Complete weight enumerators of a class of linear codes with two or three weights

TL;DR

This paper constructs a class of linear codes with at most three weights, determines their complete weight enumerators, and explores their application in secret sharing schemes, extending previous results.

Contribution

It introduces a new class of linear codes with specific weight properties and provides their complete weight enumerators, extending prior research by Wang et al. (2017).

Findings

Codes are at most three-weight codes.

Codes are suitable for secret sharing schemes.

Extension of previous results by Wang et al. (2017).

Abstract

We construct a class of linear codes by choosing a proper defining set and determine their complete weight enumerators and weight enumerators. The results show that they are at most three-weight codes and they are suitable for applications in secret sharing schemes. We give an extension of the results raised by Wang et al.(2017).

Equations

$$w[\mathsf{c}]=w_0^{k_0}{w}_1^{k_1}\cdots w_{p-1}^{k_{p-1}},$$

$$\mathrm{CWE}(C)=\sum _{\mathsf{c}\in C}w[\mathsf{c}].$$

$$C_D=\{(\mathrm{Tr}(b{d}_1),\mathrm{Tr}(b{d}_2),\cdots ,\mathrm{Tr}(b{d}_n)):b\in {\mathbb{F}}_q\}.$$

$$D_c=\{x\in {\mathbb{F}}_q^{\ast }:\mathrm{Tr}(a{x}^{p^k+1})=c\},$$

$$1+(p^{e-1}+(-1{)}^s(p-1)p^{m-1}-1)z^{(p-1)p^{e-2}}+(p-1)(p^{e-1}-(-1{)}^s{p}^{m-1})z^{(p-1)(p^{e-2}+(-1{)}^s{p}^{m-1})},$$

$$w_0^{p^{e-1}+(-1{)}^s(p-1)p^{m-1}-1}+(p^{e-1}+(-1{)}^s(p-1)p^{m-1}-1)w_0^{p^{e-2}+(-1{)}^s(p-1)p^{m-1}-1}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}}$$

$$+(p-1)(p^{e-1}-(-1{)}^s{p}^{m-1})w_0^{p^{e-2}-1}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}+(-1{)}^s{p}^{m-1}}.$$

$$1$$

$$+(p-1)(p^{e-2d-1}+(-1{)}^s{p}^{m-d-1})z^{(p-1)(p^{e-2}-(-1{)}^s{p}^{m+d-1})},$$

$$w_0^{p^{e-1}-(-1{)}^s(p-1)p^{m+d-1}-1}+(p^e-p^{e-2d})w_0^{p^{e-2}-(-1{)}^s(p-1)p^{m+d-2}-1}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}-(-1{)}^s(p-1)p^{m+d-2}}$$

$$+(p^{e-2d-1}-(-1{)}^s(p-1)p^{m-d-1}-1)w_0^{p^{e-2}-(-1{)}^s(p-1)p^{m+d-1}-1}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}}$$

$$+(p-1)(p^{e-2d-1}+(-1{)}^s{p}^{m-d-1})w_0^{p^{e-2}-1}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}-(-1{)}^s{p}^{m+d-1}}.$$

$$\displaystyle 1+\big{(}\frac{p+1}{2}p^{e-1}+\frac{p-1}{2}(-1)^{s}p^{m-1}-1\big{)}z^{(p-1)p^{e-2}}+\frac{p-1}{2}(p^{e-1}-(-1)^{s}p^{m-1})z^{(p-1)p^{e-2}-2(-1)^{s}p^{m-1}},$$

$$w_0^{p^{e-1}-(-1{)}^s{p}^{m-1}}+(p^{e-1}+(-1{)}^s(p-1)p^{m-1}-1)w_0^{p^{e-2}-(-1{)}^s{p}^{m-1}}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}}$$

$$+(p^{e-1}-(-1{)}^s{p}^{m-1})\sum _{\beta =1}^{\frac{p-1}{2}}{w}_0^{p^{e-2}+(-1{)}^s\eta (-1)p^{m-1}}{w}_{2g^{\beta }}^{p^{e-2}}{w}_{p-2g^{\beta }}^{p^{e-2}}\prod _{\begin{array}{c}\rho \in {\mathbb{F}}_p^{\ast }\\ \rho \ne 0,\pm 2g^{\beta }\end{array}}{w}_{\rho }^{p^{e-2}+(-1{)}^s{p}^{m-1}\eta ({\rho }^2-4g^{2\beta })}$$

