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Variations on seven points: An introduction to the scope and methods of coding theory and finite geometries. (English) Zbl 0546.05001

The authors play the following seven variations on the well-known seven point design S(2,3;7).
1. Geometries, in particular t-designs, with the arithmetic existence conditions and a uniqueness proof for the seven point design.
2. Groups. The authors show that Aut S(2,3;7) is the simple group \(PGL(3,2)=PSL(3,2)\) of order 168 and introduce the concepts of difference set and difference family. J. Singer’s theorem [Trans. Am. Math. Soc. 43, 377-385 (1938; Zbl 0019.00502)], as well as advanced results of U. Ott [Math. Z. 144, 195-215 (1975; Zbl 0305.50009)] and R. M. Wilson [J. Number Theory 4, 17-47 (1972; Zbl 0259.05011)], are mentioned.
3. Extensions. The extension of S(2,3;7) to the unique S(3,4;8) is discussed in detail. The extension of Hadamard designs \(S_{\lambda}(2,2\lambda +1;4\lambda +3)\) to Hadamard 3-designs \(S_{\lambda}(2,2\lambda +2;4\lambda +4)\) and E. Witt’s constructions of S(5,6;12) and S(5,8;24) [Abh. Math. Sem. Hansische Univ. 12, 256-264 and 265-275 (1938; Zbl 0019.25105 and Zbl 0019.25106)] with their automorphism groups, i.e. the Mathieu groups, are sketched.
4. Some linear algebra. The authors prove Fisher’s inequality \(b\geq v\) and introduce duality. They show that any symmetric design with a point- regular abelian automorphism group is self-dual, in particular S(2,3;7).
5. Quadratic forms. The main ideas of the Bruck-Ryser-Chowla theorem are given. H. J. Ryser’s [J. Comb. Theory, Ser. A 32, 103-105 (1982; Zbl 0472.05008)] simplified proof could not yet be known to the authors.
6. Codes. The connections between design theory and coding theory are outlined, in particular by the example of the Hamming code derived from S(2,3;7).
7. Representation theory and some applications. This variation contains more information on coding theory and introductory remarks on M. Hall’s [Duke Math. J. 14, 1079-1090 (1947; Zbl 0029.22502)] multiplier theory. After a brief Coda the authors mention some relevant books and give a list of 55 references.
The purpose of the present paper is not to give new results but enjoyment for the easy reader.
Reviewer: H.Lenz

MSC:

05-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics
05B25 Combinatorial aspects of finite geometries
94B05 Linear codes (general theory)
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
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References:

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