Almost perfect sequences based on cyclic difference sets

The perfect sequences are so ideal that all out-of-phase autocorrelation coefficients are zero, and if the sequences are used as the coding modulating signal for the phase-modulated radar, there will be no interference of side lobes theoretically. However, it has been proved that there are no binary perfect sequences of period 4 〈 n ≤ 12100. Hence, the almost perfect sequences with all out-of-phase autocorrelation coefficients as zero except one are of great practice in engineering. Currently, the cyclic difference set is one of most effective tools to analyze the binary sequences with perfect periodic autocorrelation function. In this article, two characteristic formulas corresponding to the autocorrelation and symmetric structure of almost perfect sequences are calculated respectively. All almost perfect sequences with period n, which is a multiple of 4, can be figured out by the two formulas. Several almost perfect sequences with different periods have been hunted by the program based on the two formulas and then applied to the simulation program and practical application for ionospheric sounding.

A Construction of Relative Difference Set

In this paper, we give a construction of RDS in Galois ring by using some bent function, and obtain the equivalent relationship between RDS and a kind of bent function. At the same time, its existence is demonstrated.

On Dickson Polynomials and Difference Sets

In 1998, Maschietti constructed several cyclic difference sets from monomial hyperovals. R. Evans, H.D.L. Holloman, C. Krattnthaler and Qing Xiang gave an algebraic proof of the two autocorrelation property of the related binary sequence. In this paper, we show that hyperovals are very closely related to two-to-one maps, and then we proceed to generalize Maschietti＇s result.

New Partial Geometric Difference Sets and Partial Geometric Difference Families

Using Galois rings and Galois fields, we construct several infinite classes of partial geometric difference sets, and partial geometric difference families, with new parameters. Furthermore, these partial geometric difference sets(and partial geometric difference families) correspond to new infinite families of directed strongly regular graphs. We also discuss some of the links between partially balanced designs, 2-adesigns(which were recently coined by Cunsheng Ding in "Codes from Difference Sets"(2015)), and partial geometric designs, and make an investigation into when a 2-adesign is a partial geometric design.

SOME RESULTS ON GENERALIZED DIFFERENCE SETS

Sequences with ideal correlation functions have important applications in communications such as CDMA, FDMA, etc. It has been shown that difference sets can be used to construct such sequences. The author extends Pott and Bradley＇s method to a much broader case by proposing the concept of generalized difference sets. Some necessary conditions for the existence of generalized difference sets are established by means of some Diophantine equations. The author also provides an algorithm to determine the existence of generalized difference sets in the cyclic group Zv. Some examples are presented to illustrate that our method works.

Measure of rotatability of modified five-level second-order rotatable designusing supplementary difference sets

Rotatability is a desirable quality of fitting response surface experimental designs. The propertystates that the variance of the estimated response made from the Taylor’s series expansion areconstant on circles, spheres and hyper-spheres about the centre of the design. In this article,a measure of rotatability of modified second-order rotatable design is presented. The variancefunction of a second-order response design and an infinite class of supplementary difference setsis used in coming up with the design.

On the Nonexistence of Partial Difference Sets by Projections to Finite Fields

In the study of(partial)difference sets and their generalizations in groups G,the most widely used method is to translate their definition into an equation over group ring Z[G]and to investigate this equation by applying complex representations of G.In this paper,we investigate the existence of(partial)difference sets in a different way.We project the group ring equations in Z[G]to Z[N]where N is a quotient group of G isomorphic to the additive group of a finite field,and then use polynomials over this finite field to derive some existence conditions.

Feature Selection Based on Difference and Similitude in Data Mining

Feature selection is the pretreatment of data mining. Heuristic search algorithms are often used for this subject. Many heuristic search algorithms are based on discernibility matrices, which only consider the difference in information system. Because the similar characteristics are not revealed in discernibility matrix, the result may not be the simplest rules. Although differencesimilitude（DS） methods take both of the difference and the similitude into account, the existing search strategy will cause some important features to be ignored. An improved DS based algorithm is proposed to solve this problem in this paper. An attribute rank function, which considers both of the difference and similitude in feature selection, is defined in the improved algorithm. Experiments show that it is an effective algorithm, especially for large-scale databases. The time complexity of the algorithm is O（｜ C ｜^2｜U ｜^2）.

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.

