzero $A_i$ in the sequence $(1, A_1, \ldots, A_l)$ is equal to $t$.

In coding theory, it is often desirable to know the weight distributions of the codes because they can be used to estimate the error correcting capability and the error probability of error detection and correction with respect to some algorithms. Hence weight distributions of codes are an interesting topic and were investigated in ^{[4,5,7,8,9,10,11,12,13,18,19,21,22]} etc. Moreover, those codes with few nonzero weights are of special interest in association schemes, secret sharing schemes, and frequency hopping sequences ^[2].

In this paper, let $p$ be an odd prime with $p\equiv 1\pmod 4$, $N = p^m$ a positive integer, $\operatorname{ord}_N(2) = f$, and $q = 2^f$, where $f = \frac{\phi(N)}2$ and $\phi(\cdot)$ is the Euler's function. Let $\alpha$ be a primitive element of $\Bbb F_q$ and $\beta = \alpha^{\frac {q-1}N}$ an $N$-th primitive root of unity in $\Bbb F_{q}$. We choose

as a defining set of $\mathcal{C}_{D_a}$, and

It is obvious that the dimension of $\mathcal C_{D_a}$ is $\frac{\phi(N)}2$.

For $C\in \mathcal{C}_{D_a}$, the Hamming weights of the codeword $C$ with respect to $b\in \Bbb F_q$ is denoted by $W_H(C)$.

Denote

The length of $\mathcal C_{D_a}$ is

If $b = 0$, then $W_H(C) = 0$.

If $b\ne 0$, then $W_{H}(C) = n-Z(b)$ and

It is simple to see that $\sum_{x\in \Bbb F_{q}^*}(-1)^{ \operatorname{Tr}_{q/2}(bx)} = -1$. Then

In order to obtain the length of $\mathcal C_{D_a}$ and the weight of $C\in \mathcal C_{D_a}$, we only need to calculate $S(a, 0)$ and $S(a, b)$. In general, the exact values of $S(a, 0)$ and $S(a, b)$ are hard to calculate, in Section 3, we shall consider two special cases.

This paper is organized as follows. In Section 2, we recall some concepts and several results about Gaussian sums in semi-primitive case. In Sections 3, we focus on the computation of the weight distribution of $\mathcal{C}_{D_a}$ defined as (1.1). In Section 4, we make a conclusion.

2.   Preliminaries

Let $\Bbb F_q$ be a finite field with $q$ elements and $\alpha$ a fixed primitive element of $\Bbb F_q$, i.e., $\Bbb F_q^* = \langle \alpha\rangle$. For two positive integers $n > 1$ and $N > 1$ with $q-1 = nN$, define cyclotomic cosets of order $N$ in $\Bbb F_q$: $C_i^{(N, q)} = \alpha^i\langle \alpha^N\rangle, i = 0, 1, \ldots, N -1.$ The cyclotomic numbers of order $N$ in $\Bbb F_q$ are defined as follows:

When $q = p$ is an odd prime, Lemmas 2.1–2.3 give cyclotomic numbers of order 2, 6 and order 8, respectively.

Lemma 2.1. ^[17] If $p\equiv 1\pmod4$, then $(0, 0)_2 = \frac{p-5}{4}, (0, 1)_2 = (1, 0)_2 = (1, 1)_2 = \frac{p-1}{4}.$ If $p\equiv 3\pmod4$, then $(0, 1)_2 = \frac{p+1}{4}, (0, 0)_2 = (1, 0)_2 = (1, 1)_2 = \frac{p-3}{4}.$

Lemma 2.2. ^[3] Suppose that $p\equiv 1\pmod {24}$ is a prime. Then $4p = u^2+27v^2$, $u, v\in \Bbb Z$ and $u\equiv1\pmod 3$. The possible values for the cyclotomic numbers of order 6 as follows (Table 1):

These $10$ fundamental constants $(0, 0), \ldots, (2, 4)$ are given by the relations contained in the following table (Table 2).

Lemma 2.3. ^[1,3] Suppose that $p\equiv 1\pmod {16}$ is a prime. Then $p = E^2+4F^2 = A^2+2B^2$, $E, F, A, B\in \Bbb Z$ and $E\equiv A\equiv1\pmod 4$. The possible values for the cyclotomic numbers of order 8 as follows (Table 3):

These 15 fundamental constants $(0, 0), \ldots, (2, 5)$ are given by the relations contained in the following table (Table 4).

