A group extensions approach to relative difference sets

A new approach to (normal) relative difference sets (RDSs) is presented and applied to give a new method for recursively constructing infinite families of semiregular RDSs. Our main result (Theorem 7.1) shows that any metabelian semiregular RDS gives rise to an infinite family of metabelian semiregular RDSs. The new method is applied to identify several new infinite families of non‐abelian semiregular RDSs, and new methods for constructing generalized Hadamard matrices are given. The techniques employed are derived from the general theory of group extensions. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 279–298, 2004.     

Equivalence classes of central semiregular relative difference sets

A semiregular relative difference set (RDS) in a finite group E which avoids a central subgroup C is equivalent to a cocycle which satisfies an additional condition, called orthogonality. However the basic equivalence relation, cohomology, on cocycles, does not preserve orthogonality, leading to the perception that orthogonality is essentially a combinatorial property. We show this perception is false by discovering a natural atomic structure within cohomology classes, which discriminates between orthogonal and non‐orthogonal cocycles. This atomic structure is determined by an action we term the shift action of the group G = E/C on cocycles, which defines a stronger equivalence relation on cocycles than cohomology. We prove that for each triple (C, E, G), the set of equivalence classes of semiregular RDS in E relative to C is in one to one correspondence with the set of shift‐orbits of the (Aut(C) × Aut(G))‐orbits of orthogonal cocycles. This determines a new algorithm for detecting and classifying central semiregular RDS. We demonstrate it, and propose a 7‐parameter classification scheme for equivalence classes of central semiregular relative difference sets. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 330–346, 2000     

A group extensions approach to affine relative difference sets of even order

It is shown that a group extensions approach to central relative (k+1,k-1,k,1)-difference sets of even order leads naturally to the notion of an “affine” planar map; a notion analogous to the well-known planar map corresponding to a splitting relative (m,m,m,1)-difference set. Basic properties of affine planar maps are derived and applied to give some new results regarding abelian relative (k+1,k-1,k,1)-difference sets of even order and to give new proofs, in the even order case, for some known results. The paper concludes with computational non-existence results for 10,000<k?100,000.     

Reversible relative difference sets

We investigate nontrivial (m, n, k, )-relative difference sets fixed by the inverse. Examples and necessary conditions on the existence of relative difference sets of this type are studied.     

An extension of building sets and relative difference sets

Building sets are a successful tool for constructing semi‐regular divisible difference sets and, in particular, semi‐regular relative difference sets. In this paper, we present an extension theorem for building sets under simple conditions. Some of the semi‐regular relative difference sets obtained using the extension theorem are new in the sense that their ambient groups have smaller ranks than previously known. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 50–57, 2000     

On central difference sets in certain non-abelian 2-groups

In this note, we define the class of finite groups of Suzuki type, which are non-abelian groups of exponent 4 and class 2 with special properties. A group G of Suzuki type with |G|=2^2s always possesses a non-trivial difference set. We show that if s is odd, G possesses a central difference set, whereas if s is even, G has no non-trivial central difference set.     

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 a relation between a cyclic relative difference set associated with the quadratic extensions of a finite field and the Szekeres difference sets

Letq 3 (mod 4) be a prime power and put . We consider a cyclic relative difference set with parametersq ²–1,q, 1,q–1 associated with the quadratic extension GF(q²)/GF((q). The even part and the odd part of the cyclic relative difference set taken modulon are supplementary difference sets. Moreover it turns out that their complementary subsets are identical with the Szekeres difference sets. This result clarifies the true nature of the Szekeres difference sets. We prove these results by using the theory of the relative Gauss sums.     

Semi-regular relative difference sets with large forbidden subgroups

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters (m,n,m,m/n) in groups of non-prime-power orders. Let p be an odd prime. We prove that there does not exist a (2p,p,2p,2) relative difference set in any group of order 2p², and an abelian (4p,p,4p,4) relative difference set can only exist in the group . On the other hand, we construct a family of non-abelian relative difference sets with parameters (4q,q,4q,4), where q is an odd prime power greater than 9 and . When q=p is a prime, p>9, and , the (4p,p,4p,4) non-abelian relative difference sets constructed here are genuinely non-abelian in the sense that there does not exist an abelian relative difference set with the same parameters.     

Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles

Additive Hadamard cocycles are a natural generalization of presemifields. In this paper, we study divisible designs and semi-regular relative difference sets obtained from additive Hadamard cocycles. We show that the designs obtained from additive Hadamard cocycles are flag transitive. We introduce a new product construction of Hadamard cocycles. We also study additive Hadamard cocycles whose divisible designs admit a polarity in which all points are absolute. Our main results include generalizations of a theorem of Albert and a theorem of Hiramine from presemifields to additive Hadamard cocycles. At the end, we generalize Maiorana-McFarland?s construction of bent functions to additive Hadamard cocycles.     

Optimal difference systems of sets and difference sets

Difference systems of sets (DSSs) are combinatorial structures that are generalizations of cyclic difference sets and arise in connection with code synchronization. In this paper, we give a recursive construction of DSSs with smaller redundancy from partition-type DSSs and difference sets. As applications, we obtain some new infinite classes of optimal DSSs from the known difference sets and almost difference sets.     

Renormalization and central extensions

The stochastic limit of quantum theory [1] motivated a new approach to the renormalization program. Subsequent investigations brought to light unexpected connections with conformal field theory and some subtle relationships between renormalization and central extensions. In the present paper we review the path that has lead to these connections at the light of some recent results.     

Two applications of relative difference sets: Difference triangles and negaperiodic autocorrelation functions

The well-known difference sets have various connections with sequences and their correlation properties. It is the purpose of this note to give two more applications of the (not so well known) relative difference sets: we use them to construct difference triangles (based on an idea of A. Ling) and we show that a certain nonexistence result for semiregular relative difference sets implies the nonexistence of negaperiodic autocorrelation sequences (answering a question of Parker [Even length binary sequence families with low negaperiodic autocorrelation, in: Applied Algebra, Algebraic Algorithms and Error-correcting Codes, Melbourne, 2001, Lecture Notes in Computer Science, vol. 2227, Springer, Berlin, 2001, pp. 200-209.]).     

Recursive constructions for difference matrices and relative difference families

We present a new recursive construction for difference matrices whose application allows us to improve some results by D. Jungnickel. For instance, we prove that for any Abelian p-group G of type (n₁, n₂, …, n_t) there exists a (G, p^e, 1) difference matrix with e = Also, we prove that for any group G there exists a (G, p, 1) difference matrix where p is the smallest prime dividing |G|. Difference matrices are then used for constructing, recursively, relative difference families. We revisit some constructions by M. J. Colbourn, C. J. Colbourn, D. Jungnickel, K. T. Phelps, and R. M. Wilson. Combining them we get, in particular, the existence of a multiplier (G, k, λ)-DF for any Abelian group G of nonsquare-free order, whenever there exists a (p, k, λ)-DF for each prime p dividing |G|. Then we focus our attention on a recent construction by M. Jimbo. We improve this construction and prove, as a corollary, the existence of a (G, k, λ)-DF for any group G under the same conditions as above. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 165–182, 1998     

Affine difference sets and related factor sets

In this article we study abelian affine difference sets in connection with the related group extensions and give some results on their orders.