$$+(p^{e-1}-(-1{)}^s{p}^{m-1})\sum _{\beta =1}^{\frac{p-1}{2}}{w}_0^{p^{e-2}-(-1{)}^s\eta (-1)p^{m-1}}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}+(-1{)}^s{p}^{m-1}\eta ({\rho }^2-4g^{2\beta +1})},$$

$$1$$

$$+\frac{p-1}{2}(p^{e-2d-1}+(-1{)}^s{p}^{m-d-1})z^{(p-1)p^{e-2}+2(-1{)}^s{p}^{m+d-1}},$$

$$w_0^{p^{e-1}+(-1{)}^s{p}^{m+d-1}}+(p^e-p^{e-2d})w_0^{p^{e-2}+(-1{)}^s{p}^{m+d-2}}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}+(-1{)}^s{p}^{m+d-2}}$$

$$+(p^{e-2d-1}-(-1{)}^s(p-1)p^{m-d-1}-1)w_0^{p^{e-2}+(-1{)}^s{p}^{m+d-1}}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}}$$

$$+(p^{e-2d-1}+(-1{)}^s{p}^{m-d-1})\sum _{\beta =1}^{\frac{p-1}{2}}{w}_0^{p^{e-2}-(-1{)}^s\eta (-1)p^{m+d-1}}{w}_{2g^{\beta }}^{p^{e-2}}{w}_{p-2g^{\beta }}^{p^{e-2}}\prod _{\begin{array}{c}\rho \in {\mathbb{F}}_p^{\ast }\\ \rho \ne 0,\pm 2g^{\beta }\end{array}}{w}_{\rho }^{p^{e-2}-(-1{)}^s{p}^{m+d-1}\eta ({\rho }^2-4g^{2\beta })}$$

$$+(p^{e-2d-1}+(-1{)}^s{p}^{m-d-1})\sum _{\beta =1}^{\frac{p-1}{2}}{w}_0^{p^{e-2}+(-1{)}^s\eta (-1)p^{m+d-1}}\prod _{\rho \in {\mathbb{F}}_p^{\ast }}{w}_{\rho }^{p^{e-2}-(-1{)}^s{p}^{m+d-1}\eta ({\rho }^2-4g^{2\beta +1})},$$

$$w_0^{296}+32w_0^{134}(w_1{w}_2{)}^{81}+648w_0^{98}(w_1{w}_2{)}^{99}+48w_0^{80}(w_1{w}_2{)}^{108}.$$

$$w_0^{216}+24w_0^{108}(w_1{w}_2{)}^{54}+648(w_0{w}_1{w}_2{)}^{72}+56w_0^{54}(w_1{w}_2{)}^{81}.$$

$$w_0^{144}+144w_0^{44}(w_1{w}_2{w}_3{w}_4{)}^{25}+480w_0^{24}(w_1{w}_2{w}_3{w}_4{)}^{30}.$$

$$w_0^{120}+120w_0^{30}(w_1{w}_4{)}^{25}(w_2{w}_3{)}^{20}+120w_0^{30}(w_1{w}_4{)}^{20}(w_2{w}_3{)}^{25}+120w_0^{20}(w_1{w}_4{)}^{30}(w_2{w}_3{)}^{20}$$

$$+120w_0^{20}(w_1{w}_4{)}^{20}(w_2{w}_3{)}^{30}+144w_0^{20}(w_1{w}_2{w}_3{w}_4{)}^{25}.$$

$$G({\eta }_e,{\chi }_1)=\sum _{x\in {\mathbb{F}}_q^{\ast }}{\eta }_e(x){\chi }_1(x).$$

$$\sum _{x\in {\mathbb{F}}_q}{\zeta }_p^{\mathrm{Tr}(f(x))}={\zeta }_p^{\mathrm{Tr}(a_0-a_1^2(4a_2{)}^{-1})}{\eta }_e(a_2)G_e,$$

$$\displaystyle\sum_{x\in\mathbb{F}_{q}}\eta_{e}(f(x))=\left\{\begin{array}[]{lll}-\eta_{e}(a_{2})&\textup{ if }a_{1}^{2}-4a_{0}a_{2}\neq 0,\\ (q-1)\eta_{e}(a_{2})&\textup{ if }a_{1}^{2}-4a_{0}a_{2}=0.\end{array}\right.$$

