Linear spaces with a line-transitive point-imprimitive automorphism group and Fang-Li parameter gcd(k,r) at most eight

In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial linear spaces that admit a line-transitive, point-imprimitive group action, for a given value of gcd(k,r), where k is the line size and r is the number of lines on a point. The aim of this paper is to make that result effective. We obtain a classification of all linear spaces with this property having gcd(k,r) at most 8. To achieve this we collect together existing theory, and prove additional theoretical restrictions of both a combinatorial and group theoretic nature. These are organised into a series of algorithms that, for gcd(k,r) up to a given maximum value, return a list of candidate parameter values and candidate groups. We examine in detail each of the possibilities returned by these algorithms for gcd(k,r) at most 8, and complete the classification in this case.Comment: 47 pages Version 1 had bbl file omitted. Apologie

On classifying finite edge colored graphs with two transitive automorphism groups

This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. This result generalizes the classification of doubly transitive balanced incomplete block designs with λ=1 and doubly transitive one-factorizations of complete graphs. It also provides a classification of all doubly transitive symmetric association schemes

Contributions to design of experiments

When using a fractional factorial design, some runs may be more difficult to implement than others. There may then be a need to use a fractional factorial design that does not contain any undesirable runs. As this should not go at the expense of the information that such a design can provide for the factorial effects of interest, it is important to develop techniques that provide alternative designs with the same information matrix for the effects of interest as a given design. This problem will be addressed for 2-level fractional factorial designs;One of the methods that will be considered is based on a relationship between this problem and the problem of trade-off in block designs. By exploring this relationship, the extensive results on t-trades can be used for construction of the desired factorial designs. This provides also additional incentive for the continued development of the theory of trade-off. We will also present two programs to generate fractional factorial designs that are information-equivalent to a given design;Based on a relationship between fractional factorial designs and balanced incomplete block designs, the ideas in the aforementioned programs will be used for a program to generate balanced incomplete block designs with various supports and support sizes, avoiding possible undesirable blocks. Finally, these ideas will also be used for two programs to generate [pi]PS sampling designs that satisfy requirements on the second-order inclusion probabilities, if any, and that avoid any undesirable samples in their support