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.     

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.      

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.     

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.     

Strongly regular Cayley graphs,skew Hadamard difference sets,and rationality of relative Gauss sums

In this paper, we give constructions of strongly regular Cayley graphs and skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields. Our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White (2002) [23] and several subfield examples into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.     

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 $v$?”     

Construction of relative difference sets and Hadamard groups

‘There exist normal \((2m,2,2m,m)\) relative difference sets and thus Hadamard groups of order \(4m\) for all \(m\) of the form $$\begin{aligned} m= x2^{a+t+u+w+\delta -\epsilon +1}6^b 9^c 10^d 22^e 26^f \prod _{i=1}^s p_i^{4a_i} \prod _{i=1}^t q_i^2 \prod _{i=1}^u \left( (r_i+1)/2)r_i^{v_i}\right) \prod _{i=1}^w s_i \end{aligned}$$ under the following conditions: \(a,b,c,d,e,f,s,t,u,w\) are nonnegative integers, \(a_1,\ldots ,a_r\) and \(v_1,\ldots ,v_u\) are positive integers, \(p_1,\ldots ,p_s\) are odd primes, \(q_1,\ldots ,q_t\) and \(r_1,\ldots ,r_u\) are prime powers with \(q_i\equiv 1\ (\mathrm{mod}\ 4)\) and \(r_i\equiv 1\ (\mathrm{mod}\ 4)\) for all \(i, s_1,\ldots ,s_w\) are integers with \(1\le s_i \le 33\) or \(s_i\in \{39,43\}\) for all \(i, x\) is a positive integer such that \(2x-1\) or \(4x-1\) is a prime power. Moreover, \(\delta =1\) if \(x>1\) and \(c+s>0, \delta =0\) otherwise, \(\epsilon =1\) if \(x=1, c+s=0\) , and \(t+u+w>0, \epsilon =0\) otherwise. We also obtain some necessary conditions for the existence of \((2m,2,2m,m)\) relative difference sets in partial semidirect products of \(\mathbb{Z }_4\) with abelian groups, and provide a table cases for which \(m\le 100\) and the existence of such relative difference sets is open.     

On abelian difference sets with multiplier — 1

Archiv der Mathematik -     

A two-to-one map and abelian affine difference sets

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. Set = H \ {1, ω}, where H = G/N and . Using D we define a two-to-one map g from to N. The map g satisfies g(σ ^m ) = g(σ) ^m and g(σ) = g(σ ⁻¹) for any multiplier m of D and any element σ ∈ . As applications, we present some results which give a restriction on the possible order n and the group theoretic structure of G/N.      

New necessary conditions on the existence of abelian difference sets

In this paper we develop a new method to obtain identities in a group algebraGF(p)G if an abelian difference set of ordern0 (modp) exists inG. We give an explicit formula ifp ² orp ³ is the exactp-power dividingn. This generalizes the approach of Wilbrink, Arasu and the author. The proof presented here uses some knowledge about field extensions of thep-adic numbers.     

McFarland's conjecture on abelian difference sets with multiplier −1

This paper is a continuation of the work by R.L. McFarland and S.L. Ma on abelian difference sets with –1 as a multiplier. More nonexistence results are obtained as a consequence of a theorem on the existence of sub-difference sets. In particular, nonexistence is shown for the two cases left undecided by McFarland and Ma.     

On difference sets in high exponent 2-groups

We investigate the existence of difference sets in particular 2-groups. Being aware of the famous necessary conditions derived from Turyn’s and Ma’s theorems, we develop a new method to cover necessary conditions for the existence of (2^2d+2,2^2d+1?2 ^d ,2^2d ?2 ^d ) difference sets, for some large classes of 2-groups. If a 2-group G possesses a normal cyclic subgroup 〈x〉 of order greater than 2 ^d+3+p , where the outer elements act on the cyclic subgroup similarly as in the dihedral, semidihedral, quaternion or modular groups and 2 ^p describes the size of G′∩〈x〉 or C _G (x)′∩〈x〉, then there is no difference set in such a group. Technically, we use a simple fact on how sums of 2 ⁿ -roots of unity can be annulated and use it to characterize properties of norm invariance (prescribed norm). This approach gives necessary conditions when a linear combination of 2 ⁿ -roots of unity remains unchanged under homomorphism actions in the sense of the norm.