On a geometrical method of construction of maximal t-linearly independent sets

Hamada, Noboru; Tamari, Fumikazu

doi:10.1016/0097-3165(78)90027-4

Cited by 47 publications

(11 citation statements)

References 4 publications

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“…This problem has been solved for PG (2,8) and PG(2, 9) by Hill and Ward in [9]. In the same paper, they describe five families of indecomposable (x(q + 1), x)-minihypers in the planes of square order.…”

Section: Introductionmentioning

confidence: 96%

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A study of (x(q + 1), x; 2, q)-minihypers

Landjev

Storme

2009

Des. Codes Cryptogr.

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Abstract. In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ≤ q 2 − q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q 2 − q − x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward [9], giving further results on these minihypers.

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Section: Introductionmentioning

confidence: 96%

“…The reason is that such minihypers give rise to Griesmer codes via the well-known construction of Hamada [5,8]. If F is an (x(q + 1), x; 2, q)-minihyper and s is the maximal multiplicity of a point, then sχ P − F is an (s(q 2 + q + 1) − x(q + 1), s(q + 1) − x; 2, q)-arc.…”

Section: Indecomposable Minihypersmentioning

confidence: 99%

A study of (x(q + 1), x; 2, q)-minihypers

Landjev

Storme

2009

Des. Codes Cryptogr.

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“…Definition 1.1 (Hamada and Tamari [12]). An ff ; m; N; qg-minihyper is a pair ðF; wÞ, where F is a subset of the point set of PGðN; qÞ and w is a weight function w : PGðN; qÞ !…”

Section: Introductionmentioning

confidence: 99%

A classification result on weighted {δ (p³ + 1), δ; 3, p³}‐minihypers

Ferret

Storme

2004

J of Combinatorial Designs

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“…By π r , we always denote an r-dimensional subspace of PG(n, q) and by v r+1 := q r+1 −1 q−1 , we denote the number of points of an r-dimensional projective space. Definition 1.1 (Hamada and Tamari [21,22]) An {f, m; n, q}-minihyper is a pair (F, w), where F is a subset of the point set of PG(n, q) and w is a weight function w : PG(n, q) → N : P → w(P ), satisfying 1. w(P ) > 0 ⇔ P ∈ F ,…”

Section: Introductionmentioning

confidence: 99%

Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

Beule

Govaerts

Hallez

et al. 2008

Des. Codes Cryptogr.

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Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers [8,17,18]. In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5, 3), contained in a non-singular elliptic quadric Q − (5, 3), to 56-caps contained in this elliptic quadric Q − (5, 3).

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On a geometrical method of construction of maximal t-linearly independent sets

Cited by 47 publications

References 4 publications

A study of (x(q + 1), x; 2, q)-minihypers

A study of (x(q + 1), x; 2, q)-minihypers

A classification result on weighted {δ (p³ + 1), δ; 3, p³}‐minihypers

Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

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On a geometrical method of construction of maximal t-linearly independent sets

Cited by 47 publications

References 4 publications

A study of (x(q + 1), x; 2, q)-minihypers

A study of (x(q + 1), x; 2, q)-minihypers

A classification result on weighted {δ (p3 + 1), δ; 3, p3}‐minihypers

Tight sets, weighted m-covers, weighted m-ovoids, and minihypers

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A classification result on weighted {δ (p³ + 1), δ; 3, p³}‐minihypers