Nonisomorphic Systems Research Articles

On sparse countably infinite Steiner triple systems

AbstractWe give a general construction for Steiner triple systems on a countably infinite point set and show that it yields 2 nonisomorphic systems all of which are uniform and r‐sparse for all finite r⩾4. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 115–122, 2010

Vasil’ev codes of length n = 2m and doubling of Steiner systems S(n, 4, 3) of a given rank

Extended binary perfect nonlinear Vasil'ev codes of length n = 2m and Steiner systems S(n, 4, 3) of rank n-m over F 2 are studied. The generalized concatenated construction of Vasil'ev codes induces a variant of the doubling construction for Steiner systems S(n, 4, 3) of an arbitrary rank r over F 2. We prove that any Steiner system S(n = 2m, 4, 3) of rank n-m can be obtained by this doubling construction and is formed by codewords of weight 4 of these Vasil'ev codes. The length 16 is studied in detail. Orders of the full automorphism groups of all 12 nonequivalent Vasil'ev codes of length 16 are found. There are exactly 15 nonisomorphic systems S(16, 4, 3) of rank 12 over F 2, and they can be obtained from codewords of weight 4 of the extended Vasil'ev codes. Orders of the automorphism groups of all these Steiner systems are found.

Comparison of the Impurity Effects on Lattice Dynamics in K2SnCl6 between Isomorphic and Nonisomorphic Systems Near the Structural Phase Transition Temperature Revealed by Nuclear Resonance

Nuclear resonance studies of the two different types impurity doped potassium hexachloro-stannates, the isomorphic system such as (K-Rb)2SnCl6 and K2(Sn-Re)Cl6 and the nonisomorphic system K2SnCl6:Al3+ in the high temperature cubic phase revealed contrasting features with one the other characterized by static in the former and dynamic feature in the latter case respectively. The resonance spectra of the nonisomorphic system indicate additionally a sign of the local structural transition above the conventional structural phase transition temperature. This seems to be triggered by the ligand-deficient octahedral defects and can be explained in terms of the enhanced activity of the octahedral defects for the hindered rotation.

Classification of Steiner quadruple systems of order 16 and rank at most 131

A Steiner quadruple system SQS(v) of order v is a 3-design T (v, 4, 3, ?) with ? = 1. In this paper we describe all nonisomorphic systems SQS(16) that can be obtained by the generalized concatenated construction (GC-construction). These Steiner systems have rank at most 13 over $$ \mathbb{F} $$ 2. In particular, there is one system SQS(16) of rank 11 (points and planes of the a fine geometry AG(4, 2)), fifteen systems of rank 12, and 4131 systems of rank 13. All these Steiner systems are resolvable.

A Mass Formula for Steiner Triple Systems STS(2n−1) of 2-Rank 2n−n

A formula is found for the total number of distinct Steiner triple systems on 2n−1 points whose 2-rank is one higher than the possible minimum 2n−n−1. The formula can be used for deriving bounds on the number of pairwise nonisomorphic systems for large n, and for the classification of all nonisomorphic systems of small orders. It is proved that the number of nonisomorphic Steiner triple systems on 2n−1 points of 2-rank 2n−n grows exponentially.

Essentially disconnected fully-acenoids, and fully-arenoid hydrocarbons

The class of fully-arenoid hydrocarbons is defined so that it includes the well known fully-benzenoids. A fully-arenoid B has a “full” system consisting of Kekuléan benzenoids (arenes) which cover completely all the vertices of B. Important subclasses emerge when the arenes of the full system are catacondensed or, in particular, are linear acenes with a hexagons each. These fully-[ a]acenes may be essentially disconnected. A precise definition is given of fully-[ a]acenes of class C. The number of nonisomorphic systems of this category are given in terms of a general mathematical solution.

Non-isomorphic 2-perfect 6-cycle systems of order 13

Running a computer search for new, cyclic, 2-perfect 6-cycle systems of order 13 and constructing the quasigroups which arise from such systems enabled the author to establish that there are at most two such non-isomorphic systems. Then by using two-variable laws of the quasigroups it is shown that there are exactly two non-isomorphic 2-perfect 6-cycle systems of order 13 which are cyclic.

On the covering of t-sets with ( t + 1)-sets: C(9, 5, 4) and C(10, 6, 5)

A (υ, k, t) covering system is a pair ( X, B ) where X is a υ-set of points and B is a family of k-subsets, called blocks, of X such that every t-subset of X is contained in at least one block. The minimum possible number of blocks in a (υ, k, t) covering system is denoted by C(υ, k, t). It is proven that there are exactly three non-isomorphic systems giving C(9, 5, 4) = 30, and a unique system giving C(10, 6, 5) = 50.

Cyclic large sets of Steiner triple systems of order 15

We examine a class of large sets of Steiner triple systems of order 15 having an automorphism consisting of two fixed points and a 13-cycle. We exhibit all members of this class: there are 256 nonisomorphic systems. We examined these members for initial configurations which could lead to a large set of Steiner quadruple systems of order 16 and established that no large set exists having three fixed points and a 13-cycle.

Enumeration and classification of coronoid hydrocarbons: Part V. Primitive coronoids

Some topological properties of primitive coronoids are discussed. A method of generating such systems from the corona hole is described in detail. The results from computer-aided enumerations of primitive coronoids are given for h (the number of hexagons ) up to 25. The results account for 1 075 554 nonisomorphic systems. The distributions into symmetry groups are specified. The forms of all the primitive coronoids for h ⩽ 15 are depicted and supplied with K numbers ( Kekulé structure counts ). Finally the average K values and related quantities for the systems under consideration are reported.