A new construction of zero-difference balanced functions and two applications

Abstract

Zero-difference balanced (ZDB) functions are a generalization of perfect nonlinear functions, and have received a lot of attention due to their important applications in coding theory, cryptography, combinatorics and some engineering areas. In this paper, based on cyclotomy and generalized cyclotomy, a construction of a partitioned difference family is presented, and then a class of ZDB functions is obtained. In addition, these ZDB functions are applied to construct optimal constant composition codes and optimal and perfect difference systems of sets.

Appendices

Appendix A: Proof of Lemma 1

Proof

(i) Using Eqs. (1) and (2), we have

$$\begin{aligned} \sum \limits _{I=0}^{f-1}(I,\ I) = \sum \limits _{I=0}^{f-1}(f-I,\ 0) = \sum \limits _{I=0}^{f-1}(I,\ 0) = e-1. \end{aligned}$$

(ii) Since e is odd, by Eq. (3) we have

$$\begin{aligned} \sum \limits _{I=0}^{\frac{f}{2}-1}(I,\ I) = \sum \limits _{I=0}^{\frac{f}{2}-1}\left( I+\frac{f}{2},\ I+\frac{f}{2}\right) = \sum \limits _{I=\frac{f}{2}}^{f-1}(I,\ I), \end{aligned}$$

and hence

$$\begin{aligned} \sum \limits _{I=0}^{\frac{f}{2}-1}(I,\ I) = \frac{1}{2}\sum \limits _{I=0}^{f-1}(I,\ I)=\frac{1}{2}(e-1). \end{aligned}$$

(iii) Note that for each $0\le J\le f-1$,

$$\begin{aligned} \sum \limits _{I=0}^{f-1}(I-J,\ I-J) = e-1\ \ \mathrm {and}\ \ \sum \limits _{I=0}^{\frac{f}{2}-1}(I-J,\ I-J)=\sum \limits _{I=\frac{f}{2}}^{f-1}(I-J,\ I-J). \end{aligned}$$

So

$$\begin{aligned} \sum \limits _{I=0}^{\frac{f}{2}-1}(I-J,\ I-J) =\frac{1}{2}(e-1)\ \ \mathrm {for\ each}\ 0\le J\le f-1. \end{aligned}$$

$\square$

Appendix B: Proof of Lemma 2

Proof

(i) By setting $J=(0, 0, \ldots , 0)$ and $t=0$ in Eq. (5), we immediately have

$$\begin{aligned} \sum \limits _{I\in \varPsi _{n_1}}(I,\ I) = e-1. \end{aligned}$$

(ii) Since $e\ge 3$ is odd, we observe that as the first coordinate $i_1$ of I runs over the set $\{0, \ldots , \frac{\left( F_{n_1}\right) _1}{2e}-1\}$, the first coordinate $i_1+\frac{\left( F_{n_1}\right) _1}{2e}$ of $I+\frac{F_{n_1}}{2e}$ will run over the set $\{\frac{\left( F_{n_1}\right) _1}{2e}, \ldots , \frac{\left( F_{n_1}\right) _1}{e}-1\}$. Hence, using Eq. (6), we have

$$\begin{aligned} \sum \limits _{\begin{array}{c} I=(i_1, \ldots , i_{r})\in \varPsi _{n_1},\\ 0\le i_1< \frac{\left( F_{n_1}\right) _1}{2e} \end{array}}(I,\ I)= \sum \limits _{\begin{array}{c} I=(i_1, \ldots , i_{r})\in \varPsi _{n_1},\\ 0\le i_1< \frac{\left( F_{n_1}\right) _1}{2e} \end{array}}\left( I+\frac{F_{n_1}}{2e},\ I+\frac{F_{n_1}}{2e}\right) = \sum \limits _{\begin{array}{c} I=(i_1, \ldots , i_{r})\in \varPsi _{n_1},\\ \frac{\left( F_{n_1}\right) _1}{2e}\le i_1< \frac{\left( F_{n_1}\right) _1}{e} \end{array}}(I,\ I). \end{aligned}$$

Therefore, by combining this with (i), we get

$$\begin{aligned} \sum \limits _{\begin{array}{c} I=(i_1, \ldots , i_{r})\in \varPsi _{n_1},\\ 0\le i_1< \frac{\left( F_{n_1}\right) _1}{2e} \end{array}}(I,\ I)=\frac{1}{2}\sum \limits _{I\in \varPsi _{n_1}}(I,\ I)=\frac{1}{2}(e-1). \end{aligned}$$

(iii) Note that for each $J\in \varPsi _{n_1}$,

$$\begin{aligned} \sum \limits _{I\in \varPsi _{n_1}}(I-J,\ I-J) = e-1 \end{aligned}$$

and

$$\begin{aligned} \sum \limits _{\begin{array}{c} I=(i_1, \ldots , i_{r})\in \varPsi _{n_1},\\ 0\le i_1< \frac{\left( F_{n_1}\right) _1}{2e} \end{array}}(I-J,\ I-J)=\sum \limits _{\begin{array}{c} I=(i_1, \ldots , i_{r})\in \varPsi _{n_1},\\ \frac{\left( F_{n_1}\right) _1}{2e}\le i_1< \frac{\left( F_{n_1}\right) _1}{e} \end{array}}(I-J,\ I-J), \end{aligned}$$

and so

$$\begin{aligned} \sum \limits _{\begin{array}{c} I=(i_1, \ldots , i_{r})\in \varPsi _{n_1},\\ 0\le i_1< \frac{\left( F_{n_1}\right) _1}{2e} \end{array}}(I-J,\ I-J) =\frac{1}{2}(e-1)\ \ \mathrm {for\ each}\ J\in \varPsi _{n_1}. \end{aligned}$$

$\square$