Special subsets of difference sets with particular emphasis on skew Hadamard difference sets

This article introduces a new approach to studying difference sets via their additive properties. We introduce the concept of special subsets, which are interesting combinatorial objects in their own right, but also provide a mechanism for measuring additive regularity. Skew Hadamard difference sets are given special attention, and the structure of their special subsets leads to several results on multipliers, including a categorisation of the full multiplier group of an abelian skew Hadamard difference set. We also count the number of ways to write elements as a product of any number of elements of a skew Hadamard difference set.      

A family of skew Hadamard difference sets

In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley-Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new perfect nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these perfect nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley-Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new perfect nonlinear functions has applications in cryptography, coding theory, and combinatorics.     

Pseudo-Paley graphs and skew Hadamard difference sets from presemifields

Let (K, + ,*) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, *) is a skew Hadamard difference set or a Paley type partial difference set in (K, +) according as q is congruent to 3 modulo 4 or q is congruent to 1 modulo 4. Applying this result to the Coulter–Matthews presemifield and the Ding–Yuan variation of it, we recover a recent construction of skew Hadamard difference sets by Ding and Yuan [7]. On the other hand, applying this result to the known presemifields with commutative multiplication and having order q congruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute the p-ranks of these pseudo-Paley graphs when q = 3⁴, 3⁶, 3⁸, 3¹⁰, 5⁴, and 7⁴. The p-rank results indicate that these graphs seem to be new. Along the way, we also disprove a conjecture of René Peeters [17, p. 47] which says that the Paley graphs of nonprime order are uniquely determined by their parameters and the minimality of their relevant p-ranks. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

Paley type partial difference sets in non <Emphasis Type="Italic">p</Emphasis>-groups

By modifying a construction for Hadamard (Menon) difference sets we construct two infinite families of negative Latin square type partial difference sets in groups of the form where p is any odd prime. One of these families has the well-known Paley parameters, which had previously only been constructed in p-groups. This provides new constructions of Hadamard matrices and implies the existence of many new strongly regular graphs including some that are conference graphs. As a corollary, we are able to construct Paley–Hadamard difference sets of the Stanton-Sprott family in groups of the form when is a prime power. These are new parameters for such difference sets.      

Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles

Additive Hadamard cocycles are a natural generalization of presemifields. In this paper, we study divisible designs and semi-regular relative difference sets obtained from additive Hadamard cocycles. We show that the designs obtained from additive Hadamard cocycles are flag transitive. We introduce a new product construction of Hadamard cocycles. We also study additive Hadamard cocycles whose divisible designs admit a polarity in which all points are absolute. Our main results include generalizations of a theorem of Albert and a theorem of Hiramine from presemifields to additive Hadamard cocycles. At the end, we generalize Maiorana-McFarland?s construction of bent functions to additive Hadamard cocycles.     

On the automorphisms of Paley's type II Hadamard matrix

In this paper we determine the automorphism group of Paley's type II Hadamard matrix.     

Tightening Turyn’s bound for Hadamard difference sets

This work examines the existence of (4q ²,2q ²−q,q ²−q) difference sets, for q=p ^f , where p is a prime and f is a positive integer. Suppose that G is a group of order 4q ² which has a normal subgroup K of order q such that G/K ≅ C _q ×C ₂×C ₂, where C _q ,C ₂ are the cyclic groups of order q and 2 respectively. Under the assumption that p is greater than or equal to 5, this work shows that G does not admit (4q ²,2q ²−q,q ²−q) difference sets.     

Paley type partial difference sets in abelian groups

Partial difference sets with parameters $\left(v,k,\lambda ,\mu \right)=\left(v,\left(v?1\right)/2,\left(v?5\right)/4,\left(v?1\right)/4\right)$ are called Paley type partial difference sets. In this note, we prove that if there exists a Paley type partial difference set in an abelian group of order v, where v is not a prime power, then $v=n^4$ or $9n^4$, $n>1$ an odd integer. In 2010, Polhill constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using nonzero squares of a finite field, we completely answer the following question: “For which odd positive integers $v>1$, can we find a Paley type partial difference set in an abelian group of order $v$?”     

On GMW Designs and Cyclic Hadamard Designs

We generalise results of Jackson concerning cyclic Hadamard designs admitting SL(2,2ⁿ) as a point transitive automorphism group. The generalisation concerns the designs of Gordon, Mills and Welch and we characterise these as designs admitting GM(m,qⁿ) acting in a certain way. We also generalise a construction given by Maschietti, using hyperovals, of cyclic Hadamard designs, and characterise these amongst the designs of Gordon, Mills and Welch.     

A group extensions approach to relative difference sets

A new approach to (normal) relative difference sets (RDSs) is presented and applied to give a new method for recursively constructing infinite families of semiregular RDSs. Our main result (Theorem 7.1) shows that any metabelian semiregular RDS gives rise to an infinite family of metabelian semiregular RDSs. The new method is applied to identify several new infinite families of non‐abelian semiregular RDSs, and new methods for constructing generalized Hadamard matrices are given. The techniques employed are derived from the general theory of group extensions. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 279–298, 2004.     

Linking systems of difference sets

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2‐groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in non‐2‐groups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2‐groups. We give a new construction for linking systems of difference sets in 2‐groups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2‐groups, new linking systems in other 2‐groups for which no system was previously known, and the first known examples in nonabelian groups.     

Twisted product of cocycles and factorization of semi‐regular relative difference sets

In this article, we introduce what we call twisted Kronecker products of cocycles of finite groups and show that the twisted Kronecker product of two cocycles is a Hadamard cocycle if and only if the two cocycles themselves are Hadamard cocycles. This enables us to generalize some known results concerning products and factorizations of central semi‐regular relative difference sets. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 431–441, 2008     

New difference sets in nonabelian groups of order 100

In two groups of order 100 new difference sets are constructed. The existence of a difference set in one of them has not been known. The correspondence between a (100, 45, 20) symmetric design having regular automorphism group and a difference set with the same parameters has been used for the construction. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 424–434, 2001     

Paley type group schemes and planar Dembowski–Ostrom polynomials

In this paper we give some necessary and sufficient conditions for Dembowski–Ostrom polynomials to be planar. These conditions give a simple explanation of the Coulter–Matthews and Ding–Yin commutative semifields and enable us to obtain permutation polynomials from some of the Zha–Kyureghyan–Wang commutative semifields. We then give a generalization of Feng’s construction of Paley type group schemes in extra-special p-groups of exponent p and construct a family of Paley type group schemes in what we call the flag groups of finite fields. We also determine the strong multiplier groups of these group schemes. In the last section of this paper, we give a straightforward generalization of the twin prime power construction of difference sets to a construction of Hadamard designs from twin Paley type association schemes.     

Cocyclic Generalised Hadamard Matrices and Central Relative Difference Sets

Cocyclic matrices have the form where G is a finite group, C is a finite abelian group and : G × G C is a (two-dimensional) cocycle; that is,