Tightening Turyn’s bound for Hadamard difference sets
نویسندگان
چکیده
This work examines the existence of (4q2,2q2 − q, q2 − q) difference sets, for q = p , where p is a prime and f is a positive integer. Suppose that G is a group of order 4q2 which has a normal subgroup K of order q such that G/K ∼= Cq × C2 × C2, where Cq,C2 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 (4q2,2q2 − q, q2 − q) difference sets.
منابع مشابه
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...
متن کاملInequivalence of Skew Hadamard Difference Sets and Triple Intersection Numbers Modulo a Prime
Recently, Feng and Xiang [10] found a new construction of skew Hadamard difference sets in elementary abelian groups. In this paper, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple intersection numbers modulo a prime, and discuss inequivalence between Feng-Xiang skew Hadamard difference sets and the Paley difference sets. As a consequence, we show th...
متن کاملFactoring (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 gener...
متن کاملDifference sets and doubly transitive actions on Hadamard matrices
Non-affine groups acting doubly transitively on a Hadamard matrix have been classified by Ito. Implicit in this work is a list of Hadamard matrices with non-affine doubly transitive automorphism group. We give this list explicitly, in the process settling an old research problem of Ito and Leon. We then use our classification to show that the only cocyclic Hadamard matrices developed form a dif...
متن کاملSkew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3, 32h+1)
Using a class of permutation polynomials of F32h+1 obtained from the Ree–Tits slice symplectic spreads in PG(3,32h+1), we construct a family of skew Hadamard difference sets in the additive group of F32h+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 varia...
متن کامل