$$S(\alpha ,\beta )=\sum _{x\in {\mathbb{F}}_q}{\chi }_1(\alpha x^{p^k+1}+\beta x).$$

$$\displaystyle S(\alpha,0)=\left\{\begin{array}[]{lll}(-1)^{s}p^{m}&&\textup{ if }\alpha^{(q-1)/(p^{d}+1)}\neq(-1)^{s},\\ (-1)^{s+1}p^{m+d}&&\textup{ if }\alpha^{(q-1)/(p^{d}+1)}=(-1)^{s},\end{array}\right.$$

$$S(\alpha ,\beta )=(-1{)}^s{p}^m{\chi }_1(-\alpha x_0^{p^k+1}).$$

$$S(\alpha ,\beta )=(-1{)}^{s+1}{p}^{m+d}{\chi }_1(-\alpha x_0^{p^k+1}).$$

$${\alpha }^{(q-1)/(p^d+1)}=(-1{)}^s.$$

$$C_{D_c}=\{{\mathsf{c}}_b=(\mathrm{Tr}(bx){)}_{x\in D_c}:b\in {\mathbb{F}}_q\},$$

Full text

1 Introduction

Throughout this paper, let $p$ be an odd prime and $q=p^{e}$ for a positive integer $e$. Denote by $\mathbb{F}_{q}$ a finite field with $q$ elements. An $[n,\kappa,\delta]$ linear code $C$ over $\mathbb{F}_{p}$ is a $\kappa$-dimensional subspace of $\mathbb{F}_{p}^{n}$ with minimum distance $\delta$ (see [21]). Let $A_{i}$ denote the number of codewords with Hamming weight $i$ in a linear code $C$ of length $n$. The weight enumerator of $C$ is defined by $1+A_{1}z+A_{2}z^{2}+\cdots+A_{n}z^{n}$. A code $C$ is called a $t$-weight code if there are $t$ nonzero $A_{i}$ for $1\leq i\leq n$.

The complete weight enumerator of a code $C$ over $\mathbb{F}_{p}$ enumerates the codewords according to the number of symbols of each kind contained in each codeword. Denote elements of the field by $\mathbb{F}_{p}=\{w_{0}=0,w_{1},\cdots,w_{p-1}\}$. Also, let $\mathbb{F}_{p}^{*}$ denote $\mathbb{F}_{p}\backslash\{0\}$. For a codeword $\mathsf{c}=(c_{0},c_{1},\cdots,c_{n-1})\in\mathbb{F}_{p}^{n}$, let $w[\mathsf{c}]$ be the complete weight enumerator of $\mathsf{c}$, which is defined as

[TABLE]

where $k_{j}$ is the number of components of $\mathsf{c}$ equal to $w_{j}$, $\sum_{j=0}^{p-1}k_{j}=n$. The complete weight enumerator of the code $C$ is then

[TABLE]

The weight enumerators of linear codes have been well studied in literature, such as [11, 12, 22, 24, 28, 29] and references therein. The information of the complete weight enumerators of linear codes is of vital use because they can show the frequency of each symbol appearing in each codeword. Furthermore the complete weight enumerator has close relation to the deception probabilities of certain authentication codes [7], and is used to compute the Walsh transform of monomial and quadratic bent functions over finite fields [13]. More researches can be found in [2, 3, 8, 16, 17].

We introduce a generic construction of linear codes developed in [6, 9, 10]. Set $D=\{d_{1},d_{2},\cdots,d_{n}\}\subseteq\mathbb{F}_{q}$, where $q=p^{e}$. Denote by $\mathrm{Tr}$ the absolute trace function from $\mathbb{F}_{q}$ to $\mathbb{F}_{p}$. A linear code of length $n=\#D$ is defined by

[TABLE]

The set $D$ is called the defining set of $C_{D}$. This construction technique is general and has received a good deal of attention, see [1, 14, 18, 19, 23, 25, 26] for more details.

Let $d=\gcd(k,e)$ be the greatest common divisor of positive integers $k$ and $e$. Suppose that $e/d$ is even with $e=2m$ and $m>1$. Motivated by the above construction and the idea of [23], we investigate a class of linear codes with defining set

[TABLE]

where $a\in\mathbb{F}_{q}^{*}$ and $c\in\mathbb{F}_{p}$. We will extend the results presented by Wang et al. [23] who studied the case where $a=1$.

