A characterization of Grassmann and attenuated spaces as (0,α)-geometries

We introduce distance-regular (0, α)-reguli and show that they give rise to (0, α)-geometries with a distance-regular point graph. This generalises the SPG-reguli of Thas [14] and the strongly regular (α, β)-reguli of Hamilton and Mathon [9], which yield semipartial geometries and strongly regular (α, β)-geometries, respectively. We describe two infinite classes of examples, one of which is a generalisation of the well-known semipartial geometry T * n (B) arising from a Baer subspace PG(n, q) in PG(n, q 2 ).

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“…From arguments similar as in [15] (see also [2]) it follows that the geometry is uniquely defined and hence that is isomorphic to H q (m, d).…”

mentioning

confidence: 74%

Distance-Regular (0,α)-Reguli

et al. 2006

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“…The transversals of the lines in R form another regulus R • , the so-called opposite regulus, which covers the same set of (q + 1) 2 points as R. In a similar vein, we call a set of q + 1 pairwise skew lines in PG(4, q) a regulus if they span a PG(3, q) and form a regulus in their span. For q = 2 things simplify: A regulus in PG (3,2) is just a set of three pairwise skew lines, and a regulus in PG(4, 2) is a set of three pairwise skew lines spanning a solid (3-flat). Now suppose that S forms a partial line spread of size 9 in PG(4, 2).…”

Section: Preparationsmentioning

confidence: 99%

“…The lines L i have a unique transversal line L (the intersection of the three solids 7 In PG(4, 2)/L ∼ = PG(2, 2) the solids H 1 , H 2 , H 3 form the sides of a triangle. Hence there exist a unique plane E and a unique solid H 4 in PG (4,2) such that…”

Section: Preparationsmentioning

confidence: 99%

“…, v − 1}, then C is said to be a q-ary (v, M, d; k) constant-dimension subspace code. 2 The maximum size of a q-ary (v, M, d) subspace code (a q-ary (v, M, d; k) constant-dimension subspace code) is denoted by A q (v, d) (respectively, A q (v, d; k)).…”

Section: Introductionmentioning

confidence: 99%

“…In the language of Finite Geometry, Theorem 1 says that the maximum number of planes in PG (5,2) intersecting each other in at most a point is 77, with five optimal solutions up to geometric equivalence. Theorem 1 improves on the previously known inequality 77 ≤ A 2 (6,4; 3) ≤ 81, the lower bound being due to a computer construction of a binary (6, 77, 4; 3) subspace code in [17].…”

Section: Introductionmentioning

confidence: 99%

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Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4

Honold¹,

Kiermaier²,

Kurz³

2015

Topics in Finite Fields

112

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It is shown that the maximum size of a binary subspace code of packet length v = 6, minimum subspace distance d = 4, and constant dimension k = 3 is M = 77; in Finite Geometry terms, the maximum number of planes in PG(5, 2) mutually intersecting in at most a point is 77. Optimal binary (v, M, d; k) = (6, 77, 4; 3) subspace codes are classified into 5 isomorphism types, and a computer-free construction of one isomorphism type is provided. The construction uses both geometry and finite fields theory and generalizes to any q, yielding a new family of q-ary (6, q 6 + 2q 2 + 2q + 1, 4; 3) subspace codes.2000 Mathematics Subject Classification. Primary 94B05, 05B25, 51E20; Secondary 51E14, 51E22, 51E23.

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A Q-polynomial structure for the Attenuated Space poset Aq(N,M)

Terwilliger

2024

Journal of Combinatorial Theory, Series A

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A characterization of Grassmann and attenuated spaces as (0,α)-geometries

Cited by 4 publications

References 10 publications

Distance-Regular (0,α)-Reguli

Distance-Regular (0,α)-Reguli

Optimal binary subspace codes of length 6, constant dimension 3 and minimum subspace distance 4

A Q-polynomial structure for the Attenuated Space poset Aq(N,M)

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