Some Properties of Hadamard Matrices Coming from Dihedral Groups

 In [4], one of the authors introduced a method to construct Hadamard matrices of degree 8n+4 from the dihedral group of order 2n. Here we study some properties of this construction. Received: May 7, 1999 Final version received: February 28, 2000     

Grothendieck ring of quantum double of finite groups

Let kG be a group algebra, and D(kG) its quantum double. We first prove that the structure of the Grothendieck ring of D(kG) can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of G. As a special case, we then give an application to the group algebra kD _n , where k is a field of characteristic 2 and D _n is a dihedral group of order 2n.     

Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

All Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291-303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,p^m) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71-87].     

Augmentation quotients for complex representation rings of dihedral groups

Denote by D _m the dihedral group of order 2m. Let ℛ(D _m ) be its complex representation ring, and let Δ(D _m ) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δ ⁿ (D _m )/Δ ⁿ⁺¹(D _m ) for each positive integer n.     

Adjacency preserving bijective maps on triangular matrices over any division ring

Let 𝕋 _n (D) be the set of n × n upper triangular matrices over a division ring D. We characterize the adjacency preserving bijective maps in both directions on 𝕋 _n (D) (n ≥ 3). As applications, we describe the ring semi-automorphisms and the Jordan automorphisms on upper triangular matrices over a simple Artinian ring.     

Representations of 2-Groups by Polynomials

It is known that any finite p-group can be represented by polynomials. However, how to represent p-groups and how to classify p-groups up to isomorphism are interesting and open questions. In this article, we investigate the 2-groups of order 8, and represent the dihedral group D_2n, the generalized quaternion group Q_2n, and the infinite dihedral group D.2000 Mathematics Subject Classification: 20C99, 20E99     

The cocyclic Hadamard matrices of order less than 40

In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.     

Gold codes,Hadamard partitions and the security of CDMA systems

Let V = {1, 2, . . . , M} and let be a set of Hadamard matrices with the property that the magnitude of the dot product of any two rows of distinct matrices is bounded above. A Hadamard partition is any partition of the set of all rows of the matrices H _i into Hadamard matrices. Such partitions have an application to the security of quasi-synchronous code-division multiple-access radio systems when loosely synchronized (LS) codes are used as spreading codes. A new generation of LS code can be used for each information bit to be spread. For each generation, a Hadamard matrix from some partition is selected for use in the code construction. This code evolution increases security against eavesdropping and jamming. One security aspect requires that the number of Hadamard partitions be large. Thus the number of partitions is studied here. If a Kerdock code construction is used for the set of matrices, the Hadamard partition constructed is shown to be unique. It is also shown here that this is not the case if a Gold (or Gold-like) code construction is used. In this case the number of Hadamard partitions can be enumerated, and is very large.      

The rank of a completion of a dedekind domain

Let D be a Dedekind domain with quotient field K. Let C_p be the completion of the localisationD_p , of D at a nonzero prime idealp, of D. Let r_p be the rank of C_p as a D-module, ier_p , is the dimension of the K-vector space K_p , = K? _DC_p . The following results on r_p are deduced from well-known theorems: if r_p is finite for at least one prime ideal p, then D is a discrete valuation ring; and D = C_p if _p = 1. If D is a discrete valuation ring, then r_p = dimExt(K, D) + 1. A module M is extensionless if every extension of M by M splits. The D-module r_C is an estensionless indecomposable module. If r_C is infinite for every nonzero prime ideal, it is shown that an estensionless D-module of finite rank is a direct sum or certain rank one modulcs.     

