Exhaustive construction of (255, 127, 63)-cyclic difference sets

Here, (255, 127, 63)-cyclic difference sets are exhaustively constructed. There are, in total, 64 distinct (255, 127, 63)-cyclic difference sets. They include one Singer's type and three miscellaneous types. These cyclic difference sets of miscellaneous types also provide new examples of Hadamard matrixs of order 256 and two-level autocorrelation sequences of length 255.     

Two constructions of (v, (v − 1)/2, (v − 3)/2) difference families

In this article, two constructions of (v, (v ? 1)/2, (v ? 3)/2) difference families are presented. The first construction produces both cyclic and noncyclic difference families, while the second one gives only cyclic difference families. The parameters of the second construction are new. The difference families presented in this article can be used to construct Hadamard matrices. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 164–171, 2008     

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 trace representation of binary Jacobi sequences

We determine the trace function representation, or equivalently, the Fourier spectral sequences of binary Jacobi sequences of period pq, 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.     

Some New Orders of Hadamard and Skew‐Hadamard Matrices

We construct Hadamard matrices of orders and , and skew‐Hadamard matrices of orders and . As far as we know, such matrices have not been constructed previously. The constructions use the Goethals–Seidel array, suitable supplementary difference sets on a cyclic group and a new efficient matching algorithm based on hashing techniques.     

Investigation of solvable (120, 35, 10) difference sets

Abelian difference sets with parameters (120, 35, 10) were ruled out by Turyn in 1965. Turyn's techniques do not apply to nonabelian groups. We attempt to determine the existence of (120, 35, 10) difference sets in the 44 nonabelian groups of order 120. We prove that if a solvable group admits a (120, 35, 10) difference set, then it admits a quotient group isomorphic to the cyclic group of order 24 or to U₂₄ ? 〈x,y : x⁸ = y³ = 1, xyx^?1 = y^?1〉. We describe a computer search, which rules out solutions with a ?₂₄ quotient. The existence question remains undecided in the three solvable groups admitting a U₂₄ quotient. The question also remains undecided for the three nonsolvable groups of order 120. © 2004 Wiley Periodicals, Inc.     

Exponential number of inequivalent difference sets in ℤ

Kantor [ 5 ] proved an exponential lower bound on the number of pairwise inequivalent difference sets in the elementary abelian group of order 2^2s+2. Dillon [ 3 ] generalized a technique of McFarland [ 6 ] to provide a framework for determining the number of inequivalent difference sets in 2‐groups with a large elementary abelian direct factor. In this paper, we consider the opposite end of the spectrum, the rank 2 group ? , and compute an exponential lower bound on the number of pairwise inequivalent difference sets in this group. In the process, we demonstrate that Dillon difference sets in groups ? can be constructed via the recursive construction from [ 2 ] and we show that there are exponentially many pairwise inequivalent difference sets that are inequivalent to any Dillon difference set. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 249–259, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10046     

Skew‐Hadamard matrices of orders 436, 580, and 988 exist

We construct two difference families on each of the cyclic groups of order 109, 145, and 247, and use them to construct skew‐Hadamard matrices of orders 436, 580, and 988. Such difference families and matrices are constructed here for the first time. The matrices are constructed by using the Goethals‐Seidel array. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 493–498, 2008     

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     

Short k‐radius sequences, k‐difference sequences and universal cycles

An $n$‐ary $k$‐radius sequence is a finite sequence of elements taken from an alphabet of size $n$ in which any two distinct elements occur within distance $k$ of each other somewhere in the sequence. The study of constructing short $k$‐radius sequences was motivated by some problems occurring in large data transfer. Let $f_k\left(n\right)$ be the shortest length of any $n$‐ary $k$‐radius sequence. We show that the conjecture $f_k\left(n\right)=\frac{n^2}{2k}+O\left(n\right)$ by Bondy et al is true for $k\le 4$, and determine the exact values of $f_2\left(n\right)$ for new infinitely many $n$. Further, we investigate new sequences which we call $k$‐difference, as they are related to $k$‐radius sequences and seem to be interesting in themselves. Finally, we answer a question about the optimal length of packing and covering analogs of universal cycles proposed by D?bski et al.     

Some remarks on orders of projective planes,planar difference sets and multipliers

In this paper we investigate how finite group theory, number theory, together with the geometry of substructures can be used in the study of finite projective planes. Some remarks concerning the function v(x)= x ² + x + 1are presented, for example, how the geometry of a subplane affects the factorization of v(x). The rest of this paper studies abelian planar difference sets by multipliers.Partially supported by NSA grant MDA904-90-H-1013.     

