New bounds for partial spreads of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="sans-serif">H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> and partial ovoids of the Ree–Tits octagon

Ihringer, Ferdinand; Sin, Peter; Xiang, Qing

doi:10.1016/j.jcta.2017.08.003

Cited by 6 publications

(13 citation statements)

References 12 publications

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“…Theorem 2.1 implies that every Ree-Tits octagon admits nontrivial line-domestic collineations. We note that it is erroneously stated in [11] that these octagons do not admit line-domestic collineations. The error appears to be as follows.…”

Section: Corollary 22 Every Irreducible Moufang Spherical Building Di...mentioning

confidence: 72%

“…In the case of Ree-Tits octagons, Corollary 2.2 corrects a misunderstanding from [11] (see Remark 2.3), and Corollary 3 answers a question asked to us by Barbara Baumeister.…”

Section: Theorem 1 An Automorphism Of a Split Spherical Building Of E...mentioning

confidence: 74%

“…If a thick generalised octagon admits a line-domestic colllineation θ then by [14, Theorem 2.8 and Proposition 4.1] the fixed element structure of θ is either a large full suboctagon, a distance 4-ovoid, or a ball of radius 4 in the incidence graph centred at a point. For finite Ree-Tits octagons we proved in [14,Proposition 4.4] that large full suboctagons do not exist, and in [11] it is shown that distance 4ovoids do not exist. Thus any line-domestic collineation of a finite Ree-Tits octagon necessarily fixes a ball of radius 4 centred at a point (and is thus a central collineation, the example given by Theorem 2.1).…”

Section: Corollary 22 Every Irreducible Moufang Spherical Building Di...mentioning

confidence: 99%

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Automorphisms and opposition in spherical buildings of exceptional type, I

Parkinson¹,

Maldeghem²

2020

Preprint

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To each automorphism of a spherical building there is naturally associated an opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber then the automorphism is called domestic. In this paper we give the complete classification of domestic automorphisms of split spherical buildings of types E 6 , F 4 , and G 2 . Moreover, for all split spherical buildings of exceptional type we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most 2 distinguished orbits encircled. Our results provide unexpected characterisations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

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Section: Corollary 22 Every Irreducible Moufang Spherical Building Di...mentioning

confidence: 72%

“…In the case of Ree-Tits octagons, Corollary 2.2 corrects a misunderstanding from [11] (see Remark 2.3), and Corollary 3 answers a question asked to us by Barbara Baumeister.…”

Section: Theorem 1 An Automorphism Of a Split Spherical Building Of E...mentioning

confidence: 74%

Section: Corollary 22 Every Irreducible Moufang Spherical Building Di...mentioning

confidence: 99%

See 1 more Smart Citation

Automorphisms and opposition in spherical buildings of exceptional type, I

Parkinson¹,

Maldeghem²

2020

Preprint

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“…This bound is slightly better than the corresponding bound |Y | ≤ q 2n−1 − q n + q n−1 of Theorem 2. Some improved bounds for n-codes in X(n, q) in the case that q is not a prime can be obtained from [13]. Our final result of this section gives the inner distribution of a d-code, provided that it is also an (n − d)-design.…”

Section: Combinatorial Properties Of Subsets Of X(n Q)mentioning

confidence: 99%

Hermitian rank distance codes

Schmidt

2017

Des. Codes Cryptogr.

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Abstract. Let X = X(n, q) be the set of n × n Hermitian matrices over F q 2 . It is well known that X gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study d-codes in this scheme, namely subsets Y of X with the property that, for all distinct A, B ∈ Y , the rank of A − B is at least d. We prove bounds on the size of a d-code and show that, under certain conditions, the inner distribution of a d-code is determined by its parameters. Except if n and d are both even and 4 ≤ d ≤ n−2, constructions of d-codes are given, which are optimal among the d-codes that are subgroups of (X, +). This work complements results previously obtained for several other types of matrices over finite fields.

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“…If n is odd, an upper bound for the largest partial spreads of H(2n − 1, q 2 ) is q n + 1 [29] and there are examples of partial spreads of that size [18]. If n is even the situation is less clear: upper bounds can be found in [15], as for lower bounds there is a partial spread of H(2n − 1, q 2 ) of size (3q 2 − q)/2 + 1 for n = 2, q > 13, [1, p. 32] and of size q n + 2 for n ≥ 4 [11].…”

Section: Introductionmentioning

confidence: 99%

On symmetric and Hermitian rank distance codes

Cossidente¹,

Marino²,

Pavese³

2020

Preprint

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Let M denote the set S n,q of n × n symmetric matrices with entries in GF(q) or the set H n,q 2 of n × n Hermitian matrices whose elements are in GF(q 2 ). Then M equipped with the rank distance d r is a metric space. We investigate d-codes in (M, d r ) and construct d-codes whose sizes are larger than the corresponding additive bounds. In the Hermitian case, we show the existence of an n-code of M, n even and n/2 odd, of size 3q n − q n/2 /2, and of a 2-code of size q 6 + q(q − 1)(q 4 + q 2 + 1)/2, for n = 3. In the symmetric case, if n is odd or if n and q are both even, we provide better upper bound on the size of a 2-code. In the case when n = 3 and q > 2, a 2-code of size q 4 + q 3 + 1 is exhibited. This provides the first infinite family of 2-codes of symmetric matrices whose size is larger than the largest possible additive 2-code and an answer to a question posed in [25, Section 7], see also [23, p. 176].

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New bounds for partial spreads of H(2d−1,q2) and partial ovoids of the Ree–Tits octagon

Cited by 6 publications

References 12 publications

Automorphisms and opposition in spherical buildings of exceptional type, I

Automorphisms and opposition in spherical buildings of exceptional type, I

Hermitian rank distance codes

On symmetric and Hermitian rank distance codes

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