Zero-difference balanced (ZDB) functions integrate a number of subjects in combinatorics and algebra, and have many applications in coding theory, cryptography, and communications engineering. In this paper, three new families of ZDB functions are presented. The first construction gives ZDB functions defined on the abelian groups (GF(q1)×,...,×GF(qk),+) with new and flexible parameters. The other two constructions are based on 2-cyclotomic cosets and yield ZDB functions on \BBZn with new parameters. The parameters of optimal constant composition codes, optimal, and perfect difference systems of sets obtained from these new families of ZDB functions are also summarized.

Three New Families of Zero-Difference Balanced Functions With Applications

In the present paper, by generalizing the definition of the zero-difference balanced (ZDB) function to be the G-ZDB function, several classes of G-ZDB functions are constructed based on properties of cyclotomic numbers. Furthermore, some special constant composition codes are obtained directly from G-ZDB functions.

On generalized zero-difference balanced functions

Zero-difference balanced (ZDB) functions integrate a number of subjects in combinatorics and algebra, and have many applications in coding theory, cryptography and communications engineering. In this paper, three new families of ZDB functions are presented. The first construction, inspired by the recent work \cite{Cai13}, gives ZDB functions defined on the abelian groups $(\gf(q_1) \times \cdots \times \gf(q_k), +)$ with new and flexible parameters. The other two constructions are based on $2$-cyclotomic cosets and yield ZDB functions on $\Z_n$ with new parameters. The parameters of optimal constant composition codes, optimal and perfect difference systems of sets obtained from these new families of ZDB functions are also summarized.

Three New Families of Zero-difference Balanced Functions with Applications

Perfect nonlinear functions are of importance in cryptography. By using Galois rings and investigating the character values of corresponding relative difference sets, we construct a perfect nonlinear function from $\mathbb{Z}^{n}_{p_{2}}$ to $\mathbb{Z}^{m}_{p_{2}}$ where 2m is possibly larger than the largest divisor of n. Meanwhile we prove that there exists a perfect nonlinear function from $\mathbb{Z}^{2}_{2_{p}}$ to $\mathbb{Z}_{2_{p}}$ if and only if p=2, and that there doesn't exist a perfect nonlinear function from $\mathbb{Z}^{2n}_{2k_{l}}$ to $\mathbb{Z}^{m}_{2k_{l}}$ if m>n and l(l is odd) is self-conjugate modulo 2k(k≥1) .

Jinjiang Yuan

Papers

A note of perfect nonlinear functions