Let $q$ be odd, define the quadratic multiplicative character of $\Bbb F_q$ denoted by $\eta$ as follows: $\eta(c) = 1$ if $c$ is the square element of $\Bbb F_q^*$ and $\eta(c) = -1$ otherwise. If $q$ is an odd prime, then for $c\in \Bbb F_q^*$, we have $\eta(c) = ({c\over q})$, where $({\cdot\over q})$ is the Legendre symbol.

Let $\operatorname{Tr}_{q/2}$ be the trace function from $\Bbb F_{q}$ to $\Bbb F_{2}$ defined by $\operatorname{Tr}_{q/2}(x) = x+x^2+\cdots+x^{q/2}$, $x\in \Bbb F_{q}$, and $\chi$ is the canonical additive character of $\Bbb F_{q}$: For $c\in \Bbb F_q$, $\chi(c) = (-1)^{ \operatorname{Tr}_{q/2}(c)}$. It is a well-known fact that $\sum_{c\in \Bbb F_q}\chi(c) = 0$. In the following, we list a useful result, which is called the semi-primitive case.

Lemma 2.4. ^[15]HY__HY, Theorem 1] Let $q = 2^{2sd}$ and $N\mid (2^d+1)$, where $s$ and $d$ are positive integers. Let $\alpha$ be a primitive element of $\Bbb F_q$. Then for $a = \alpha^b\in \Bbb F_q^*$ and $\operatorname{Ind}_\alpha(a) = b$,

Lemma 2.5. ^[3] Suppose that $p$ is a prime and $p\equiv 1\pmod {24}$. Then $p = A^2+3B^2$ and $4p = u^2+27v^2$, where $A, B\in \Bbb Z$ and $A\equiv1\pmod 3$, $u, v\in \Bbb Z$, $u\equiv1\pmod 3$ and $v = \frac{A-B}3$.

3.   Main results

In this section, let $p$ be an odd prime with $p\equiv 1\pmod 4$, $N = p^m$ a positive integer, $\operatorname{ord}_N(2) = f$, and $q = 2^f$, where $f = \frac{\phi(N)}2$ and $\phi(\cdot)$ is the Euler's function. We always suppose that $\alpha$ is a primitive element of $\Bbb F_q$, $\beta = \alpha^{\frac {q-1}N}$ and $\gamma = \beta^{p^{m-1}}$ is a primitive $N$-th and $p$-th root of unity in $\Bbb F_{q}$.

Recall that the length of $\mathcal C_{D_a}$ is equal to

and for $b\in \Bbb F_q^*$, the weight $W_H(C), C\in \mathcal C_{D_a}$ is

The length of $\mathcal C_{D_a}$ and the weight $W_H(C), C\in \mathcal C_{D_a}$ is relate to the value $S(a, 0)$ and $S(a, b)$, respectively.

Let $S(a) = \sum_{i = 0}^{N-1}(-1)^{ \operatorname{Tr}_{q/2}(a\beta^i)}$. Then

Recall that for $b\in \Bbb F_q^*$,

Let $H = \langle \alpha^N\rangle$, $\Bbb F_q^* = \cup_{i = 0}^{N-1}\alpha^iH$. Then

By $p\equiv 1\pmod 4$ and Lemma 2.4,

In the following, for two cases about $a\in \Bbb F_q$, we calculate the values $S(a, 0)$ and $S(a, b), b\in \Bbb F_q^*$. Let $\Bbb F_p^* = \langle \theta\rangle = H_0\cup H_1$, $H_0 = \langle \theta^2\rangle$ is a subgroup consisting of all square elements of index 2 in $\Bbb F_p^*$, and $H_1 = \theta H_0$ consists of all non-square elements in $\Bbb F_p^*$. First, we give an lemma.

