Two classes of p-ary bent functions and linear codes with three or four weights

Abstract

In this paper, a secondary construction of p-ary bent functions is presented. Two classes of p-ary bent functions of algebraic degree p are constructed by modifying the values of some known bent functions on some set of $\mathbb {F}_{p^{2k}}$. Furthermore, the resulted p-ary bent functions are employed to construct a class of linear codes with three or four weights.

Appendix :

The number $A_{2}^{\bot }$ of codewords with weight 2 in the dual code of $\mathcal {C}_{D}$ of Theorem 3.

Let D be the set defined in (15). Since 0∉D, the minimum distance of $\mathcal {C}_{D}^{\bot }$ cannot be one. Assume that x∈D. We claim that if c x∈D for any $c \in \mathbb {F}_{p}^{*}\setminus \{1\}$, then $\text {Tr}^{k}_{1}(\lambda x^{p^{k}+1})=0$ and $\text {Tr}_{1}^{n}(ux)\text {Tr}_{1}^{n}(x)^{p-1}=0$. Otherwise, it follows from x∈D and c x∈D that c=1. This is a contradiction with $c \in \mathbb {F}_{p}^{*}\setminus \{1\}$. To give the number $A_{2}^{\bot }$ of codewords with weight 2 in $\mathcal {C}_{D}^{\bot }$, we define the following set

$$D_{1}=\{x\in \mathbb{F}_{p^{n}}^{*}: \text{Tr}^{k}_{1}(\lambda x^{p^{k}+1})=0\ \text{and} \ \text{Tr}_{1}^{n}(ux)\text{Tr}_{1}^{n}(x)^{p-1}=0\}. $$

Indeed, $D_{1}=\{x\in \mathbb {F}_{p^{n}}^{*}: x\in D ~\text {and} ~cx\in D\}\subseteq D$ for any $c \in \mathbb {F}_{p}^{*}\setminus \{1\}$. By the orthogonal property of additive characters and Theorem 1, we have

$$\begin{array}{@{}rcl@{}} |D_{1}|+1&=&\frac{1}{p^{2}}\sum\limits_{x\in\mathbb{F}_{p^{n}}}(\sum\limits_{y\in\mathbb{F}_{p}}\omega_{p}^{y\text{Tr}^{k}_{1}(\lambda x^{p^{k}+1})})(\sum\limits_{z\in\mathbb{F}_{p}}\omega_{p}^{z\text{Tr}_{1}^{n}(ux)\text{Tr}_{1}^{n}(x)^{p-1}}) \\ &=&p^{n-2}+\frac{1}{p^{2}}\sum\limits_{y\in\mathbb{F}_{p}^{*}}\sum\limits_{x\in\mathbb{F}_{p^{n}}}\omega_{p}^{y\text{Tr}^{k}_{1}(\lambda x^{p^{k}+1})}+\frac{1}{p^{2}}\sum\limits_{z\in\mathbb{F}_{p}^{*}}\sum\limits_{x\in\mathbb{F}_{p^{n}}}\omega_{p}^{z\text{Tr}_{1}^{n}(ux)\text{Tr}_{1}^{n}(x)^{p-1}} \\ &&+\frac{1}{p^{2}}\sum\limits_{y\in \mathbb{F}_{p}^{*}}\sum\limits_{z\in \mathbb{F}_{p}^{*}}\sum\limits_{x\in\mathbb{F}_{p^{n}}}\omega_{p}^{y\text{Tr}^{k}_{1}(\lambda x^{p^{k}+1})+z\text{Tr}_{1}^{n}(ux)\text{Tr}_{1}^{n}(x)^{p-1}} \\ &=&p^{n-2}+\frac{1}{p^{2}}\sum\limits_{y\in\mathbb{F}_{p}^{*}}\widehat{\chi}_{f_{y\lambda,0}}(0)+\frac{1}{p^{2}}\sum\limits_{y\in \mathbb{F}_{p}^{*}}\sum\limits_{z\in \mathbb{F}_{p}^{*}}\widehat{\chi}_{f_{y\lambda,zu}}(0)+\frac{1}{p^{2}}\sum\limits_{z\in\mathbb{F}_{p}^{*}}\sum\limits_{x\in\mathbb{F}_{p^{n}}}\omega_{p}^{z\text{Tr}_{1}^{n}(ux)\text{Tr}_{1}^{n}(x)^{p-1}} \\ &=&p^{n-2}-p^{k-2}(p-1)-p^{k-2}(p-1)^{2}+\frac{1}{p^{2}}{\Omega}, \end{array} $$

(24)

where

$${\Omega}=\sum\limits_{z\in\mathbb{F}_{p}^{*}}\sum\limits_{x\in\mathbb{F}_{p^{n}}}\omega_{p}^{z\text{Tr}_{1}^{n}(ux)\text{Tr}_{1}^{n}(x)^{p-1}}. $$

Let T _i be the set defined in (4) for i=0,1,⋯ ,p−1. By a similar process of calculating the Walsh transform coefficient of f(x) in Lemma 5, we have

$$\begin{array}{@{}rcl@{}} {\Omega}=(p-1)p^{n-1}. \end{array} $$

(25)

Combining (24) and (25), we get

$$|D_{1}|=(2p-1)p^{n-3}-(p-1)p^{k-1}-1. $$

Note that if x∈D ₁, then c x∈D ₁ for any $c\in \mathbb {F}_{p}^{*}$. This implies that the minimum distance of $\mathcal {C}^{\bot }$ is equal to 2. In fact, for each p−1 distinct elements x,2x,⋯ ,(p−1)x∈D ₁, we can obtain ${\ p-1\choose 2}(p-1)$ codewords with weight 2 in $\mathcal {C}_{D}^{\perp }$. Therefore, the number $A_{2}^{\perp }$ of codewords with weight 2 in $\mathcal {C}_{D}^{\perp }$ is

$$A_{2}^{\perp}=\frac{|D_{1}|}{p-1}{\ p-1 \choose 2}(p-1)={\ p-1 \choose 2}|D_{1}|. $$