Divisible designs with dual translation group

Many different divisible designs are already known. Some of them possess remarkable automorphism groups, so called dual translation groups. The existence of such an automorphism group enables us to characterize its associated divisible design as being isomorphic to a substructure of a finite affine space.      

Some constructions of divisible designs from Laguerre geometries

In the nineties, A.G. Spera introduced a construction principle for divisible designs. Using this method, we get series of divisible designs from finite Laguerre geometries. We show a close connection between some of these divisible designs and divisible designs whose construction was based on a conic in a plane of a three-dimensional projective space.     

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.     

On the twisted q-Euler numbers and polynomials associated with basic q-l-functions

One of the purposes of this paper is to construct the twisted q-Euler numbers by using p-adic invariant integral on Zp in the fermionic sense. Moreover, we consider the twisted Euler q-zeta functions and q-l-functions which interpolate the twisted q-Euler numbers and polynomials at a negative integer.     

Degenerate Sklyanin algebras and generalized twisted homogeneous coordinate rings

In this work, we introduce the point parameter ring B, a generalized twisted homogeneous coordinate ring associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of Artin–Tate–van den Bergh, namely that B is generated in degree one and thus is a factor of the corresponding degenerate Sklyanin algebra.     

Self-dual codes from orbit matrices and quotient matrices of combinatorial designs

In this paper we give constructions of self-orthogonal and self-dual codes, with respect to certain scalar products, with the help of orbit matrices of block designs and quotient matrices of symmetric (group) divisible designs (SGDDs) with the dual property. First we describe constructions from block designs and their extended orbit matrices, where the orbit matrices are induced by the action of an automorphism group of the design. Further, we give some further constructions of self-dual codes from symmetric block designs and their orbit matrices. Moreover, in a similar way as for symmetric designs, we give constructions of self-dual codes from SGDDs with the dual property and their quotient matrices.     

On coloured constant composition designs

This paper deals with a particular class of coloured designs: the incidence vectors of all blocks of these designs have the same composition, and the same is true for the incidence vectors of all points. For this reason, we call these designs constant composition designs, or CC-designs for short. We will derive necessary and sufficient conditions on the existence and conclude the presentation with a collection of examples.     

Orthogonal designs,self‐dual codes,and the Leech lattice

Symmetric designs and Hadamard matrices are used to construct binary and ternary self‐dual codes. Orthogonal designs are shown to be useful in construction of self‐dual codes over large fields. In this paper, we first introduce a new array of order 12, which is suitable for any set of four amicable circulant matrices. We apply some orthogonal designs of order 12 to construct new self‐dual codes over large finite fields, which lead us to the odd Leech lattice by Construction A. © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 184–194, 2005.     

Mutually disjoint designs and new 5‐designs derived from groups and codes

The article gives constructions of disjoint 5‐designs obtained from permutation groups and extremal self‐dual codes. Several new simple 5‐designs are found with parameters that were left open in the table of 5‐designs given in (G. B. Khosrovshahi and R. Laue, t‐Designs with t⩾3, in “Handbook of Combinatorial Designs”, 2nd edn, C. J. Colbourn and J. H. Dinitz (Editors), Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 79–101), namely, 5−(v, k, λ) designs with (v, k, λ)=(18, 8, 2m) (m=6, 9), (19, 9, 7m) (m=6, 9), (24, 9, 6m) (m=3, 4, 5), (25, 9, 30), (25, 10, 24m) (m=4, 5), (26, 10, 126), (30, 12, 440), (32, 6, 3m) (m=2, 3, 4), (33, 7, 84), and (36, 12, 45n) for 2⩽n⩽17. These results imply that a simple 5−(v, k, λ) design with (v, k)=(24, 9), (25, 9), (26, 10), (32, 6), or (33, 7) exists for all admissible values of λ. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 305–317, 2010     

Some infinite families of 2‐designs from PG (n,q)

In this paper, we establish the existence of some infinite families of 2‐designs from ‐dimensional projective geometry , which admit ‐dimensional projective special linear group as their flag‐transitive automorphism group.     

Solving isomorphism problems about 2‐designs from disjoint difference families

Recently, two new constructions of disjoint difference families in Galois rings were presented by Davis, Huczynska, and Mullen and Momihara. Both were motivated by a well‐known construction of difference families from cyclotomy in finite fields by Wilson. It is obvious that the difference families in the Galois ring and the difference families in the finite field are not equivalent. A related question, which is in general harder to answer, is whether the associated designs are isomorphic or not. In our case, this problem was raised by Davis, Huczynska, and Mullen. In this paper, we show that, in most cases, the 2‐ designs arising from the difference families in Galois rings and those arising from the difference families in finite fields are nonisomorphic by comparing their block intersection numbers.     

Projective embeddings of dual polar spaces arising from a class of alternative division rings

We discuss some recent results of us regarding a class of polar spaces which includes the nonembeddable polar spaces introduced by Tits [Tits, J., “Buildings of spherical type and finite BN-pairs,” Lecture Notes in Mathematics 386, Springer-Verlag, Berlin-New York, 1974]. These results include an elementary construction of the polar space, a construction of a polarized embedding of the corresponding dual polar space and the determination whether this projective embedding is universal and unique (as a polarized embedding).