The remainder of this paper is organized as follows. In Section 2, we describe the main results of this paper, additionally we give some examples. In Section 3, we briefly recalls some definitions and results on exponential sums, then proves the main results. In Section 4, we make a conclusion.

2 Main results

In this section, we only introduce the complete weight enumerator and weight enumerator of $C_{D_{c}}$ with defining set $D_{c}$. The main results of this paper are presented below, whose proofs will be given in Section 3. Denote $s=m/d$.

Theorem 1**.**

If $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$, then the code $C_{D_{0}}$ of (1) is a $[p^{e-1}+(-1)^{s}(p-1)p^{m-1}-1,e]$ linear code with weight enumerator

[TABLE]

and its complete weight enumerator is

[TABLE]

Theorem 2**.**

If $m>d+1$ and $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$, then the code $C_{D_{0}}$ of (1) is a $[p^{e-1}-(-1)^{s}(p-1)p^{m+d-1}-1,e]$ linear code with weight enumerator

[TABLE]

and its complete weight enumerator is

[TABLE]

When $m=d+1$ or $m=d$, the corresponding results are described below.

Corollary 1**.**

If $m=2$, $d=1$ and $a^{(q-1)/(p^{d}+1)}=1$, the set $C_{D_{0}}$ of (1) is a multi set with each codewords appear $p^{2}$ times. It is a $[p^{2}-1,2,(p-1)p]$ linear code with only one nonzero weight $(p-1)p$.

Corollary 2**.**

If $m=d$ and $a^{(q-1)/(p^{d}+1)}=-1$, then the code $C_{D_{0}}$ of (1) is a $[p^{e}-1,e,(p-1)p^{e-1}]$ linear code with only one nonzero weight $(p-1)p^{e-1}$.

Theorem 3**.**

Let $g$ be a generator of $\mathbb{F}_{p}^{*}$ and $c\in\mathbb{F}_{p}^{*}$. If $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$, then the code $C_{D_{c}}$ of (1) is a $[p^{e-1}-(-1)^{s}p^{m-1},e]$ linear code with weight enumerator

[TABLE]

and its complete weight enumerator is

[TABLE]

where $\eta$ is the quadratic character over $\mathbb{F}_{p}^{*}$.

Theorem 4**.**

Let $g$ be a generator of $\mathbb{F}_{p}^{*}$ and $c\in\mathbb{F}_{p}^{*}$. If $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$, then the code $C_{D_{c}}$ of (1) is a $[p^{e-1}+(-1)^{s}p^{m+d-1},e]$ linear code with weight enumerator

[TABLE]

and its complete weight enumerator is

[TABLE]

where $\eta$ is the quadratic character over $\mathbb{F}_{p}^{*}$.

Some concrete examples are provided to illustrate our results.

Example 1**.**

Let $(p,m,k)=(3,3,1)$, $\mathbb{F}_{3^{6}}^{*}=\langle\theta\rangle$ and $a=\theta^{2}$. Then $d=1$, $s=3$ and $a^{(q-1)/(p^{d}+1)}=-1$. If $c=0$, by Theorem 2, the corresponding code $C_{D_{0}}$ is a $[296,6,162]$ linear code. Its weight enumerator is $1+32z^{162}+648z^{198}+48z^{216}$, and its complete weight enumerator is

[TABLE]

If $c\neq 0$, by Theorem 4, the corresponding code $C_{D_{1}}$ has parameters $[216,6,108]$ with weight enumerator $1+24z^{108}+648z^{144}+56z^{162}$ and complete weight enumerator

[TABLE]

These results coincide with the numerical computation by Magma.

Example 2**.**

Let $(p,m,k)=(5,2,1)$, $\mathbb{F}_{5^{4}}^{*}=\langle\theta\rangle$ and $a=\theta^{3}$. Then $d=1$, $s=2$ and $a^{(q-1)/(p^{d}+1)}\neq 1$. By Theorem 1, the code $C_{D_{0}}$ has parameters $[144,4,100]$ with weight enumerator $1+144z^{100}+480z^{120}$ and complete weight enumerator

[TABLE]

By Theorem 3, the code $C_{D_{1}}$ is a $[120,4,90]$ linear code. Its weight enumerator is $1+240z^{90}+384z^{100}$, and its complete weight enumerator is

[TABLE]

These results coincide with the numerical computation by Magma.

