4,114 research outputs found
On the volumes and affine types of trades
A -trade is a pair of disjoint collections of subsets
(blocks) of a -set such that for every , any -subset of
is included in the same number of blocks of and of . It follows
that and this common value is called the volume of . If we
restrict all the blocks to have the same size, we obtain the classical
-trades as a special case of -trades. It is known that the minimum
volume of a nonempty -trade is . Simple -trades (i.e., those
with no repeated blocks) correspond to a Boolean function of degree at most
. From the characterization of Kasami--Tokura of such functions with
small number of ones, it is known that any simple -trade of volume at most
belongs to one of two affine types, called Type\,(A) and Type\,(B)
where Type\,(A) -trades are known to exist. By considering the affine
rank, we prove that -trades of Type\,(B) do not exist. Further, we derive
the spectrum of volumes of simple trades up to , extending the
known result for volumes less than . We also give a
characterization of "small" -trades for . Finally, an algorithm to
produce -trades for specified , is given. The result of the
implementation of the algorithm for , is reported.Comment: 30 pages, final version, to appear in Electron. J. Combi
Recommended from our members
Configurations and colouring problems in block designs
A Steiner triple system of order v (STS(v)) is called x-chromatic if x is the smallest number of colours needed to avoid monochromatic blocks. Amongst our results on colour class structures we show that every STS (19) is 3- or 4-chromatic, that every 3-chromatic STS(19) has an equitable 3-colouring (meaning that the colours are as uniformly distributed as possible), and that for all admissible v > 25 there exists a 3-chromatic STS(v) which does not admit an equitable 3-colouring. We obtain a formula for the number of independent sets in an STS(v) and use it to show that an STS(21) must contain eight independent points. This leads to a simple proof that every STS(21) is 3- or 4-chromatic. Substantially extending existing tabulations, we provide an enumeration of STS trades of up to 12 blocks, and as an application we show that any pair of STS(15)s must be 3-1-isomorphic. We prove a general theorem that enables us to obtain formulae for the frequencies of occurrence of configurations in triple systems. Some of these are used in our proof that for v > 25 no STS(u) has a 3-existentially closed block intersection graph. Of specific interest in connection with a conjecture of Erdos are 6-sparse and perfect Steiner triple systems, characterized by the avoidance of specific configurations. We describe two direct constructions that produce 6-sparse STS(v)s and we give a recursive construction that preserves 6-sparseness. Also we settle an old question concerning the occurrence of perfect block transitive Steiner triple systems. Finally, we consider Steiner 5(2,4, v) designs that are built from collections of Steiner triple systems. We solve a longstanding problem by constructing such systems with v = 61 (Zoe’s design) and v = 100 (the design of the century)
New Steiner 2-designs from old ones by paramodifications
Techniques of producing new combinatorial structures from old ones are commonly called trades. The switching principle applies for a broad class of designs: it is a local transformation that modifies two columns of the incidence matrix. In this paper, we present a construction, which is a generalization of the switching transform for the class of Steiner 2-designs. We call this construction paramodification of Steiner 2-designs, since it modifies the parallelism of a subsystem. We study in more detail the paramodifications of affine planes, Steiner triple systems, and abstract unitals. Computational results show that paramodification can construct many new unitals
- …