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Two-character sets as subsets of parabolic quadrics

Bart De Bruyn (UGent)
(2016) ARS COMBINATORIA. 127. p.125-132
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Abstract
A two-character set is a set of points of a finite projective space that has two intersection numbers with respect to hyperplanes. Two-character sets are related to strongly regular graphs and two-weight codes. In the literature, there are plenty of constructions for (non-trivial) two-character sets by considering suitable subsets of quadrics and Hermitian varieties. Such constructions exist for the quadrics Q(+)(2n-1, q) subset of PG(2n - 1, q), Q(-)(2n + 1, q) subset of PG(2n + 1, q) and the Hermitian varieties H(2n - 1, q(2)) subset of PG(2n - 1, q(2)), H(2n, q(2)) subset of PG(2n, q2). In this note we show that every two-character set of PG(2n, q) that is contained in a given nonsingular parabolic quadric Q(2n, q) subset of PG(2n, q) is a subspace of PG(2n, q). This offers some explanation for the absence of the parabolic quadrics in the above mentioned constructions.
Keywords
Two-character set, parabolic quadric, STRONGLY REGULAR GRAPHS, POLAR SPACES, M-SYSTEMS, 2-WEIGHT CODES, TIGHT SETS, M-OVOIDS, GEOMETRY, CAMERON, LIEBLER, Q)

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Please use this url to cite or link to this publication:

Chicago
De Bruyn, Bart. 2016. “Two-character Sets as Subsets of Parabolic Quadrics.” Ars Combinatoria 127: 125–132.
APA
De Bruyn, B. (2016). Two-character sets as subsets of parabolic quadrics. ARS COMBINATORIA, 127, 125–132.
Vancouver
1.
De Bruyn B. Two-character sets as subsets of parabolic quadrics. ARS COMBINATORIA. 2016;127:125–32.
MLA
De Bruyn, Bart. “Two-character Sets as Subsets of Parabolic Quadrics.” ARS COMBINATORIA 127 (2016): 125–132. Print.
@article{8510640,
  abstract     = {A two-character set is a set of points of a finite projective space that has two intersection numbers with respect to hyperplanes. Two-character sets are related to strongly regular graphs and two-weight codes. In the literature, there are plenty of constructions for (non-trivial) two-character sets by considering suitable subsets of quadrics and Hermitian varieties. Such constructions exist for the quadrics Q(+)(2n-1, q) subset of PG(2n - 1, q), Q(-)(2n + 1, q) subset of PG(2n + 1, q) and the Hermitian varieties H(2n - 1, q(2)) subset of PG(2n - 1, q(2)), H(2n, q(2)) subset of PG(2n, q2). In this note we show that every two-character set of PG(2n, q) that is contained in a given nonsingular parabolic quadric Q(2n, q) subset of PG(2n, q) is a subspace of PG(2n, q). This offers some explanation for the absence of the parabolic quadrics in the above mentioned constructions.},
  author       = {De Bruyn, Bart},
  issn         = {0381-7032},
  journal      = {ARS COMBINATORIA},
  language     = {eng},
  pages        = {125--132},
  title        = {Two-character sets as subsets of parabolic quadrics},
  volume       = {127},
  year         = {2016},
}

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