3 The proofs of the main results

3.1 Auxiliary results

In order to prove the results proposed in Section 2, we will use several results which are depicted and proved in the sequel. We start with group characters and exponential sums. For each $b\in\mathbb{F}_{q}$, an additive character $\chi_{b}$ of $\mathbb{F}_{q}$ is defined by $\chi_{b}(x)=\zeta_{p}^{\text{Tr}(bx)}$ for all $x\in\mathbb{F}_{q}$, where $\zeta_{p}=\exp\left(\frac{2\pi\sqrt{-1}}{p}\right)$ and Tr is the simplification of the trace function $\text{Tr}^{e}_{1}$ from $\mathbb{F}_{q}$ to $\mathbb{F}_{p}$. For $b=1$, $\chi_{1}$ is called the canonical additive character of $\mathbb{F}_{q}$.

Let $\eta_{e}$ denote the quadratic character of $\mathbb{F}_{q}^{*}$ and is extended by $\eta_{e}(0)=0$. The quadratic Gauss sum $G(\eta_{e},\chi_{1})$ is defined by

[TABLE]

We denote $G_{e}=G(\eta_{e},\chi_{1})$ and $G=G(\eta,\chi_{1}^{\prime})$, where $\eta$ and $\chi_{1}^{\prime}$ are the quadratic character and canonical additive character of $\mathbb{F}_{p}$, respectively. Moreover, it is well known that $G_{e}=(-1)^{e-1}\sqrt{p^{*}}^{e}$ and $G=\sqrt{p^{*}}$, where $p^{*}=\eta(-1)p$. See [10, 20] for more information.

The following lemmas will be of special use in the sequel.

Lemma 1** (Theorem 5.33, [20]).**

Let $q=p^{e}$ be odd and $f(x)=a_{2}x^{2}+a_{1}x+a_{0}\in\mathbb{F}_{q}[x]$ with $a_{2}\neq 0$. Then

[TABLE]

where $\eta_{e}$ is the quadratic character of $\mathbb{F}_{q}$.

Lemma 2** (Theorem 5.48, [20]).**

With the notation of Lemma 1, we have

[TABLE]

For $\alpha,\beta\in\mathbb{F}_{q}$ and any integer $k$, the exponential sum $S(\alpha,\beta)$ is defined by

[TABLE]

We recall some results of $S(\alpha,\beta)$ for $\alpha\neq 0$ and $q$ odd.

Lemma 3** (Theorem 2, [4]).**

Let $d=\gcd(k,e)$ and $e/d$ be even with $e=2m$. Then

[TABLE]

where $s=m/d$.

Lemma 4** (Theorem 4.7, [5]).**

Let $\beta\neq 0$ and $e/d$ be even with $e=2m$. Then $S(\alpha,\beta)=0$ unless the equation $\alpha^{p^{k}}X^{p^{2k}}+\alpha X=-\beta^{p^{k}}$ is solvable. There are two possibilities.

$(i)$* If $\alpha^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$, then for any choice of $\beta\in\mathbb{F}_{q}$, the equation has a unique solution $x_{0}$ and*

[TABLE]

$(ii)$* If $\alpha^{(q-1)/(p^{d}+1)}=(-1)^{s}$ and if the equation is solvable with some solution $x_{0}$ say, then*

[TABLE]

Lemma 5** (Theorem 4.1, [4]).**

For $e=2m$ the equation $\alpha^{p^{k}}X^{p^{2k}}+\alpha X=0$ is solvable for $X\in\mathbb{F}_{q}^{*}$ if and only if $e/d$ is even and

[TABLE]

In such cases there are $p^{2d}-1$ non-zero solutions.

It follows that $f_{\alpha}$ is a permutation polynomial of $\mathbb{F}_{q}$ with $q=p^{e}$ if and only if $e/d$ is odd or $e/d$ is even with $e=2m$ and $\alpha^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$.

