Perfect Difference Systems Of Sets Research Articles

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

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.

Optimal FHSs and DSSs via near zero-difference balanced functions

Zero-difference balanced (ZDB) functions were introduced by Ding in connection with constructions of optimal constant composition codes and optimal and perfect difference systems of sets. Based on such functions, people have constructed optimal constant weight codes and optimal frequency-hopping sequences. In order to obtain more optimal cryptographic objects, the zero-difference balanced (ZDB) function is generalized to the near zero-difference balanced (N-ZDB) function in the present paper, whose characterizations are partially given. Furthermore, we prove that near zero-difference balanced (N-ZDB) functions are equivalent to partitioned almost difference families (PADFs) in design theory. As the main contribution of this paper, three classes of the N-ZDB functions are proposed by means of the partition of Zn, where n is an odd positive integer. Employing these N-ZDB functions, we obtain at the same time optimal frequency-hopping sequences and optimal difference systems of sets with flexible parameters.

Zero-Difference Balanced Functions With New Parameters and Their Applications

As an optimal combinatorial object, zero-difference balanced (ZDB) functions introduced by Ding in 2008, are a generalization of the well-known perfect nonlinear functions. ZDB functions have received much attention in recent years due to its important applications in coding theory and sequence design. One objective of this paper is to present a construction of ZDB functions based on a kind of generalized cyclotomy. It generates ZDB functions over cyclic group with new parameters which can not be produced by earlier constructions. Another objective of this paper is to employ these ZDB functions to obtain at the same time: 1) optimal constant-composition codes; 2) perfect difference systems of sets; and 3) optimal frequency-hopping sequences, all with new parameters.

Cyclotomic Constructions of Zero-Difference Balanced Functions With Applications

Based on a recent work of Ding, Wang, and Xiong, we present some cyclotomic constructions of zero-difference balanced (ZDB) functions, which interpret the previous construction proposed by Cai, Zeng, Helleseth, Tang, and Yang. The resulted ZDB functions can be used to generate optimal constant composition codes and perfect difference systems of sets with new parameters.

Three New Families of Zero-Difference Balanced Functions With Applications

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.

Sets of zero-difference balanced functions and their applications

Zero-difference balanced (ZDB) functions can be employed in many applications, e.g., optimal constant composition codes, optimal and perfect difference systems of sets, optimal frequency hopping sequences, etc. In this paper, two results are summarized to characterize ZDB functions, among which a lower bound is used to achieve optimality in applications and determine the size of preimage sets of ZDB functions. As the main contribution, a generic construction of ZDB functions is presented, and many new classes of ZDB functions can be generated. This construction is then extended to construct a set of ZDB functions, in which any two ZDB functions are related uniformly. Furthermore, some applications of such sets of ZDB functions are also introduced.

A New Construction of Zero-Difference Balanced Functions and Its Applications

In this paper, a new construction of zero-difference balanced functions defined on is given, where is an odd positive integer. Based on the generic constructions proposed by Ding, optimal constant composition codes and perfect difference systems of sets with new parameters can be generated from the zero-difference balanced functions constructed in this paper.

Some New Classes of Zero-Difference Balanced Functions

Zero-difference balanced (ZDB) functions were introduced recently by Ding for the construction of optimal constant-composition codes, and optimal and perfect difference systems of sets. They are closely related to partitioned difference families. In this paper, we present generic constructions of ZDB functions from functions with difference-balanced property. In particular, two classes of ZDB functions with new and flexible parameters are reported. Employing these new ZDB functions, we obtain at the same time optimal (1) constant-composition codes, (2) constant-weight codes, and (3) perfect difference systems of sets, all with new and flexible parameters.

Partitioned difference families and almost difference sets

Partitioned difference families (PDFs) were first studied by Ding and Yin in conjunction with the construction of constant composition codes (CCCs). In 2008, Yin et al. presented the constructions of a number of infinite classes of PDFs based on known difference sets in GF(q). In this paper, we further investigate the constructions of PDFs by using known almost difference sets in GF(q), and establish some recursive constructions of PDFs. As their applications, we also get a number of perfect difference systems of sets (DSSs) over Zq2 with q odd prime.

Perfect difference systems of sets and the unit group of [formula omitted

A perfect ( v , { | D i | : 0 ≤ i ≤ s − 1 } , ρ ) difference system of sets is a collection of s disjoint subsets D i , 0 ≤ i ≤ s − 1 of Z v , such that every non-zero element of Z v appears exactly ρ times in the multiset { a − b : a ∈ D i , b ∈ D j , 0 ≤ i ≠ j ≤ s − 1 } . In this paper, we give two recursive constructions of perfect DSS s by using the unit group of Z p n , and obtain some infinite classes of perfect DSS s over Z p n .

Optimal and perfect difference systems of sets from q-ary sequences with difference-balanced property

Difference systems of sets (DSS) are important for the construction of codes for synchronization. In this paper, a general construction of optimal and perfect difference systems of sets based on q-ary sequences of period n = −1 (mod q) with difference- balanced property is presented, where q is a prime power. This works for all the known q-ary sequences with ideal autocorrelation, and generalizes the earlier construction based on ternary sequences with ideal autocorrelation. In addition, we construct another class of optimal and perfect difference systems of sets, employing decimation of q-ary d-form sequences of period q m −1 with difference-balanced property, which generalizes the previous construction from power functions.

Perfect difference systems of sets and Jacobi sums

A perfect ( v , { k i ∣ 1 ≤ i ≤ s } , ρ ) difference system of sets (DSS) is a collection of s disjoint k i -subsets D i , 1 ≤ i ≤ s , of any finite abelian group G of order v such that every non-identity element of G appears exactly ρ times in the multiset { a − b ∣ a ∈ D i , b ∈ D j , 1 ≤ i ≠ j ≤ s } . In this paper, we give a necessary and sufficient condition in terms of Jacobi sums for a collection { D i ∣ 1 ≤ i ≤ s } defined in a finite field F q of order q = e f + 1 to be a perfect ( q , { k i ∣ 1 ≤ i ≤ s } , ρ ) -DSS, where each D i is a union of cyclotomic cosets of index e (and the zero 0 ∈ F q ). Also, we give numerical results for the cases e = 2 , 3 , and 4.

Optimal and perfect difference systems of sets

Difference systems of sets (DSS) were introduced in 1971 by Levenstein for the construction of codes for synchronization, and are closely related to cyclic difference families. In this paper, algebraic constructions of difference systems of sets using functions with optimum nonlinearity are presented. All the difference systems of sets constructed in this paper are perfect and optimal. One conjecture on difference systems of sets is also presented.