On Hadamard matrices of order 2(p+1) with an automorphism of odd prime order p

In this paper, we investigate Hadamard matrices of order 2(p + 1) with an automorphism of odd prime order p. In particular, the classification of such Hadamard matrices for the cases p = 19 and 23 is given. Self‐dual codes related to such Hadamard matrices are also investigated. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 367–380, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10052     

On non-normal arc-transitive 4-valent dihedrants

Let X be a connected non-normal 4-valent arc-transitive Cayley graph on a dihedral group Dn such that X is bipartite, with the two bipartition sets being the two orbits of the cyclic subgroup within Dn. It is shown that X is isomorphic either to the lexicographic product Cn[2K1] with n 〉 4 even, or to one of the five sporadic graphs on 10, 14, 26, 28 and 30 vertices, respectively.     

Almost split sequences of the quantum double of a finite group

Let k be a field of characteristic p > 0, and G be a finite group of order divisible by p. We prove that the almost split sequences of the quantum double D(kG) can be constructed from those of group algebras, where the groups run over all centralizer subgroups of representatives of conjugate classes of G. As a special case, we give an application to the quantum double of dihedral groups.     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

Finite presentability of <Emphasis Type="Italic">SL</Emphasis><Subscript>1</Subscript>(<Emphasis Type="Italic">D</Emphasis>)

Let D be a finite dimensional division algebra over a local field of characteristic p and let SL ₁(D) denote the group of elements of reduced norm 1 in D. In this paper we prove that SL ₁(D) is finitely presented as a profinite group. This work is part of the author’s Ph.D. Thesis at Yale University.     

Automorphism Group of Representation Ring of the Weak Hopf Algebra $$\widetilde{H_8 }$$

Let H₈ be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra \(\widetilde{H_8 }\)based on H₈, then we investigate the structure of the representation ring of \(\widetilde{H_8 }\). Finally, we prove that the automorphism group of \(r\left( {\widetilde{H_8 }} \right)\)is just isomorphic to D₆, where D₆ is the dihedral group with order 12.     

A family of non class-regular symmetric transversal designs of spread type

Many classes of symmetric transversal designs have been constructed from generalized Hadamard matrices and they are necessarily class regular. In (Hiramine, Des Codes Cryptogr 56:21–33, 2010) we constructed symmetric transversal designs using spreads of \mathbbZ_p²ⁿ{\mathbb{Z}_p^{2n}} with p a prime. In this article we show that most of them admit no class regular automorphism groups. This implies that they are never obtained from generalized Hadamard matrices. As far as we know, this is the first infinite family of non class-regular symmetric transversal designs.     

The generic character table of a Sylow p-subgroup of a finite Chevalley group of type D4

Let U be a Sylow p-subgroup of the finite Chevalley group of type D₄ over the field of q elements, where q is a power of a prime p. We describe a construction of the generic character table of U.     

The structure of weighing matrices having large weights

We examine the structure of weighing matricesW(n, w), wherew=n–2,n–3,n–4, obtaining analogues of some useful results known for the casen–1. In this setting we find some natural applications for the theory ofsigned groups and orthogonal matrices with entries from signed groups, as developed in [3]. We construct some new series of Hadamard matrices from weighing matrices, including the following:W(n, n–2) implies an Hadamard matrix of order2n ifn0 mod 4 and order 4n otherwise;W(n, n–3) implies an Hadamard matrix of order 8n; in certain cases,W(n, n–4) implies an Hadamard matrix of order 16n. We explicitly derive 117 new Hadamard matrices of order 2 ^t p, p<4000, the smallest of which is of order 2³·419.Supported by an NSERC grant     

Random Walks on Dihedral Groups

In this paper, we look at the lower bounds of two specific random walks on the dihedral group. The first theorem discusses a random walk generated with equal probabilities by one rotation and one flip. We show that roughly p ² steps are necessary for the walk to become close to uniformly distributed on all of D _2p where p≥3 is an integer. Next we take a random walk on the dihedral group generated by a random k-subset of the dihedral group. The latter theorem shows that it is necessary to take roughly p ^2/(k−1) steps in the typical random walk to become close to uniformly distributed on all of D _2p . We note that there is at least one rotation and one flip in the k-subset, or the random walk generated by this subset has periodicity problems or will not generate all of D _2p .