3.2 The proofs of Theorems in Section 2

In this subsection, we will give the proofs of our main results presented in Section 2. Recall that $q=p^{e}$, $d=\gcd(k,e)$, $e/d$ is even with $e=2m$. The code $C_{D_{c}}$ with $c\in\mathbb{F}_{p}$, is defined by

[TABLE]

where $D_{c}=\{x\in\mathbb{F}_{q}^{*}:\mathrm{Tr}(ax^{p^{k}+1})=c\}$, with $a\in\mathbb{F}_{q}^{*}$ and $c\in\mathbb{F}_{p}$. For convenience we define $n_{c}=\#\{x\in\mathbb{F}_{q}:\mathrm{Tr}(ax^{p^{k}+1})=c\}$. Then the length of the code is $n_{0}-1$ for $c=0$ and otherwise $n_{c}$ for $c\neq 0$.

Lemma 6**.**

Let $c\in\mathbb{F}_{p}$. Denote $s=m/d$.

If $c=0$, then

[TABLE]

• 2.

If $c\neq 0$, then

[TABLE]

Proof.

It follows that

[TABLE]

A straightforward calculation gives that $y^{(q-1)/(p^{d}+1)}=1$ for $y\in\mathbb{F}_{p}^{*}$. Then the desired conclusion follows from Lemma 3. ∎

In order to investigate the weight enumerators of $C_{D_{c}}$, we need to do some preparations. Observe that $b=0$ gives the zero codeword. Hence, we may assume that $b\neq 0$ in the rest of this subsection. Let $N_{\rho}(b,c)$ denote the number of components $\mathrm{Tr}(bx)$ of $\mathsf{c}_{b}$ that are equal to $\rho$, where $\rho\in\mathbb{F}_{p}$, $b\in\mathbb{F}_{q}^{*}$ and $c\in\mathbb{F}_{p}$. That is

[TABLE]

Then it is easy to obtain the Hamming weight of $\mathsf{c}_{b}$, that is

[TABLE]

So we only consider $\rho\in\mathbb{F}_{p}^{*}$ and $b\in\mathbb{F}_{q}^{*}$ in the sequel.

By the definition of $N_{\rho}(b,c)$, we have

[TABLE]

where

[TABLE]

We are going to determine the values of $B(b,c)$ in Lemmas 7 and 8. For later use, we set $f_{a}(X)=a^{p^{k}}X^{p^{2k}}+aX\in\mathbb{F}_{q}[X]$ for $a\in\mathbb{F}_{q}^{*}$ in the sequel.

Lemma 7**.**

If $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$, then $f_{a}(X)=-b^{p^{k}}$ has a solution $\gamma$ in $\mathbb{F}_{q}$ and the following assertions hold.

If $c=0$, then

[TABLE]

• 2.

If $c\neq 0$, then

[TABLE]

Proof.

If $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$, by Lemma 5, we know the equation $f_{a}(X)=a^{p^{k}}X^{p^{2k}}+aX$ is a permutation polynomial over $\mathbb{F}_{q}$ and $f_{a}(X)=-b^{p^{k}}$ has a unique solution $\gamma$ in $\mathbb{F}_{q}$. Thus $y^{-1}z\gamma$ is the unique solution for $f_{ay}(X)=-{(bz)}^{p^{k}}$ for any $y,z\in\mathbb{F}_{p}^{*}$. According to Lemma 4, $S(ay,bz)=(-1)^{s}p^{m}\chi_{1}(-ay(y^{-1}z\gamma)^{p^{k}+1})$. It follows from (4) that

[TABLE]

If $\mathrm{Tr}(a\gamma^{p^{k}+1})=0$, then

[TABLE]

leading to the desired results.

Now we assume that $\mathrm{Tr}(a\gamma^{p^{k}+1})\neq 0$ in the rest of the proof. It follows that

[TABLE]

If $c\neq 0$, we have from Lemma 1 that

[TABLE]

This gives the desired assertion, completing the whole proof. ∎

Lemma 8**.**

Let $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$. If $f_{a}(X)=-b^{p^{k}}$ has no solution in $\mathbb{F}_{q}$, then $B(b,c)=0$ for $c\in\mathbb{F}_{p}$. Suppose that $f_{a}(X)=-b^{p^{k}}$ has a solution $\gamma$ in $\mathbb{F}_{q}$, we have

If $c=0$, then

[TABLE]

• 2.

If $c\neq 0$, then

[TABLE]

Proof.

