Special sets of the Hermitian surface and indicator sets

Cossidente, Antonio; Marino, Giuseppe; Polverino, Olga

doi:10.1002/jcd.20148

Cited by 5 publications

(6 citation statements)

References 13 publications

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“…Fix the points R = (α q − β q+1 θ , β, θ) 2 of σ and R 1 = (−θy q+1 1 , 2ξy 1 , 2ξθ) 2 of p 1 = l y 1 with y 1 satisfying Eq. (5). Since σ corresponds to an internal point for the non-singular conic (B ⊥ ∩ Π ∩ Q − (5, q))/B of the quotient space (B ⊥ ∩ Π)/B, the line p 2 is the unique totally singular line l y , y = y 1 , intersecting the line R, R 1 .…”

Section: Construction Of the Subsets Of Type U P 1 Pmentioning

confidence: 99%

“…together with the condition Eq. (5). By plugging x = x 0 + θx 1 , α = a 0 + θa 1 , x i , a i ∈ GF(q), into (6) (note that x 1 = 0 otherwise the intersection point would coincide with B), we rewrite (6) in the equivalent form…”

Section: Construction Of the Subsets Of Type U P 1 Pmentioning

confidence: 99%

“…Theorem 3.1 in [4] gives a characterisation of special sets of CP-type in terms of the unitary form defining H(3, q 2 ). By [5,Theorem 2.1], a special set of CP-type corresponds to a pseudo-conic. This paper is on characterisations of pseudo-conics from an algebraic combinatorial point of view.…”

Section: Introductionmentioning

confidence: 99%

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Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes

Bamberg¹,

Giusy²,

Siciliano³

2021

Preprint

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A pseudo-oval of a finite projective space over a finite field of odd order q is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order (q n , q n ) and a Laguerre plane of order q n (for some n).In setting out a programme to construct new generalised quadrangles, Shult and Thas [17] asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric Q − (5, q), non-equivalent to the classical example, a so-called pseudo-conic. To date, every known pseudo-oval of lines of Q − (5, q) is projectively equivalent to a pseudoconic. Thas [18] characterised pseudo-conics as pseudo-ovals satisfying the perspective * The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM).property, and this paper is on characterisations of pseudo-conics from an algebraic combinatorial point of view. In particular, we show that pseudo-ovals in Q − (5, q) and pseudo-conics can be characterised as certain Delsarte designs of an interesting fiveclass association scheme. These association schemes are introduced and explored, and we provide a complete theory of how pseudo-ovals of lines of Q − (5, q) can be analysed from this viewpoint.

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Section: Construction Of the Subsets Of Type U P 1 Pmentioning

confidence: 99%

Section: Construction Of the Subsets Of Type U P 1 Pmentioning

confidence: 99%

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Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes

Bamberg¹,

Giusy²,

Siciliano³

2021

Preprint

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“…In what follows we recall that the same set of

q^{2} goodbreakinfix+ 1

lines of

PG (5, q)

can be seen as the classical BLT‐set of

Q^{-} (5, q)

and the unique

1

‐system of

Q^{+} (5, q)

(see , [23, Section 4.3], , , p. 1012]).…”

Section: Projective Paley Sets In Boldpg(bold5bold-italicq)mentioning

confidence: 99%

“…Consider

B

embedded in the elliptic quadric

E_{i}

scriptP

. Under the Plücker map the points of the elliptic quadric

E_{i}

correspond to the lines of a Hermitian surface

scriptH (3, q^{2})

PG (3, q^{2})

and, from , the

q^{2} goodbreakinfix+ 1

lines of

B

correspond to the points of an elliptic quadric

Q^{-} (3, q)

contained in a Baer subgeometry

scriptB

PG (3, q^{2})

. In particular

scriptB goodbreakinfix\cap scriptH (3, q^{2}) goodbreakinfix= Q^{-} (3, q)

and the points of

scriptB

are those admitting the same polar plane with respect to both the unitary polarity associated with

scriptH (3, q^{2})

and the orthogonal polarity associated with

Q^{-} (3, q)

.…”

Section: Projective Paley Sets In Boldpg(bold5bold-italicq)mentioning

confidence: 99%

Projective Paley sets

Cossidente

Marino

Pavese

2019

J of Combinatorial Designs

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A two‐character set in PG(r,q) is a set scriptX of points with the property that the intersection number with any hyperplane only takes two values. A projective Paley set of PG(2n−1,q)goodbreakinfix,0.33emq odd, is a subset scriptX of points such that every hyperplane of PG(2n−1,q) meets scriptX in either (qn+1)(qn−1−1)∕2(q−1) or (qn−1)(qn−1+1)∕2(q−1) points. A quasi‐quadric in PG(2n−1,q) is a two‐character set that has the same size and the same intersection numbers with respect to hyperplanes as a nondegenerate quadric. Here we construct projective Paley sets of PG(3,q) left invariant by a cyclic group of order q2goodbreakinfix+1 and of PG(5,q) admitting PSL(2,q2) as an automorphism group. Also infinite families of quasi‐quadrics of PG(5,q) are provided.

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