On the asymptotic existence of cocyclic Hadamard matrices

Let q be an odd natural number. We prove there is a cocyclic Hadamard matrix of order ²10+tq whenever . We also show that if the binary expansion of q contains N ones, then there is a cocyclic Hadamard matrix of order ²4N−2q.     

Ruling out (160, 54, 18) difference sets in some nonabelian groups

We prove the following theorems. Theorem A. Let G be a group of order 160 satisfying one of the following conditions. (1) G has an image isomorphic to D₂₀ × Z₂ (for example, if G ≃ D₂₀ × K). (2) G has a normal 5‐Sylow subgroup and an elementary abelian 2‐Sylow subgroup. (3) G has an abelian image of exponent 2, 4, 5, or 10 and order greater than 20. Then G cannot contain a (160, 54, 18) difference set. Theorem B. Suppose G is a nonabelian group with 2‐Sylow subgroup S and 5‐Sylow subgroup T and contains a (160, 54, 18) difference set. Then we have one of three possibilities. (1) T is normal, |ϕ(S)| = 8, and one of the following is true: (a) G = S × T and S is nonabelian; (b) G has a D₁₀ image; or (c) G has a Frobenius image of order 20. (2) G has a Frobenius image of order 80. (3) G is of index 6 in A Γ L(1, 16). To prove the first case of Theorem A, we find the possible distribution of a putative difference set with the stipulated parameters among the cosets of a normal subgroup using irreducible representations of the quotient; we show that no such distribution is possible. The other two cases are due to others. In the second (due to Pott) irreducible representations of the elementary abelian quotient of order 32 give a contradiction. In the third (due to an anonymous referee), the contradiction derives from a theorem of Lander together with Dillon's “dihedral trick.” Theorem B summarizes the open nonabelian cases based on this work. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 221–231, 2000     

The existence of (p,q)-extended Rosa sequences

A (p,q)-extended Rosa sequence is a sequence of length 2n+2 containing each of the symbols 0,1,…,n exactly twice, and such that two occurrences of the integer j>0 are separated by exactly j-1 symbols. We prove that, with two exceptions, the conditions necessary for the existence of a (p,q)-extended Rosa sequence with prescribed positions of the symbols 0 are sufficient. We also extend the result to λ-fold (p,q)-extended Rosa sequences; i.e., the sequences where every pair of numbers is repeated exactly λ times.     

Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

Let p be a prime, q=p^m and F_q be the finite field with q elements. In this paper, we will consider q-ary sequences of period qⁿ-1 for q>2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period qⁿ-1 is difference-balanced and has the “cyclic” array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period qⁿ-1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q=p.     

Relative difference sets fixed by inversion (III)—Cocycle theoretical approach

In this paper, we give a characterization of a group G which contains a semiregular relative difference set R relative to a central subgroup N containing the commutator subgroup [G,G] of G such that 1∈R and rRr=R for all r∈R. In particular, these relative difference sets are fixed by inversion and inner automorphisms of the group are multipliers. We also present a construction of such relative difference sets.     

Relative difference sets fixed by inversion (ii)—character theoretical approach

In this paper, we continue our investigation of relative difference sets fixed by inversion. We exclusively focus our attention on abelian groups. New necessary conditions are obtained and a new family of such relative difference sets with forbidden subgroup Z/4Z is constructed. The methods we use are character theory of abelian groups and Galois rings over Z/4Z.     

The existence of disjoint (hooked) near‐Rosa sequences and applications

We show that the necessary conditions are sufficient for the existence of two disjoint near (hooked) Rosa sequences, with all admissible orders $n\ge 6$ and all possible defects. Further, we apply this result for the existence of new types of cyclic and simple GDDs.     

Solving the equation −uxx − ϵuyy = f(x,y, u) by an O(h4) finite difference method

The semi‐linear equation −u_xx − ϵu_yy = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h⁴) finite difference method, which has local truncation error O(h²) at the mesh points neighboring the boundary and O(h⁴) at most interior mesh points. It is proved that the finite difference method is O(h⁴) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000     

Tridiagonal pairs of shape (1, 2, 1)

Let denote a field and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V that satisfies the following conditions: (i) each of A,A^* is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^*V_iV_i-1+V_i+V_i+1 for 0id, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^* such that for 0iδ, where and ; (iv) there is no subspace W of V such that AWW, A^*WW, W≠0,W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and that for 0id the dimensions of coincide; we denote this common value by ρ_i. The sequence is called the shape of the pair. In this paper we assume the shape is (1,2,1) and obtain the following results. We describe six bases for V; one diagonalizes A, another diagonalizes A^*, and the other four underlie the split decompositions for A,A^*. We give the action of A and A^* on each basis. For each ordered pair of bases among the six, we give the transition matrix. At the end we classify the tridiagonal pairs of shape (1,2,1) in terms of a sequence of scalars called the parameter array.