Let $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$. By (4),

[TABLE]

For $y,z\in\mathbb{F}_{p}^{*}$, it follows from Lemma 4 that $S(ay,bz)=0$ unless the equation $f_{a}(X)=-b^{p^{k}}$ is solvable. Note that $f_{a}(X)=aX^{p^{2k}}+aX$ is not a permutation polynomial over $\mathbb{F}_{q}$ by Lemma 5. By a similar argument as above, we obtain the desired conclusions. The details are omitted. ∎

Lemma 9**.**

With the notation introduced above, we denote

[TABLE]

If $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$, then $\#S=p^{e-2d}$.

Proof.

Taking into account that $e/d$ is even and $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$, we get from Lemma 5 that $f_{a}(X)=-b^{p^{k}}$ has $p^{2d}$ solutions in $\mathbb{F}_{q}$, where $b\in S$. For two distinct elements $b_{1}$ and $b_{2}$ in $\mathbb{F}_{q}$, there are no common solutions for equations $f_{a}(X)=-b_{1}^{p^{k}}$ and $f_{a}(X)=-b_{2}^{p^{k}}$. On the other hand, for each $x\in\mathbb{F}_{q}$, we know $f_{a}(x)$ is in $\mathbb{F}_{q}$ and there exits an element $b$ such that $f_{a}(x)=-b^{p^{k}}$. Hence we must have $\#S\cdot p^{2d}=p^{e}$ giving the desired conclusion. ∎

With the above preparations, we can prove our main results listed in Section 2. There are four cases to consider:

• (1)

$c=0$ and $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$,

• (2)

$c=0$ and $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$,

• (3)

$c\neq 0$ and $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$,

• (4)

$c\neq 0$ and $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$.

3.2.1 The first case where $c=0$ and $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$

In this subsection, we assume that $c=0$ and $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$. Recall that $s=m/d$ and $\gamma$ is the unique solution for the equation $f_{a}(X)=-b^{p^{k}}$. By (2), (3.2), Lemmas 6 and 7, we have the following two lemmas.

Lemma 10**.**

If $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$, $b\in\mathbb{F}_{q}^{*}$ and $\rho\in\mathbb{F}_{p}^{*}$, then

[TABLE]

Lemma 11**.**

If $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$ and $b\in\mathbb{F}_{q}^{*}$, then we have

[TABLE]

Now we are in a position to prove Theorem 1.

Proof of Theorem 1.

Denote

[TABLE]

The code $C_{D_{0}}$ has length $n_{0}-1=p^{e-1}+(-1)^{s}(p-1)p^{m-1}-1$ and dimension $e$, since $wt(\mathsf{c}_{b})>0$ for each $b\in\mathbb{F}_{q}^{*}$. By the first two Pless Power Moments (see [15], page 260) the frequency $A_{w_{i}}$ of $w_{i}$ satisfies the following equations

[TABLE]

Solving the equations gives that

[TABLE]

This leads to the weight enumerator and complete weight enumerator given in Theorem 1. ∎

3.2.2 The second case where $c=0$ and $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$

In this subsection, we assume that $c=0$ and $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$. By (2), (3.2), Lemmas 6 and 8, we have the following two lemmas.

Lemma 12**.**

Let $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$, $b\in\mathbb{F}_{q}^{*}$ and $\rho\in\mathbb{F}_{p}^{*}$. If $f_{a}(X)=-b^{p^{k}}$ has no solution in $\mathbb{F}_{q}$, then

[TABLE]

If $f_{a}(X)=-b^{p^{k}}$ has a solution $\gamma$ in $\mathbb{F}_{q}$, then

[TABLE]

Lemma 13**.**

Let $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$ and $b\in\mathbb{F}_{q}^{*}$. If $f_{a}(X)=-b^{p^{k}}$ has no solution in $\mathbb{F}_{q}$, then

[TABLE]

If $f_{a}(X)=-b^{p^{k}}$ has a solution $\gamma$ in $\mathbb{F}_{q}$, then we have

[TABLE]

Now it comes to prove Theorem 2.

Proof of Theorem 2.

Denote

[TABLE]

The code $C_{D_{0}}$ has length $n_{0}-1=p^{e-1}-(-1)^{s}(p-1)p^{m+d-1}-1$ and dimension $e$. It follows from Lemma 9 that $A_{w_{1}}=p^{e}-p^{e-2d}$. By the first two Pless Power Moments (see [15], page 260) the frequency $A_{w_{i}}$ of $w_{i}$ satisfies the following equations

[TABLE]

Solving the equations gives that

[TABLE]

This leads to the weight enumerator and complete weight enumerator given in Theorem 2. ∎

3.2.3 The third case where $c\neq 0$ and $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$

In this subsection, we assume that $c\neq 0$ and $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$. Recall that $\gamma$ is the unique solution for the equation $f_{a}(X)=-b^{p^{k}}$. By (3.2) and Lemma 7, we have the values of $N_{\rho}(b,c)$.