Lemma 3.1. ^[14] Suppose that $H_1$ is the set consisting of all non-square elements in $\Bbb F_p^\ast$, then for $0\le i\le N -1$,

Suppose that $r|(p-1)$ and $a = \sum_{j = 0}^{r-1}\gamma^{w\zeta_r^j}\in \Bbb F_q$, where $\gamma = \beta^{p^{m-1}}$, $w\in \Bbb F_p^*$, and $\zeta_r$ is a primitive $r$-th root of unity in $\Bbb F_p$.

For $0\le i\le N-1$, let $i = kp^{m-1}+l$, $0\le k\le p-1$, $0\le l\le p^{m-1}-1$. By Lemma 3.1 and note that if $l\ne 0$, then $\operatorname{Tr}_{q/2}(\beta^{ (k+w\zeta_r^j)p^{m-1}+l}) = 0$. Then

where

Let $W = \{ -w, -w\zeta_r, \ldots, -w\zeta_r^{r-1}\}$. By Lemma 3.1 and by the fact that the product of any two square elements or any two non-square elements is a square element and the product of a square element with a non-square element is a non-square element, we can easily check that if $x\in \Bbb F_p\setminus W$, then $\sum_{j = 0}^{r-1} \operatorname{Tr}_{q/2}(\gamma^{ x+w\zeta_r^j})$ and $\operatorname{Tr}_{q/2}(\gamma^{\Pi_{j = 0}^{r-1}(x+w\zeta_r^j)})$ have the same parity. Then

Theorem 3.2. The notations are as above. Let $p\equiv 1\pmod {24}$ be a prime. Then $4p = u^2+27v^2$, $u, v\in \Bbb Z$ and $u\equiv1\pmod 3$. Suppose that $a = \sum_{j = 0}^5\gamma^{w\zeta_6^j}\in \Bbb F_q$.

(1) If $({2\over p})_3 = 1$, then

(1) If $({2\over p})_3\ne1$, then

Proof. By (3.1), we only need to calculate $\Omega$ denoted by (3.2). Let

where $W = \{ -w, -w\zeta_6, -w\zeta_6^{2}, -w\zeta_6^{3}, -w\zeta_6^{4}, -w\zeta_6^{5}\}$.

Let $\Bbb F_p^* = \langle \theta\rangle$, $C_i^{(2, p)} = \theta^i \langle \theta^2\rangle$, $i = 0, 1$, and $C_i^{(6, p)} = \theta^j\langle \theta^6\rangle, j = 0, 1, 2, 3, 4, 5$. It is clear that $C_0^{(2, p)} = C_0^{(6, p)}\cup C_2^{(6, p)}\cup C_4^{(6, p)}$ and $C_1^{(2, p)} = C_1^{(6, p)}\cup C_3^{(6, p)}\cup C_5^{(6, p)}$. For $w\in \Bbb F_p^*$, $\Pi_{j = 0}^5(x+w\zeta_6^j) = x^6-w^6$. Now we count the number :

If $x = 0$, then $x^6-w^6 = y^2$ has a solution $y\in \Bbb F_p$, i.e., when $x = 0$, there exists a $y\in \Bbb F_p$, but not unique, such that $x^6-w^6 = y^2$.

If $x\ne 0$, $x^6-w^6 = y^6$ is equivalent to $1+(-(\frac w x)^6) = (\frac yx)^6$, then the number of $\frac w x$ such that $1+(-(\frac w x)^6) = (\frac yx)^6$ is equal to $|(1+C_0^{(6, q)}) \cap C_0^{(6, q)}| = (0, 0)_6$ and the number of $x$ such that $x^6-w^6 = y^6$ is equal to $6(0, 0)_6$. Similarly, the number of $x$ such that $x^6-w^6 = \gamma^2y^6$ is equal to $6(0, 2)_6$ and the number of $x$ such that $x^6-w^6 = \gamma^4y^6$ is equal to $6(0, 4)_6$.

