Hyperplane partitions and difference systems of sets

Difference Systems of Sets (DSS) are combinatorial configurations that arise in connection with code synchronization. This paper gives new constructions of DSS obtained from partitions of hyperplanes in a finite projective space, as well as DSS obtained from balanced generalized weighing matrices and partitions of the complement of a hyperplane in a finite projective space.     

Difference systems of sets based on cosets partitions

Difference systems of sets （DSS） are combinatorial configurations that arise in connection with code synchronization. This paper proposes a new method to construct DSSs, which uses known DSSs to partition some of the cosets of Zv relative to subgroup of order k, where v = km is a composite number. As applications, we obtain some new optimal DSSs.     

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=ef+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.     

On some new difference systems of sets constructed from the cyclotomic classes of order 12

Difference system of sets (DSS), introduced by Levenshtein, has an interesting connection with the construction of comma-free codes. In this paper, we construct two new families of DSS from the cyclotomic classes of order 12.     

Two applications of relative difference sets: Difference triangles and negaperiodic autocorrelation functions

The well-known difference sets have various connections with sequences and their correlation properties. It is the purpose of this note to give two more applications of the (not so well known) relative difference sets: we use them to construct difference triangles (based on an idea of A. Ling) and we show that a certain nonexistence result for semiregular relative difference sets implies the nonexistence of negaperiodic autocorrelation sequences (answering a question of Parker [Even length binary sequence families with low negaperiodic autocorrelation, in: Applied Algebra, Algebraic Algorithms and Error-correcting Codes, Melbourne, 2001, Lecture Notes in Computer Science, vol. 2227, Springer, Berlin, 2001, pp. 200-209.]).     

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.     

Difference Sets and Recursion Theory

There is a recursive set of natural numbers which is the difference set of some recursively enumerable set but which is not the difference set of any recursive set.     

On the constructions of MDS self-dual codes via cyclotomy

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.     

Constructions of optimal difference systems of sets

Difference systems of sets (DSSs) are combinatorial configurations which were introduced in 1971 by Levenstein for the construction of codes for synchronization. In this paper, we present two kinds of constructions of difference systems of sets by using disjoint difference families and a special type of difference sets, respectively. As a consequence, new infinite classes of optimal DSSs are obtained.     

Difference sets and three-weight linear codes from trinomials

We confirm a conjecture of Cunsheng Ding claiming that the punctured value-sets of a list of eleven trinomials over odd-degree extensions of the binary field give rise to difference sets with Singer parameters. In the course of confirming the conjecture, we show that these trinomials share the remarkable property that every element of the value-set of each trinomial has either one or four preiamges. We also give the partial resolution of another conjecture of Cunsheng Ding claiming that linear codes constructed from those eleven trinomials are three-weight.     

Constructions of Partial Difference Sets and Relative DifferenceSets Using Galois Rings

We use Galois rings to construct partial difference sets and relative difference sets in non-elementary abelianp-groups. As an example, we also use Galois ringG R(4, 2) to construct a (96,20,4) difference set in Z₄ × Z₄ × Z₆.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

New identification and control methods of sine-function Julia sets

In this paper, we propose two new methods to realize drive-response system synchronization control and parameter identification for two kinds of sine-function Julia sets. By means of these two methods, the zero asymptotic sliding variables and the stability theory in difference equations are applied to control the fractal identification. Furthermore, the problem of synchronization control is solved in the case of a drive system with unknown parameters, where the unknown parameters of the drive system can be identified in the asymptotic synchronization process. The results of simulation examples demonstrate the effectiveness of the new methods.     

A Note on Certain 2-Groups with Hadamard Difference Sets

Nontrivial difference sets in 2-groups are part of the family of Hadamarddifference sets. An abelian group of order 2^2d+2 has a difference setif and only if the exponent of the group is less than or equal to2 ^d+2. We provide an exponent bound for a more general type of 2-groupwhich has a Hadamard difference set. A recent construction due to Davis and Iiamsshows that we can attain this bound in at least half of the cases.     

Negative Latin square type partial difference sets in nonabelian groups of order 64

There exist few examples of negative Latin square type partial difference sets (NLST PDSs) in nonabelian groups. We present a list of 176 inequivalent NLST PDSs in 48 nonisomorphic, nonabelian groups of order 64. These NLST PDSs form 8 nonisomorphic strongly regular graphs. These PDSs were constructed using a combination of theoretical techniques and computer search, both of which are described. The search was run exhaustively on 212/267 nonisomorphic groups of order 64.     

Forbidden sets and stability in some rational difference equations

In this paper, we solve open problem (5) submitted by Sedaghat in his paper, On third order rational difference equations with quadratic terms, J. Differ. Equ. Appl., 14(8) (2008), pp. 889–897. We also confirm conjecture (6) in the mentioned paper.