Lemma 14**.**

Assume that $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$, $b\in\mathbb{F}_{q}^{*}$, $c\in\mathbb{F}_{p}^{*}$ and $\rho\in\mathbb{F}_{p}^{*}$. Then

[TABLE]

Lemma 15**.**

Assume that $a^{(q-1)/(p^{d}+1)}\neq(-1)^{s}$, $b\in\mathbb{F}_{q}^{*}$ and $c\in\mathbb{F}_{p}^{*}$. Then we have

[TABLE]

Proof.

It follows from (2), Lemmas 6 and 14 that

[TABLE]

where $n_{c}=p^{e-1}-(-1)^{s}p^{m-1}$. Taking into account that $\sum_{\rho\neq 0}\eta(\rho^{2}-4c\mathrm{Tr}(a\gamma^{p^{k}+1}))=-1-\eta(-c\mathrm{Tr}(a\gamma^{p^{k}+1}))$ by Lemma 2, we get the desired results. ∎

The proof of Theorem 3 is given below.

Proof of Theorem 3.

In this case, it follows from Lemmas 14 and 15 that the weight of $\mathsf{c}_{b}$ has possible values $w_{1}=(p-1)p^{e-2}$ and $w_{2}=(p-1)p^{e-2}-2(-1)^{s}p^{m-1}$. By a similar argument as above, we get the desired conclusions. The details are omitted here. ∎

3.2.4 The fourth case where $c\neq 0$ and $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$

In this subsection, we assume that $c\neq 0$ and $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$. By (2), (3.2), Lemmas 2 and 7, we have the values of $N_{\rho}(b,c)$ and $N_{0}(b,c)$.

Lemma 16**.**

Let $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$, $b\in\mathbb{F}_{q}^{*}$, $c\in\mathbb{F}_{p}^{*}$ and $\rho\in\mathbb{F}_{p}^{*}$. If $f_{a}(X)=-b^{p^{k}}$ has no solution in $\mathbb{F}_{q}$, then

[TABLE]

If $f_{a}(X)=-b^{p^{k}}$ has a solution $\gamma$ in $\mathbb{F}_{q}$, then

[TABLE]

Lemma 17**.**

Let $a^{(q-1)/(p^{d}+1)}=(-1)^{s}$, $b\in\mathbb{F}_{q}^{*}$ and $c\in\mathbb{F}_{p}^{*}$. If $f_{a}(X)=-b^{p^{k}}$ has no solution in $\mathbb{F}_{q}$, then

[TABLE]

If $f_{a}(X)=-b^{p^{k}}$ has a solution $\gamma$ in $\mathbb{F}_{q}$, then we have

[TABLE]

Now we begin to prove Theorem 4.

Proof of Theorem 4.

In this case, it follows from Lemmas 16 and 17 that $wt(\mathsf{c}_{b})$ takes values in $\{w_{1}=(p-1)(p^{e-2}+(-1)^{s}p^{m+d-2}),w_{2}=(p-1)p^{e-2},w_{3}=(p-1)p^{e-2}+2(-1)^{s}p^{m+d-1}\}$. By a similar argument as above, we get the desired conclusions. The details are omitted here. ∎

4 Concluding remarks

In this paper, we employed exponential sums to present the complete weight enumerator and weight enumerator of $C_{D_{c}}$ with defining set $D_{c}$. As introduced in [27], any linear code over $\mathbb{F}_{p}$ can be employed to construct secret sharing schemes with interesting access structures provided that

[TABLE]

where $w_{min}$ and $w_{max}$ denote the minimum and maximum nonzero weights in $C_{D}$, respectively. For the linear codes in Theorems 1 and 3, we have $\frac{w_{min}}{w_{max}}>\frac{p-1}{p}$ if $m>2$. Similarly for the linear codes in Theorems 2 and 4, we have $\frac{w_{min}}{w_{max}}>\frac{p-1}{p}$ if $m>d+2$. We remark that the dimension of the code of this paper is small compared with its length and this makes it suitable for the application in secret sharing schemes with interesting access structures.