Suppose that $2$ is a cubic residue modulo $p$, i.e, $({2\over p})_3 = 1$. By Lemma 2.2,

Suppose that $2$ is not a cubic residue modulo $p$, i.e., $({2\over p})_3\ne1$. By Lemma 2.2,

Moreover, by $p\equiv 1\pmod {24}$,

Similarly,

Thus

Note that

Hence

□

Theorem 3.3. The notations are as above. Let $p\equiv 1\pmod {16}$ be a prime. Then $p = E^2+4F^2 = A^2+2B^2$, $E, F, A, B\in \Bbb Z$ and $E\equiv A\equiv1\pmod 4$. Suppose that $a = \sum_{j = 0}^7\gamma^{w\zeta_8^j}\in \Bbb F_q$, then

Proof. By (3.1), we only need to calculate $\Omega$ denoted by (3.2). Let

where $W = \{ -w, -w\zeta_8, -w\zeta_8^{2}, -w\zeta_8^{3}, -w\zeta_8^{4}, -w\zeta_8^{5}, -w\zeta_8^{6}, -w\zeta_8^{7}\}$.

Let $\Bbb F_p^* = \langle \theta\rangle$, $C_i^{(2, p)} = \theta^i \langle \theta^2\rangle$, $i = 0, 1$, and $C_i^{(8, p)} = \theta^j\langle \theta^8\rangle, j = 0, 1, 2, 3, 4, 5, 6, 7$. It is clear that $C_0^{(2, p)} = C_0^{(8, p)}\cup C_2^{(8, p)}\cup C_4^{(8, p)}\cup C_6^{(8, p)}$. For $w\in \Bbb F_p^*$, $\Pi_{j = 0}^7(x+w\zeta_8^j) = x^8-w^8$. Now we count the number :

If $x = 0$, then $x^8-w^8 = y^2$ has a solution $y\in \Bbb F_p$, i.e., when $x = 0$, there exists a $y\in \Bbb F_p$, but not unique, such that $x^6-w^6 = y^2$.

If $x\ne 0$, similar to the discussion of Theorem 3.2, the number of $x$ such that $x^8-w^8 = y^8$, $x^8-w^8 = \gamma^2y^8$, $x^8-w^8 = \gamma^4y^8$, and $x^8-w^8 = \gamma^6y^8$ is equal to $8(0, 0)_8$, $8(0, 2)_8$, $8(0, 4)_8$, and $8(0, 6)_8$, respectively.

Suppose that $({2\over p})_4 = 1$ or suppose that $({2\over p})_4\ne1$. By Lemma 2.3,

Moreover, by $p\equiv 1\pmod {16}$, it is easy to check that if $x\in W$, then

Thus

Hence

and

□

Recall that the length of $\mathcal C_{D_a}$ defined as (1.1) is equal to $n = \frac12 (q-1+S(a, 0))$ and $S(a, 0) = \frac{q-1}{N}S(a)$. Then the following results are obtained.

Theorem 3.4. The notations are as Theorem 3.2.

(1) If $({2\over p})_3\ne1$, then the length of $\mathcal C_{D_a}$ defined as (1.1) is equal to

(2) If $({2\over p})_3 = 1$, then the length of $\mathcal C_{D_a}$ defined as (1.1) is equal to

Theorem 3.5. The notations are as Theorem 3.3. The length of $\mathcal C_{D_a}$ defined as (1.1) is equal to

Now we return to investigate the weight $W_H(C)$ of $C\in \mathcal C_{D_a}$.

Theorem 3.6. Let $p\equiv 1\pmod {24}$ be a prime. Then $4p = u^2+27v^2$, $u, v\in \Bbb Z$ and $u\equiv1\pmod 3$. Suppose that $a = \sum_{j = 0}^5\gamma^{w\zeta_6^j}\in \Bbb F_q$, where $w\in H_0$.

$(1)$ If $({2\over p})_3\ne1$, then $\mathcal{C}_{D_a}$ defined as (1.1) is a two weight binary linear code with length $\frac{(q-1)(2p^m-p+5-u-9v)}{2p^m}$ and its weight distributions are given by Table $5$.

$(2)$ If $({2\over p})_3 = 1$, then $\mathcal{C}_{D_a}$ defined as (1.1) is a two weight binary linear code with length $\frac{(q-1)(2p^m-p+5+2u)}{2p^m}$ and its weight distributions are given by Table $6$.

Proof. Suppose that $a = \sum_{j = 0}^5\gamma^{w\zeta_6^j}\in \Bbb F_q$, $w\in H_0$. By Theorem 3.2,

If $b\in \Bbb F_q^*$, then

We only need to count the number of $b\in \Bbb F_q^*$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 0$ or $1$. It is clear that $\operatorname{ord}(b^{-\frac{q-1}{N}}) = p^t, 0\le t\le m$.

