Sets of even type in PG(3, 4), alias the binary (85, 24) projective geometry code

Hirschfeld, James J.W.P.; Hubaut, Xavier

doi:10.1016/0097-3165(80)90051-5

Cited by 23 publications

(23 citation statements)

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“…If H 1 and H 2 are two distinct pseudo-hyperplanes of S, then the complement H 1 ∆H 2 := P \ (H 1 ∆H 2 ) of the symmetric difference H 1 ∆H 2 of H 1 and H 2 is again a pseudohyperplane of S. The fact that H 1 ∆H 2 is again a pseudo-hyperplane is a well-known fact for hyperplanes of point-line geometries with three points per line and was an important tool in the papers of Hirschfeld & Hubaut [14] and Sherman [22] to obtain their desired classification results.…”

Section: Introductionmentioning

confidence: 99%

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Pseudo-embeddings and pseudo-hyperplanes

Bruyn¹

2013

Advances in Geometry

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We generalize some known results regarding hyperplanes and projective embeddings of point-line geometries with three points per line to geometries with an arbitrary but finite number of points on each line. In order to generalize these results, we need to introduce the new notions of pseudo-hyperplane, (universal) pseudo-embedding, pseudo-embedding rank and pseudo-generating rank. We prove several connections between these notions and address the problem of the existence of (certain) pseudo-embeddings. We apply this to several classes of point-line geometries. We also determine the pseudo-embedding rank and the pseudo-generating rank of the projective space PG(n, 4) and the affine space AG(n, 4).

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Section: Introductionmentioning

confidence: 99%

“…This is the case for geometries with three points per line, where the pseudohyperplanes are precisely the hyperplanes. In this context it is also worth mentioning the work of Hirschfeld & Hubaut [14] and Sherman [22], who obtained a classification of all pseudo-hyperplanes (also known as sets of odd type) of PG(n, 4).…”

Section: Introductionmentioning

confidence: 99%

Pseudo-embeddings and pseudo-hyperplanes

Bruyn¹

2013

Advances in Geometry

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“…[4] and [5] is based on results of Tallini Scafati who studied more general problems in her papers [11,12,13].…”

Section: The Pseudo-hyperplanes Of Ag(n 4)mentioning

confidence: 99%

The pseudo-hyperplanes and homogeneous pseudo-embeddings of AG(n, 4) and PG(n, 4)

Bruyn

2011

Des. Codes Cryptogr.

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We determine all homogeneous pseudo-embeddings of the affine space AG(n, 4) and the projective space PG(n, 4). We give a classification of all pseudo-hyperplanes of AG(n, 4). We also prove that the two homogeneous pseudo-embeddings of the generalized quadrangle Q(4, 3) are induced by the two homogeneous pseudo-embeddings of AG(4, 4) into which Q(4, 3) is fully embeddable.Keywords: homogeneous pseudo-embedding, pseudo-hyperplane MSC2000: 51E20, 05B25 Basic definitions and main resultsThe aim of this section is to state the main results of this paper and to define the basic notions which are necessary to understand these results. Throughout this section, S = (P, L, I) is a point-line geometry with the property that the number of points on each line is finite and at least three.Suppose V is a vector space over the field F 2 of order 2. A pseudo-embedding of S into the projective space Σ = PG(V ) is a mapping e from P to the point set of Σ satisfying: (1) < e(P) > Σ = Σ; (2) if L is a line of S with points x 1 , x 2 , . . . , x k , then the points e(x 1 ), e(x 2 ), . . . , e(x k−1 ) of Σ are linearly independent and e(x k ) = wherev i , i ∈ {1, 2, . . . , k − 1}, is the unique vector of V for which e(x i ) = Σ . Two pseudo-embeddings e 1 : S → Σ 1 and e 2 : S → Σ 2 of S are called isomorphic (e 1 ∼ = e 2 ) if there exists an isomorphism φ : Σ 1 → Σ 2 such that e 2 = φ • e 1 . The notion pseudoembedding was introduced in De Bruyn [1].Suppose e : S → PG(V ) is a pseudo-embedding of S and G is a group of automorphisms of S. We say that e is G-homogeneous if for every θ ∈ G, there exists a (necessarily unique) projectivity η θ of PG(V ) such that e(x θ ) = e(x) η θ for every point x of S. If G is the full automorphism group of S, then e is also called a homogeneous pseudo-embedding.Suppose e : S → Σ is a pseudo-embedding of S and α is a subspace of Σ satisfying the following two properties: 1

