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Cyclotomic integers and finite geometry.

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Cyclotomic integers and finite geometry.

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dc.contributor.author Bernhard, Schmidt.
dc.date.accessioned 2009-08-11T01:33:27Z
dc.date.available 2009-08-11T01:33:27Z
dc.date.copyright 1999
dc.date.issued 2009-08-11T01:33:27Z
dc.identifier.citation Schmidt, B. (1999). Cyclotomic Integers and Finite Geometry. Journal of American Mathematical Society, 12(4), 929-952.
dc.identifier.issn 0894-0347
dc.identifier.uri http://hdl.handle.net/10220/6029
dc.description.abstract We obtain an upper bound for the absolute value of cyclotomic integers which has strong implications on several combinatorial structures including (relative) difference sets, quasiregular projective planes, planar functions, and group invariant weighing matrices. Our results are of broader applicability than all previously known nonexistence theorems for these combinatorial objects. We will show that the exponent of an abelian group G containing a (v,k,ג,n)-difference set cannot exceed ((2^(s-1).F(v,n))/n)^0.5where is the number of odd prime divisors of v and F(v,n) is a number-theoretic parameter whose order of magnitude usually is the squarefree part of . One of the consequences is that for any finite set P of primes there is a constant C such that exp(G) ≤ C|G|^0.5for any abelian group G containing a Hadamard difference set whose order is a product of powers of primes in P. Furthermore, we are able to verify Ryser's conjecture for most parameter series of known difference sets. This includes a striking progress towards the circulant Hadamard matrix conjecture. A computer search shows that there is no Barker sequence of length l with 13< l <4x10^12. Finally, we obtain new necessary conditions for the existence of quasiregular projective planes and group invariant weighing matrices including asymptotic exponent bounds for cases which previously had been completely intractable.
dc.format.extent 37 p.
dc.language.iso en
dc.relation.ispartofseries Journal of american mathematical society
dc.rights Journal of American Mathematical Society © copyright 1999 American Mathematical Society. The journal's website is located at http://www.ams.org/jams/1999-12-04/S0894-0347-99-00298-2/S0894-0347-99-00298-2.pdf.
dc.subject DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics.
dc.title Cyclotomic integers and finite geometry.
dc.type Journal Article
dc.contributor.school School of Physical and Mathematical Sciences
dc.identifier.doi http://dx.doi.org/10.1090/S0894-0347-99-00298-2
dc.description.version Published version

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