Constructions for almost perfect binary sequence pairs with even length

The concept of the binary sequence pair is generalized from a single binary sequence. Binary sequence pairs are applied in many fields of radar, sonar or communication systems, in which signals with optimal periodic correlation are required. Several types of almost perfect binary sequence pairs of length T = 2q are constructed, where q is an odd number. These almost perfect binary sequence pairs are based on binary ideal sequence or binary ideal two-level correlation sequence pairs by using Chinese remainder theorem. For these almost perfect binary sequence pairs with good balanced property, their corresponding divisible difference set pairs(DDSPs) are also derived.

A Kind of Diophantine Equations in Finite Simple Groups

In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation.

SOME NEW FAMILIES OF SDSS AND HADAMARD MATRICES

In this paper we prove that for any prime power q equivalent to 3 (mod 8) there exist 4 - {q(2); k, k, k, k; lambda} supplementary difference sets (SDSs) with k = q(q - 1)/2, lambda 4k - q(2), and Hadamard matrices of order 4q(2), and give several constructions of these SDSs. Moreover, combining the results of reference [1], we conclude that for any prime p equivalent to 3 (mod 8) and integer r greater than or equal to 1 there exists an Hadamard matrix of order 4p(2r).

ON THE CLASS

An infinite family of is obtained, that is,(ν)≠￠ forυ ∈ N=N1UN2 U N3 where Using(υ),we give 2-(2υ;k1,k2;k1 + k2 -υ} supplementary difference sets withυ=(k1-υ)2 + (k2 - υ)2. Finally. we prove that if there exists an orthogonal design OD(4t;t,t,t,t) and(υ) ≠￠ then a Hadamard rnatrix of order 4tυ can be constructed.

Diference 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.

The ranks of Maiorana-McFarland bent functions

In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t 6.

Constructions for Anonymous Secret Sharing Schemes Using Combinatorial Designs

In an anonymous secret sharing scheme the secret can be reconstructed without knowledge of which participants hold which shares. In this paper some constructions of anonymous secret sharing schemes with 2 thresholds by using combinatorial designs are given. Let v（t, w, q） denote the minimum size of the set of shares of a perfect anonymous （t, w） threshold secret sharing scheme with q secrets. In this paper we prove that v（t, w, q） - Θ（q） if t and w are fixed and that the lower bound of the size of the set of shares in [4] is not optimal under certain condition.

A SURVEY OF ORTHOGONAL ARRAYS OF STRENGTH TWO

The purpose of this paper is to survey the construction of orthogonal arrays of strength two by using difference sets. Some methods for constructing difference set D(2p.2p,p,2), where p is a prime or a prime power, are given. It is shown that the Kronecker sum of a difference set D(λ1p, k1, p, 2) and an orthogonal array(λ2p2, k2, p, 2) leads to another orthogonal array (λ1λ2p3 .k1k2+1 ,p, 2). This enables us to construct orthogonal arrays[2p(n+1)、1+2(p+p2 +…+pn),p,2],[4p(n+2),1+2p+4(p2+p3+…+p(n+1)),p, 2],and [8p(n+3),1+2P+4p2+8(p3+p4+…+p(n+2)),p,2]where p is a prime or a prime power.

New Results on Nonexistence of Perfect p-Ary Sequences and Almost p-Ary Sequences

Complex periodical sequences with lower antocorrelation values are used in CDMA comnmnieation systems and cryptography. In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyelotomic fields and their subfields.

Some Results on the Multiplier Conjecture for the Case n=5n_1

Using the method presented in [1], we obtain some new results which improve on the result of MacFarland’s theorem (see [2]) in this case.

Non-Uniformity and Generalised Sacks Splitting

We show that there do not exist computable fimetions f_1(e,i).f_2(e,i).g_1(e,i),g_2(e,i)such that for all e,i ∈ω, (1)(W_(f_1)(e,i)-W_(f_2)(e,i))≤T(W_e-W_1): (2)(W_(g_1)(e,i)-W_(g_2)(e,i))≤T(W_e-W_i): (3)(W_w-W_i)≤T(W_(f_1)(e,i)-W_(f_2)(e,i))⊕(W_(g_1)(e,i)-W_(g_2)(e,i)): (4)(W_e-W_i)T(W_(f_1)(e,i)-W_(f_2)(e,i))uuless(W_e-W_i)≤T:and (5)(W_e-W_i)T(E_(g_1)(e,i)-W_(g_2)(e,i))unless(W_w-W_i)≤T. It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.