1. On Minimal Defining Sets of Full Designs and Self-Complementary Designs, and a New Algorithm for Finding Defining Sets of t-Designs.
- Author
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Kolotoğlu, Emre and Yazıcı, Emine Şule
- Subjects
- *
ALGORITHMS , *SET theory , *ARBITRARY constants , *ALGEBRA , *MATHEMATICS - Abstract
A defining set of a t-(υ, k, λ) design is a partial design which is contained in a unique t-design with the given parameters. A minimal defining set is a defining set, none of whose proper partial designs is a defining set. This paper proposes a new and more efficient algorithm that finds all nonisomorphic minimal defining sets of a given t-design. The complete list of minimal defining sets of 2-(6, 3, 6) designs, 2-(7, 3, 4) designs, the full 2-(7, 3, 5) design, a 2-(10, 4, 4) design, 2-(10, 5, 4) designs, 2-(13, 3, 1) designs, 2-(15, 3, 1) designs, the 2-(25, 5, 1) design, 3-(8, 4, 2) designs, the 3-(12, 6, 2) design, and 3-(16, 8, 3) designs are given to illustrate the efficiency of the algorithm. Also, corrections to the literature are made for the minimal defining sets of four 2-(7, 3, 3) designs, two 2-(6, 3, 4) designs and the 2-(21, 5, 1) design. Moreover, an infinite class of minimal defining sets for 2-(υ3) designs, where υ ≥ 5, has been constructed which helped to show that the difference between the sizes of the largest and the smallest minimal defining sets of 2-(υ3) designs gets arbitrarily large as υ→∞. Some results in the literature for the smallest defining sets of t-designs have been generalized to all minimal defining sets of these designs.We have also shown that all minimal defining sets of t-(2n, n, λ) designs can be constructed from the minimal defining sets of their restrictions when t is odd and all t-(2n, n, λ) [ABSTRACT FROM AUTHOR]
- Published
- 2010
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