If $t\ge 2$, then $\operatorname{ord}(\gamma^{w\zeta_6^j} b^{-\frac{q-1}{N} }) = p^t > p$, where $j = 0, \ldots, 5$. So by Lemma 3.1, $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 0$. Moreover, $b = \alpha ^{p^{m-t}\mu}$, where $0 < \mu\le \frac{q-1}{p^{m-t}}-1$ and $\gcd(\mu, p) = 1$. Hence there are $\sum_{t = 2}^m (\frac{q-1}{p^{m-t}}-\frac{q-1}{p^{{m-t+1}}}) = q-1-\frac{q-1}{p^{m-1}}$ such elements $b\in \Bbb F_q^*$ such that $\operatorname{ord}(b^{-\frac{q-1}{N}}) > p$.

If $0\le t\le 1$, i.e, $b^{-\frac{q-1}{N}} = \gamma^{x}$, $0\le x\le p-1$, it is obvious that there are $\frac{q-1}{p^m}$ elements $b$.

Moreover,

Suppose that $x\in W = \{ -w, -w\zeta_6, -w\zeta_6^2, -w\zeta_6^3, -w\zeta_6^4, -w\zeta_6^5\}$. Then

Hence there are $\frac{6 (q-1)}{p^{m}}$ such elements $b\in \Bbb F_q^*$ such that $b^{-\frac{q-1}N} = \gamma^x$.

Suppose that $x\in \Bbb F_p\setminus W$. Then

By the proof of Theorem 3.2, there are $\Delta = \frac{p-7-u-9v}2$ such elements $x\in \Bbb F_p\setminus W$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 0$; there are $p-6-\Delta = \frac{p-5+u +9v}2$ such elements $x\in \Bbb F_p\setminus W$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 1$, where $4p = u^2+27v^2$, $u, v\in \Bbb Z$ and $u\equiv1\pmod 3$.

Therefore there are

elements $b\in\Bbb F_q^*$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 0$; there are $\frac{p-5+u +9v}2\cdot \frac{q-1}{p^{m}}$ such elements $b\in\Bbb F_q^*$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 1$. Then the Table $5$ is given.

Suppose that $({2\over p})_3 = 1$, we obtain the Table $6$ similarly.□

Theorem 3.7. Let $p\equiv 1\pmod {16}$ be a prime. Then $p = E^2+4F^2 = A^2+2B^2$, $E, F, A, B\in \Bbb Z$ and $E\equiv A\equiv1\pmod 4$. Suppose that $a = \sum_{j = 0}^7\gamma^{w\zeta_8^j}\in \Bbb F_q$, where $w\in H_1$. Then $\mathcal{C}_{D_a}$ defined in (1.1) is a two weight binary linear code with length $\frac{(q-1)(2p^m-p-2E-4A-9)}{2p^m}$ and its weight distributions are given by Table $7$.

Proof. Suppose that $a = \sum_{j = 0}^7\gamma^{w\zeta_8^j}\in \Bbb F_q$, $w\in H_1$. By Theorem 3.3,

If $b\in \Bbb F_q^*$, then

We only need to count the number of $b\in \Bbb F_q^*$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 0$ or $1$. It is clear that $\operatorname{ord}(b^{-\frac{q-1}{N}}) = p^t, 0\le t\le m$.

If $t\ge 2$, then $\operatorname{ord}(\gamma^{w\zeta_8^j} b^{-\frac{q-1}{N} }) = p^t > p$, where $j = 0, \ldots, 7$. So by Lemma 3.1, $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 0$. Moreover, $b = \alpha ^{p^{m-t}\mu}$, where $0 < \mu\le \frac{q-1}{p^{m-t}}-1$ and $\gcd(\mu, p) = 1$. Hence there are $\sum_{t = 2}^m (\frac{q-1}{p^{m-t}}-\frac{q-1}{p^{{m-t+1}}}) = q-1-\frac{q-1}{p^{m-1}}$ such elements $b\in \Bbb F_q^*$ such that $\operatorname{ord}(b^{-\frac{q-1}{N}}) > p$.

