Partial Difference Sets with Paley Parameters

Partial difference sets with parameters (,k,,µ) = (,(– 1)/2, ( – 5)/4,( – 1)/4) are called Paleypartial difference sets. By using finite local rings, we constructa family of Paley PDSs for abelian p-groups with any given exponent.Furthermore, we prove some non-existence results on Paley PDSs.Using these results, we prove that Paley PDSs exist in a rank2 abelian group if and only if the group is isomorphic to Z_p^r x Z_{p ^r} where p is an odd prime.     

On Storer Difference Sets

Let p, q be distinct odd primes, and let a, b be positive integers.In this paper we prove that if S(p^a, q^b) is a Storer differenceset with the parameters = p^aq^b, k = (–1)/4 and =(–5)/16,then we have a = b = 1, and , where , and r is a positiveinteger. 1991 Mathematics Subject Classification 05B10.     

On Central Cantor Sets with Self-Arithmetic Difference of Positive Lebesgue Measure

We prove the existence of C^r - (but not C^r+1-) regular centralCantor sets with zero Lebesgue measure such that their selfarithmetic difference is a Cantor set with positive Lebesguemeasure. This is motivated by a conjecture in the field of bifurcationsof dynamical systems posed by Jacob Palis.     

Cyclic Relative Difference Sets and their p-Ranks

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q ^3k –1)/(q–1), q–1, q ^3k–1, q ^3k–2) relative difference sets, where q=3 ^e . These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.     

Difference Sets Corresponding to a Class of Symmetric Designs

We study difference sets with parameters(v, k, ) = (p ^s(r ^2m - 1)/(r - 1), p ^s-1 r ^2m-2 r - 1)r ^{2m -2}, where r = r ^s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p,s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z ₃ × Z ₉ × Z ₇.     

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     

On the p-Ranks and Characteristic Polynomials of Cyclic Difference Sets

In this paper, the p-ranks and characteristic polynomials of cyclic difference sets are derived by expanding the trace expressions of their characteristic sequences. Using this method, it is shown that the 3-ranks and characteristic polynomials of the Helleseth–Kumar–Martinsen (HKM) difference set and the Lin difference set can be easily obtained. Also, the p-rank of a Singer difference set is reviewed and the characteristic polynomial is calculated using our approach.     

Negative Latin Square type Partial Difference Sets in Nonelementary Abelian 2-Groups

Combining results on quadrics in projective geometries withan algebraic interplay between finite fields and Galois rings,the first known family of partial difference sets with negativeLatin square type parameters is constructed in nonelementaryabelian groups, the groups x for all k when is odd andfor all k < when is even. Similarly, partial differencesets with Latin square type parameters are constructed in thesame groups for all k when is even and for all k< when is odd. These constructions provide the first example wherethe non-homomorphic bijection approach outlined by Hagita andSchmidt can produce difference sets in groups that previouslyhad no known constructions. Computer computations indicate thatthe strongly regular graphs associated to the partial differencesets are not isomorphic to the known graphs, and it is conjecturedthat the family of strongly regular graphs will be new.     

Abelian Difference Sets Without Self-conjugacy

We obtain some results that are useful to the study of abelian difference sets and relative difference sets in cases where the self-conjugacy assumption does not hold. As applications we investigate McFarland difference sets, which have parameters of the form v=q^d+1( q^d+ q^d-1 +...+ q+2) ,k=q^d( q^d+q^d-1+...+q+1) , = q^d ( q^(d-1)+q^(d-2)+...+q+1), where q is a prime power andd a positive integer. Using our results, we characterize those abelian groups that admit a McFarland difference set of order k- = 81. We show that the Sylow 3-subgroup of the underlying abelian group must be elementary abelian. Our results fill two missing entries in Kopilovich's table with answer no.     

Constructions of Nested Partial Difference Sets with Galois Rings

There have been several recent constructions of partial difference sets (PDSs) using the Galois rings for p a prime and t any positive integer. This paper presents constructions of partial difference sets in where p is any prime, and r and t are any positive integers. For the case where 2$$ " align="middle" border="0"> many of the partial difference sets are constructed in groups with parameters distinct from other known constructions, and the PDSs are nested. Another construction of Paley partial difference sets is given for the case when p is odd. The constructions make use of character theory and of the structure of the Galois ring , and in particular, the ring × . The paper concludes with some open related problems.     

One-Dimensional p-Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalyticsets, where p is a fixed prime number, Q_p is the field of p-adicnumbers and Z_p is the ring of p-adic integers. One of thesetheorems [2, 3.32] says that each subanalytic subset of Z_p issemialgebraic. This is extended here as follows.     