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“…If Q contains subsets of both types (ii) and (iii), then necessarily n = %q +1. This special case was first studied by Hirschfeld and Thas [3,4] under the stronger assumption that Q is of class (l,n,q + l): an infinite family of examples is constructed and it is proved that, for q > 4 and d > 3, these are the only possible examples; if d = 3 and q > 4, the same result remains valid [1]; if q = 4, the situation is much more complicated: an exhaustive list is obtained in PG(3,g) by Hirschfeld and Hubaut [2] and, in higher dimensions an algebraic description is given by Sherman [11]. The general problem of classifying sets of class (0, l,n,q + l) having plane sections of both types (ii) and (iii), thus with n = -\q + \, remains open.…”

Section: Background and Main Theoremsmentioning

confidence: 99%

Characterization of Finite Hermitian Semi-Quadrics

Lefevre-Percsy

1985

Journal of the London Mathematical Society

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Hermitian semi-quadrics of PG (d,q) are characterized as sets of class (0, l,n,q + l) having sufficiently large cardinality. Moreover a complete classification of sets of class (0, 1, n, q+ 1) in P G ( 3 , q) is achieved. Definitions and preliminary resultsLet P = PG {d, q) be the finite projective space of dimension d ^ 2 and order q > 2. Let be the subspace spanned by some subset X of points of P.If Q is a subset of points of P and if n is an integer such that 2 ^ n ^ q, we say that Q is a set of class (0, l , n , g + l) if every line of P meets Q in 0 , 1 , w or g + 1 points. If, furthermore, each line meets Q in at least one point, then we say more precisely that Q is of class (\,n,q+l). Sets of class (0,1, n) of PG (2, a), having more than one point, are special {k, «}-arcs. Recall that a {&,«}-arc in PG (2, q) is a set K of k points of PG (2, q) such that some line of the plane meets K in n points, but such that no line meets K in more than n points.Two distinct points a, b are adjacent if the line is contained in Q; we write a ~ b and we assume that each point is adjacent to itself. Q is degenerate if there is a point that is adjacent to all points of Q. A line of Q is & line which is contained in Q. A secanf line is a line that is not a line of Q and meets Q in more than one point. A line is exterior to Q if it does not intersect Q.It is obvious that if Q is a set of class (0,1, n, q +1) in P and if S is some subspace of P, then S n Q is a set of class (0,1, n, q +1) of S. Let us recall that degenerate sets can be described from non-degenerate ones: see [6]. Recall also that [9, Theorem III] implies the following well-known classification of sets of class (0, l,n,q + l) in a plane. THEOREM 0. IfQ is a set of class (0, l,n,q + l) spanning PG (2, q) and admitting some secant line, then Q is one of the following:(i) the union of n lines having a common point;(ii) the complement of a maximal {(q -n){q + l)+l,q -n + l}-arc;(iii) the union of a maximal {(n -2)(

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Sets of even type in PG(3, 4), alias the binary (85, 24) projective geometry code

Cited by 23 publications

References 3 publications

Pseudo-embeddings and pseudo-hyperplanes

Pseudo-embeddings and pseudo-hyperplanes

The pseudo-hyperplanes and homogeneous pseudo-embeddings of AG(n, 4) and PG(n, 4)

Characterization of Finite Hermitian Semi-Quadrics

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