ABSTRACT Let be a non‐trivial 2‐ design with , admitting a flag‐transitive automorphism group which is also properly innately transitive on points. Then is of almost simple quotient type. As a corollary, we obtain that is innately transitive on points if and only if it is primitive when . And we proved that there exists a unique 2‐ design for with a automorphism group acting transitively on flags and properly innately transitively on points, that is, the unique symmetric 2‐ design with and its plinth as in [14, Proposition 1.5].
ABSTRACT A symmetric ‐design is said to be ‐pyramidal, with , under the action of a group if acts as an automorphism group fixing points and acting sharply transitively on the remaining ones. We show that, necessarily, . In particular, for a Hadamard design with parameters , for some , it follows that . Recently, working on the complement design, the family of Hadamard ‐designs admitting an abelian ‐pyramidal automorphism group has been completely determined in the case . In this paper, we generalize that result by showing that coincides with the family of Hadamard ‐designs admitting a ‐pyramidal automorphism group, without assuming either that the group is abelian or that is a power of 2.
A mixed Steiner triple system is a 3-GDD which is viewed as a code with minimum Hamming distance 3. These codes are the minimum weight codewords of a 1-perfect code over a mixed alphabet, when the related codes exist, and provide the connection between 3-GDDs and coding theory. We prove that a 3-GDD of type , where , with minimum distance 3 exists for every and such that , or , or , and . These designs are of the shortest possible length (smallest number of elements) for given and . Other constructions for such triple systems are also presented.
A design is said to be super-simple if the intersection of any two blocks contains at most two elements. In the statistical design of experiments, super-simple designs yield samples in which the maximum pairwise block intersection is minimized. Moreover, such designs play a valuable role in the construction of codes, including optical orthogonal codes and superimposed codes, other combinatorial designs such as group divisible designs and pairwise balanced designs, and can be used to determine some exact Zarankiewicz numbers. In this paper, we investigate the existence of a super-simple -BIBD and show that such a design exists if and only if and .
ABSTRACT A design is said to be super‐simple if any two of its blocks have at most two common points. A design with index is said to be ‐ decomposable , if its blocks can be partitioned into nonempty collections , such that each with the point set forms a design with index . In this paper, it is proved that there exists a ‐DSS ‐GDD (decomposable super‐simple group divisible design) of type if and only if and , and except possibly for .
