207,500 research outputs found
Nontrivial t-Designs over Finite Fields Exist for All t
A - design over \F_q is a collection of -dimensional
subspaces of \F_q^n, called blocks, such that each -dimensional subspace
of \F_q^n is contained in exactly blocks. Such -designs over
\F_q are the -analogs of conventional combinatorial designs. Nontrivial
- designs over \F_q are currently known to exist only for
. Herein, we prove that simple (meaning, without repeated blocks)
nontrivial - designs over \F_q exist for all and ,
provided that and is sufficiently large. This may be regarded as
a -analog of the celebrated Teirlinck theorem for combinatorial designs
Conformal Designs based on Vertex Operator Algebras
We introduce the notion of a conformal design based on a vertex operator
algebra. This notation is a natural analog of the notion of block designs or
spherical designs when the elements of the design are based on self-orthogonal
binary codes or integral lattices, respectively. It is shown that the subspaces
of fixed degree of an extremal self-dual vertex operator algebra form conformal
11-, 7-, or 3-designs, generalizing similar results of Assmus-Mattson and
Venkov for extremal doubly-even codes and extremal even lattices. Other
examples are coming from group actions on vertex operator algebras, the case
studied first by Matsuo. The classification of conformal 6- and 8-designs is
investigated. Again, our results are analogous to similar results for codes and
lattices.Comment: 35 pages with 1 table, LaTe
Conway groupoids, regular two-graphs and supersimple designs
A design is said to be supersimple
if distinct lines intersect in at most two points. From such a design, one can
construct a certain subset of Sym called a "Conway groupoid". The
construction generalizes Conway's construction of the groupoid . It
turns out that several infinite families of groupoids arise in this way, some
associated with 3-transposition groups, which have two additional properties.
Firstly the set of collinear point-triples forms a regular two-graph, and
secondly the symmetric difference of two intersecting lines is again a line. In
this paper, we show each of these properties corresponds to a group-theoretic
property on the groupoid and we classify the Conway groupoids and the
supersimple designs for which both of these two additional properties hold.Comment: 17 page
The Intersection problem for 2-(v; 5; 1) directed block designs
The intersection problem for a pair of 2-(v, 3, 1) directed designs and 2-(v,
4, 1) directed designs is solved by Fu in 1983 and by Mahmoodian and Soltankhah
in 1996, respectively. In this paper we determine the intersection problem for
2-(v, 5, 1) directed designs.Comment: 17 pages. To appear in Discrete Mat
Hadamard matrices modulo 5
In this paper we introduce modular symmetric designs and use them to study
the existence of Hadamard matrices modulo 5. We prove that there exist
5-modular Hadamard matrices of order n if and only if n != 3, 7 (mod 10) or n
!= 6, 11. In particular, this solves the 5-modular version of the Hadamard
conjecture.Comment: 7 pages, submitted to JC
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