Cyclotomic integers and finite geometry

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 applicabi...

Full description

Bibliographic Details
Main Author: Bernhard, Schmidt.
Other Authors: School of Physical and Mathematical Sciences
Format: Journal Article
Language:English
Published: 2009
Subjects:
Online Access:https://hdl.handle.net/10356/92085
http://hdl.handle.net/10220/6029
_version_ 1826115232182829056
author Bernhard, Schmidt.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Bernhard, Schmidt.
author_sort Bernhard, Schmidt.
collection NTU
description 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.
first_indexed 2024-10-01T03:51:51Z
format Journal Article
id ntu-10356/92085
institution Nanyang Technological University
language English
last_indexed 2024-10-01T03:51:51Z
publishDate 2009
record_format dspace
spelling ntu-10356/920852023-02-28T19:28:15Z Cyclotomic integers and finite geometry Bernhard, Schmidt. School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics 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. Published version 2009-08-11T01:33:27Z 2019-12-06T18:17:06Z 2009-08-11T01:33:27Z 2019-12-06T18:17:06Z 1999 1999 Journal Article Schmidt, B. (1999). Cyclotomic Integers and Finite Geometry. Journal of American Mathematical Society, 12(4), 929-952. 0894-0347 https://hdl.handle.net/10356/92085 http://hdl.handle.net/10220/6029 10.1090/S0894-0347-99-00298-2 en Journal of american mathematical society 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. 37 p. application/pdf
spellingShingle DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics
Bernhard, Schmidt.
Cyclotomic integers and finite geometry
title Cyclotomic integers and finite geometry
title_full Cyclotomic integers and finite geometry
title_fullStr Cyclotomic integers and finite geometry
title_full_unstemmed Cyclotomic integers and finite geometry
title_short Cyclotomic integers and finite geometry
title_sort cyclotomic integers and finite geometry
topic DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics
url https://hdl.handle.net/10356/92085
http://hdl.handle.net/10220/6029
work_keys_str_mv AT bernhardschmidt cyclotomicintegersandfinitegeometry