If $0\le t\le 1$, i.e, $b^{-\frac{q-1}{N}} = \gamma^{x}$, $0\le x\le p-1$. it is obvious that there are $\frac{q-1}{p^m}$ elements $b$.

Moreover,

Suppose that $x\in W = \{ -w, -w\zeta_8, -w\zeta_8^2, -w\zeta_8^3, -w\zeta_8^4, -w\zeta_8^5, -w\zeta_8^6, -w\zeta_8^7\}$. Then

Hence there are $\frac{8 (q-1)}{p^{m}}$ such elements $b\in \Bbb F_q^*$ such that $b^{-\frac{q-1}N} = \gamma^x$.

Suppose that $x\in \Bbb F_p\setminus W$. Then

By the proof of Theorem 3.3, there are $\Delta = \frac{p-9-2E-4A}2$ such elements $x\in \Bbb F_p\setminus W$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 0$; there are $p-8-\Delta = \frac{p-7+2E +4A}2$ such elements $x\in \Bbb F_p\setminus W$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 1$, where $p = E^2+4F^2 = A^2+2B^2$, $E, F, A, B\in \Bbb Z$ and $E\equiv A\equiv1\pmod 4$.

Therefore there are

such elements $b\in\Bbb F_q^*$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 0$; there are $\frac{p+9+2E +4A}2\cdot \frac{q-1}{p^{m}}$ such elements $b\in\Bbb F_q^*$ such that $\operatorname{Tr}_{q/2}(ab^{-\frac{q-1}N}) = 1$.

The desired result follows.□

In the following, we give some examples.

Example 3.8. Let $p = 17$. Then $\mathcal{C}_{D_a}$ is an $[135, 8, 64]$ two-weight binary linear code with weight enumerator $1+135z^{64}+120z^{72}$. The dual is an $[135, 127, 3]$ binary linear code and is optimal, due to ^[6]. The code is also obtained from Theorem 3.2 in ^[20].

Example 3.9. Let $p = 97$. Then $97\equiv 1\pmod {24}$ and $97\equiv 1\pmod {16}$. By Theorem 3.6 (1), $4p = 19^2+27v^2$, by Lemma 2.5, $v = 1$. By Theorem 3.7, $p = 9^2+4F^2 = 5^2+2B^2$, $F^2 = 4$ and $B^2 = 36$. The weight distributions of $\mathcal{C}_{D_a}$ are given by Table $8$.

From the above Table $8$, we can see that the minimum Hamming distance of the line code in Theorems 3.6 (1) is larger than that of in Theorem 3.7.

Example 3.10. Let $p = 193$. Then $193\equiv 1\pmod {24}$ and $193\equiv 1\pmod {16}$. By Theorem 3.6 (1), $4p = (-23)^2+27v^2$, by Lemma 2.5, $v = 3$. By Theorem 3.7, $p = (-7)^2+4F^2 = (-11)^2+2B^2$, $F^2 = 36$ and $B^2 = 36$. The weight distributions of $\mathcal{C}_{D_a}$ are given by Table $9$.

From the above Table 9, we can see that the minimum Hamming distance of the line code in Theorems 3.7 is larger than that of in Theorem 3.6.

4.   Conclusions

Suppose that $p\equiv 1\pmod 4$ is a prime and $\frac{\phi(p^m)}2$ is the multiplicative order of $2$ modulo $p^m$. Let $q = 2^{ \frac{\phi(p^m)}2}$, in this paper, we constructed two classes of two-weight linear codes over $\Bbb F_q$ and obtained their weight distributions. The main work was the calculations of two classes of exponential sums, which were special forms of exponential sums defined by Moisio in ^[16]. The technique that we adopted was to count the number of the square elements by cyclotomic numbers over $\Bbb F_p$. By this method, other problems such as cross correlations of sequences and Walsh spectrums of functions can also be investigated.

Acknowledgments

We are very grateful to the reviewers and the editor for their valuable comments and suggestions that much improved the quality of this paper. The work was supported by National Natural Science Foundation of China under Grant 12171420 and Natural Science Foundation of Shandong Province under Grant ZR2021MA046.

Conflict of interest

The authors declare no conflicts of interest.