Hadamard Difference Sets Research Articles

Cyclotomic constructions of skew Hadamard difference sets

We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields by using union of cyclotomic classes of F q of order N = 2 p 1 m , where p 1 is a prime and m a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.

Using rational idempotents to show Turyn’s bound is sharp

Davis, Dillon, and Jedwab all showed the existence of difference sets in groups $${C_{2^{r+2}}\times C_{2^{r}}}$$ . Turyn's bound had previously shown that abelian 2-groups with higher exponents could not admit difference sets. We give a new construction technique that utilizes character values, rational idempotents, and tiling structures to produce Hadamard difference sets in the group $${C_{2^{r+2}}\times C_{2^{r}}}$$ to replicate the result.

Some REALLY beautiful Hadamard matrices

While all Hadamard matrices are beautiful in the eyes of the mathematician, those which arise as the ±1-incidence matrix of a design with a regular group of automorphisms are especially appealing. This paper reviews some of the history of the development of Hadamard difference sets, illustrating some aspects with representative matrices whose beauty makes them appropriate decorations as we celebrate the birthday and the work of our friend and colleague Warwick de Launey.

Paley partial difference sets in groups of order [formula omitted] and [formula omitted] for any odd [formula omitted

A partial difference set with parameters ( v , v − 1 2 , v − 5 4 , v − 1 4 ) is said to be of Paley type. In this paper, we give a recursive theorem that for all odd n > 1 constructs Paley partial difference sets in certain groups of order n 4 and 9 n 4 . We are also able to construct Paley–Hadamard difference sets of the Stanton–Sprott family in groups of order n 4 ( n 4 ± 2 ) when n 4 ± 2 is a prime power and 9 n 4 ( 9 n 4 ± 2 ) when 9 n 4 ± 2 is a prime power. Many of these are new parameters for such difference sets, and also give new Hadamard designs and matrices.

ARRAY ANTENNAS OF SIZE 8x8 BASED ON HADAMARD DIFFERENCE SETS

The problem of synthesizing the optimized equi-amplitude array antenna (AA) on the 8 8 × grid based on a Hadamard difference set is considered. By using newly found sets of this type on the 8 8 × grid the 28and 36-element AAs having a low sidelobe level are obtained. A numerical experiment showed that by a small alteration of the structure of such a set, further reduction of the AA sidelobe radiation is possible.

Non-abelian skew Hadamard difference sets fixed by a prescribed automorphism

Let p be a prime larger than 3 and congruent to 3 modulo 4, and let G be the non-abelian group of order p 3 and exponent p. We study the structure of a putative difference set with parameters ( p 3 , p 3 − 1 2 , p 3 − 3 4 ) in G which is fixed by a certain element of order p in Aut ( G ) . We then give a construction of skew Hadamard difference set in the group G for each prime p > 3 that is congruent to 3 modulo 4. This is the first infinite family of non-abelian skew Hadamard difference sets. Finally, we show that the symmetric designs derived from these new difference sets are not isomorphic to the Paley designs.

Strongly regular graphs with parameters [formula omitted] exist for all [formula omitted

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4 m 4 for every positive integer m . If m > 1 , such a Hadamard matrix is equivalent to a strongly regular graph with parameters ( 4 m 4 , 2 m 4 + m 2 , m 4 + m 2 , m 4 + m 2 ) . Strongly regular graphs with these parameters have been called max energy graphs, because they have maximal energy (as defined by Gutman) among all graphs on 4 m 4 vertices. For odd m ≥ 3 the strongly regular graphs seem to be new.

A Hybrid Approach Based on PSO and Hadamard Difference Sets for the Synthesis of Square Thinned Arrays

A hybrid approach for the synthesis of planar thinned antenna arrays is presented. The proposed solution exploits and combines the most attractive features of a particle swarm algorithm and those of a combinatorial method based on the noncyclic difference sets of Hadamard type. Numerical experiments validate the proposed solution, showing improvements with respect to previous results.

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.

