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By Peter J. Cameron

ISBN-10: 0511662106

ISBN-13: 9780511662102

ISBN-10: 0521211603

ISBN-13: 9780521211604

Those notes current an research of a just like Euclid's parallel axiom for subsets of finite units. The heritage fabric to the speculation of parallelisms is brought and the writer then describes the hyperlinks this concept has with different issues from the full diversity of combinatorial thought and permutation teams. those contain community flows, ideal codes, Latin squares, block designs and multiply-transitive permutation teams, and lengthy and specified appendices are supplied to function introductions to those numerous topics. the various effects are released for the 1st time.

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The distance between two vectors is defined to be the number of coordinate places where they differ, that is, the number of errors required to transform one into the other. If, for any reason, we can be reasonably sure that not more than e errors have occurred, we use a code with the property that any vector lies at distance at most e from at most one codeword. Then, if our confidence is justified, the receiver can decode the received word (that is, decide uniquely which codeword was transmitted).

4 gives v ? e2 + 4e + 2 (provided v > 2e + 1). This is the result (Theorem 2A. 3) needed in the proof of the Perfect Code Theorem, since the existence of a perfect e-error-correcting code of length m implies that of a Steiner system S(e + 1, 2e + 1, m). Indeed, we know that this system is extendable; for the extension S(e + 2, 2e + 2, m + 1), Theorems 3A. 4 and 3A. 5 both give the result. Examples of Steiner systems Projective geometries. Let V be a vector space of rank d + 1 over GF(q). ) Let the point set 1.

This result is referred to as the 'divisibility conditions'. If P E P, the contraction (or derivation) Sp of S at p is the system with point set P - {p), and as blocks all sets of the form B - 1p), where B is a block of S containing p. If t ? 1, then Sp is a Steiner system S(t - 1, k - 1, v - 1). Note that different contractions of S may not be isomorphic. This process can be continued: if t ? i, then any 1th contraction of S is a Steiner system S(t - i, k - i, v - i). An extension of S is a Steiner system which has a contraction isomorphic to S.

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Parallelisms of Complete Designs by Peter J. Cameron

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