Small maximal partial $t$-spreads

Govaerts, Patrick

doi:10.36045/bbms/1133793347

Cited by 9 publications

(10 citation statements)

References 2 publications

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“…Lemma 5 (Govaerts [13]) Let n and t > 1 be integers such that n ≥ 2t. Then there exist (see page 610 in [13] for a construction) maximal partial t-spreads of V (n + t − 1, q) of size σ q (n, t). Consequently, τ q (n + t − 1, t) ≤ σ q (n, t).…”

Section: Application To Maximal Partial T-spreadsmentioning

confidence: 99%

“…In particular, we will use Theorem 1 in Section 1. We first introduce the relevant definitions and a useful Lemma due to Govaerts [13]. A set of points B, i.e., 1-spaces of V , is called a blocking set with respect to the t-spaces of V if W ∩ B = {0} for any t-space W in V .…”

Section: Application To Maximal Partial T-spreadsmentioning

confidence: 99%

“…Lemma 6 (Govaerts [13]) Let n and t > 1 be integers such that n ≥ 2t. If S is a minimum size maximal partial t-spread of V (n, q), then W ∈S W contains a trivial blocking set.…”

Section: Application To Maximal Partial T-spreadsmentioning

confidence: 99%

“…After some preliminary results in Section 2, we will prove our theorem in Section 3 and Section 4. Finally, in Section 5, we combine our result on σ q (n, t) with a construction of P. Govaerts [13] to show that the minimum size of a maximal partial t-spread in V (n+t−1, q) is σ q (n, t) for any integer n ≥ 2t.…”

Section: Introductionmentioning

confidence: 97%

See 3 more Smart Citations

Extremal sizes of subspace partitions

Heden

Lehmann

Nastase

et al. 2011

Des. Codes Cryptogr.

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A subspace partition of V = V (n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of . The size of is the number of its subspaces. Let σ q (n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρ q (n, t) denote the maximum size of a subspace partition of V in which the smallest subspace has dimension t. In this article, we determine the values of σ q (n, t) and ρ q (n, t) for all positive integers n and t. Furthermore, we prove that if n ≥ 2t, then the minimum size of a maximal partial t-spread in V (n + t − 1, q) is σ q (n, t).

show abstract

Section: Application To Maximal Partial T-spreadsmentioning

confidence: 99%

Section: Application To Maximal Partial T-spreadsmentioning

confidence: 99%

“…Lemma 6 (Govaerts [13]) Let n and t > 1 be integers such that n ≥ 2t. If S is a minimum size maximal partial t-spread of V (n, q), then W ∈S W contains a trivial blocking set.…”

Section: Application To Maximal Partial T-spreadsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Extremal sizes of subspace partitions

Heden

Lehmann

Nastase

et al. 2011

Des. Codes Cryptogr.

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show abstract

“…Theorem 5 [Govaerts (2005)]. Let P be a projective space PG(p − 1, 2), with p = kt + r for k ≥ 1, 0 < r < t < p, and let S be a partial (t − 1)-spread of P with |S| = 2 r 2 kt −1 2 t −1 − s, where s is known as the deficiency.…”

mentioning

confidence: 99%

Existence and construction of randomization defining contrast subspaces for regular factorial designs

Ranjan¹,

Bingham²,

Dean³

2009

Ann. Statist.

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Regular factorial designs with randomization restrictions are widely used in practice. This paper provides a unified approach to the construction of such designs using randomization defining contrast subspaces for the representation of randomization restrictions. We use finite projective geometry to determine the existence of designs with the required structure and develop a systematic approach for their construction. An attractive feature is that commonly used factorial designs with randomization restrictions are special cases of this general representation. Issues related to the use of these designs for particular factorial experiments are also addressed.

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A non‐existence result on Cameron–Liebler line classes

Beule

Hallez

Storme

2007

J of Combinatorial Designs

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Cameron-Liebler line classes are sets of lines in PG(3, q) that contain a fixed number x of lines of every spread. Cameron and Liebler classified Cameron-Liebler line classes for x ∈ {0, 1, 2, q 2 − 1, q 2 , q 2 + 1} and conjectured that no others exist. This conjecture was disproven by Drudge for q = 3 [8] and his counterexample was generalised to a counterexample for any odd q by Bruen and Drudge [4]. A counterexample for q even was found by Govaerts and Penttila [9]. Non-existence results on Cameron-Liebler line classes were found for different values of x. In this paper, we improve the non-existence results on Cameron-Liebler line classes of Govaerts and Storme [11], for q not a prime. We prove the non-existence of Cameron-Liebler line classes for 3 ≤ x < q 2 .

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Small maximal partial $t$-spreads

Cited by 9 publications

References 2 publications

Extremal sizes of subspace partitions

Extremal sizes of subspace partitions

Existence and construction of randomization defining contrast subspaces for regular factorial designs

A non‐existence result on Cameron–Liebler line classes

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