Browsing by Author "Zebene Girma"

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    New Existence Results on Group Divisible Designs and 3-GDDs
    (Addis Ababa University, 2024-06) Zebene Girma; Dinesh G. Sarvate; Samuel Assefa
    Combinatorial design theory offers a framework for efficiently and systematically arranging objects with given criterion and studying these arrangement (combinatorial designs) and their applications is various fields. For example, combinatorial designs are used in statistical design theory, computer science, cryptography and coding theory. Two specific combinatorial designs are group divisible designs (GDDs) and t-GDDs. The aim of our study is to find new existence results on GDDs and t-GDDs when t = 3. In particular, we gave some new constructions and necessary conditions for GDDs with two groups and block size four. An extension of the concepts of group divisible designs is called t-GDD. It is obtained by combining the concepts of group divisible designs and t-design. So for example, a 3-GDD(n, 2, k, λ1, λ2) is defined by combining the concepts of GDDs and a 3-design. 3-GDDs with 2 groups and block size 4 have been studied in the literature. We obtain new results on 3-GDDs with three groups and block size five, as well as with four groups and block size five. In this work, certain required necessary conditions for these GDDs to exist are designed, several new constructions are given, and specific instances of non-existence are proved.
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    New Existence Results on Group Divisible Designs and 3-GDDs
    (Addis Ababa University, 2024-06) Zebene Girma; Dinesh G. Sarvate; Samuel Asefa (Professor)
    Combinatorial design theory offers a framework for efficiently and systematically arranging objects with given criterion and studying these arrangement (combinatorial designs) and their applications is various fields. For example, combinatorial designs are used in statistical design theory, computer science, cryptography and coding theory. Two specific combinatorial designs are group divisible designs (GDDs) and t-GDDs. The aim of our study is to find new existence results on GDDs and t-GDDs when t = 3. In particular, we gave some new constructions and necessary conditions for GDDs with two groups and block size four. An extension of the concepts of group divisible designs is called t-GDD. It is obtained by combining the concepts of group divisible designs and t-design. So for example, a 3-GDD(n, 2, k, λ1, λ2) is defined by combining the concepts of GDDs and a 3-design. 3-GDDs with 2 groups and block size 4 have been studied in the literature. We obtain new results on 3-GDDs with three groups and block size five, as well as with four groups and block size five. In this work, certain required necessary conditions for these GDDs to exist are designed, several new constructions are given, and specific instances of non-existence are proved.