Riesz Transforms and Harmonic Lip1-Capacity in Cantor Sets

We estimate the L²-norm of the s-dimensional Riesz transformson some Cantor sets in R^d. Towards this end, we show that theRiesz transforms truncated at different scales behave in a quasiorthogonalway. As an application, we obtain some precise numerical estimatesfor the Lipschitz harmonic capacity of these sets. 2000 MathematicsSubject Classification 42B20, 42B25.     

On Relative Difference Sets in Dihedral Groups

In this paper, we study extensions of trivial difference sets in dihedral groups. Such relative difference sets have parameters of the form (uλ,u,uλ, λ) or (uλ+2,u, uλ+1, λ) and are called semiregular or affine type, respectively. We show that there exists no nontrivial relative difference set of affine type in any dihedral group. We also show a connection between semiregular relative difference sets in dihedral groups and Menon–Hadamard difference sets. In the last section of the paper, we consider (m, u, k, λ) difference sets of general type in a dihedral group relative to a non-normal subgroup. In particular, we show that if a dihedral group contains such a difference set, then m is neither a prime power nor product of two distinct primes.     

Curves which Intersect Lines in Finite Sets

We study closed subsets in the plane which intersect each linein at least m points and at most n points, for which we tryto minimize the difference n – m. It is known that m cannotbe equal to n. The results in this paper show that for everyeven number n there exist closed sets in the plane for whichm = n – 2.     

Foldings of Difference Sets in Abelian Groups

 In this paper, we show that under some conditions the existence of a difference set in G implies the existence of another difference set with the same parameters in G′, where G and G′ are abelian groups of the same order. This explains why there are more difference sets in abelian groups of low exponent and high rank than in those of high exponent and low rank. Received: September 1, 1997 / Revised: March 24, 1998     

Combining Perfect Systems of Difference Sets

We use sharply 2-transitive permutation groups to constructan additive sequence of permutations from a system of differencesets, each component of which has size one less than a primepower. This allows us to combine perfect systems of differencesets to form other perfect systems. In particular, if thereexists a perfect (m, n + 1, 1)-system and a perfect (q, n +1, 1)-system then there exists a perfect (mqn(n + 1) + m + q,n + 1, 1)-system.     

Two Generalized Constructions of Relative Difference Sets

We give two generalizations of some known constructions of relative difference sets. The first one is a generalization of a construction of RDS by Chen, Ray-Chaudhuri and Xiang using the Galois ring GR(4, m). The second one generalizes a construction of RDS by Ma and Schmidt from the setting of chain rings to a setting of more general rings.     

A Sharp Exponent Bound for McFarland Difference Sets withp=2

We show that under the self-conjugacy condition a McFarland difference set withp=2 andf2 in an abelian groupGcan only exist, if the exponent of the Sylow 2-subgroup does not exceed 4. The method also works for oddp(where the exponent bound ispand is necessary and sufficient), so that we obtain a unified proof of the exponent bounds for McFarland difference sets. We also correct a mistake in the proof of an exponent bound for (320, 88, 24)-difference sets in a previous paper.     

Singularities of Centre Symmetry Sets

The center symmetry set (CSS) of a smooth hypersurface S inan affine space Rⁿ is the envelope of lines joining pairs ofpoints where S has parallel tangent hyperplanes. The idea stemsfrom a definition of Janeczko, in an alternative version dueto Giblin and Holtom. For n = 2 the envelope is always real,while for n > 3 the existence of a real envelope dependson the geometry of the hypersurface. In this paper we make alocal study of the CSS, some results applying to n 5 and othersto the cases n = 2,3. The method is to construct a generatingfunction whose bifurcation set contains the CSS and possiblysome other redundant components. Focal sets of smooth hypersurfacesare a special case of the construction, but the CSS is an affineand not a euclidean invariant. Besides the familiar local formsof focal sets there are other local forms corresponding to boundarysingularities, and yet others which do not appear to have arisenelsewhere in a geometrical context. There are connections withFinsler geometry. This paper concentrates on the theory andthe proof of the local normal forms for the CSS. 2000 MathematicsSubject Classification 57R45, 58K40, 32S25, 58B20.     

On the Asymptotic Behaviour of a Certain Non-linear Difference Equation

Arising from the study of the convergence properties of a rationalapproximation method for determining a zero of the functionf(x)is a certain non-linear difference equation. This equation hasthe form v_n+1 = g_p–1(v_n)/g_p(v_n), Where g_p(v_n) is a polynomialin v_n whose coefficients depend on a parameter p, the orderof the zero of f The asymptotic behaviour of the differenceequation is studied and it is shown that if there is a limitorder of convergence it is always linear for multiple zeros.