Paley type partial difference sets in non p-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 $${\mathbb {Z}_3^2 \times \mathbb {Z}_p^{4t}}$$ 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 $${\mathbb {Z}_3^2 \times \mathbb {Z}_p^{4t} \times {\it EA} (9p^{4t} \pm 2)}$$ when $${9p^{4t} \pm 2}$$ is a prime power. These are new parameters for such difference sets.

Strongly Regular Graphs with Parameters (4m^4, 2m^4 + m^2, m^4 + m^2, m^4 + m2) Exist for All m>1

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m^4 for every positive integer m. If m is greater than 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m^4, 2m^4 + m^2, m^4 + m^2, m^4 + m^2). Strongly regular graphs with these parameters have been called max energy graphs, because they have maximal energy (as defined by Gutman) among all graphs on 4m^4 vertices. For odd m >= 3, the strongly regular graphs seem to be new.

Factoring $(16, 6, 2)$ Hadamard Difference Sets

We describe a "factoring" method which constructs all twenty-seven Hadamard $(16,6,2)$ difference sets. The method involves identifying perfect ternary arrays of energy 4 (PTA(4)) in homomorphic images of a group $G$, studying the image of difference sets under such homomorphisms and using the preimages of the PTA(4)s to find the "factors" of difference sets in $G$. This "factoring" technique generalizes to other parameters, offering a general mechanism for creating Hadamard difference sets.

Some results on skew Hadamard difference sets

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v 1 and v 2 with |v 1 ? v 2| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte's theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 35 or 37, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 39. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ? 311.

A trace representation of binary Jacobi sequences

We determine the trace function representation, or equivalently, the Fourier spectral sequences of binary Jacobi sequences of period p q , where p and q are two distinct odd primes. This includes the twin-prime sequences of period p ( p + 2 ) whenever both p and p + 2 are primes, corresponding to cyclic Hadamard difference sets.

Square Array Antennas Based on Hadamard Difference Sets

A regular method for synthesizing a planar aperiodic thinned array antenna (AA) with a low peak sidelobe level is suggested. It is based on using specific combinatorial constructions - noncyclic difference sets (DSs). The method uses the fact that when the elements of an equiamplitude AA are arranged according to a DS law, its power pattern takes constant value in the net of uniformly located space points in the sidelobe region, and this value is less than , where is the element number. In distinction to the method using cyclic DSs (see Leeper, 1999) which enables one to build planar AAs only on rectangular grids with co-prime sidelengths, the represented method omits such a constraint. The most important class of the noncyclic 2-D DSs is represented by the sets of Hadamard type (sets). Based on such sets, rectangular and square aperiodic roughly half-filled AAs can be built. Here, the numerical results obtained for the square AAs, with the element number in the array up to 300, are presented.

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.

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 Rene 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.

Skew Hadamard difference sets from the Ree–Tits slice symplectic spreads in [formula omitted

Using a class of permutation polynomials of F 3 2 h + 1 obtained from the Ree–Tits slice symplectic spreads in PG ( 3 , 3 2 h + 1 ) , we construct a family of skew Hadamard difference sets in the additive group of F 3 2 h + 1 . With the help of a computer, we show that these skew Hadamard difference sets are new when h = 2 and h = 3 . We conjecture that they are always new when h > 3 . Furthermore, we present a variation of the classical construction of the twin prime power difference sets, and show that inequivalent skew Hadamard difference sets lead to inequivalent difference sets with twin prime power parameters.

Symmetric Bush-type Hadamard matrices of order $4m^4$ exist for all odd $m$

Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order 4m 4 for all odd integers m.

Automorphisms and Equivalence of Bent Functions and of Difference Sets in Elementary Abelian 2-Groups

ABSTRACT The problem of computing the automorphism groups of an elementary Abelian Hadamard difference set or equivalently of a bent function seems to have attracted not much interest so far. We describe some series of such sets and compute their automorphism group. For some of these sets the construction is based on the nonvanishing of the degree 1-cohomology of certain Chevalley groups in characteristic two. We also classify bent functions f such that Aut(f) together with the translations from the underlying vector space induce a rank 3 group of automorphisms of the associated symmetric design. Finally, we discuss computational aspects associated with such questions.