By Peter J. Cameron

ISBN-10: 0511662106

ISBN-13: 9780511662102

ISBN-10: 0521211603

ISBN-13: 9780521211604

Those notes current an research of a situation just like Euclid's parallel axiom for subsets of finite units. The heritage fabric to the idea of parallelisms is brought and the writer then describes the hyperlinks this thought has with different subject matters from the complete variety of combinatorial conception and permutation teams. those comprise community flows, ideal codes, Latin squares, block designs and multiply-transitive permutation teams, and lengthy and unique appendices are supplied to function introductions to those a variety of matters. some of the effects are released for the 1st time.

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**Extra info for Parallelisms of Complete Designs**

**Sample text**

The reader can verify that this is a representation; it is enough to show that the matrices M. satisfy the same recursion as the polynomials 0i(A). ) It follows that Mj=0j(M1), 0:5 jsm, and if N. is the matrix representing Ej(0), then j(M1), 0

We are given some kind of geometrical 'system', and are also given a particularly pleasant system S0 of this kind which has no proper subsystems except for trivial ones; we want to find all those systems S with the property that all their 'minimally generated' non-trivial subsystems are isomorphic to So. Theorem 2. 1 answers this question for parallelX is ms of (t) in the case where S0 is the parallelism with n = 2t. (For other theorems of this type, see the Veblen-Young axiomatization of projective geometries [VY], and also Buekenhout [6], Hall [14], Young [37], and Gleason's theorem in Dembowski [D], p.

Theorem 3A. 15. S(4, 7, 23) is unique (up to isomorphism). Proof. If S' is such a system and Sp = S', we can identify S with the system in the last theorem in such a way that the points p correspond. // The perfect 3-error-correcting linear binary code of length 23 is unique (up to isomorphism). // Theorem 3A. 16.

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