Finite semifields and nonsingular tensors

Lavrauw, Michel

doi:10.1007/s10623-012-9710-6

Cited by 23 publications

(23 citation statements)

References 62 publications

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“…These semifields are classified and the only examples are the so-called twisted fields. Following the notation of [6], the multiplication in such a semifield S corresponds to a tensor T S in F containing T S and generated by its fundamental tensors. In order to make the search feasible, we used the computer-algebra system GAP [3] and its package FinInG [4], dedicated to finite incidence geometry.…”

Section: Resultsmentioning

confidence: 99%

“…The tensor rank is also of interest in the theory of semifields, due to the relation between bilinear maps and three-fold tensors. In [6] the tensor rank of a semifield is defined as the rank of the three-fold tensor which corresponds to the multiplication in the semifield, and it is shown that the tensor rank of a semifield is an invariant of the isotopism class of the semifield. (Here a semifield is a division algebra with an identity element for multiplication, but not necessarily associative.)…”

Section: Introductionmentioning

confidence: 99%

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On the rank of 3×3×3-tensors

Lavrauw

Pavan

Zanella

2013

Linear and Multilinear Algebra

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Abstract. Let U , V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U ⊗ V ⊗ W is the minimum dimension of a subspace of U ⊗ V ⊗ W containing τ and spanned by fundamental tensors, i.e. tensors of the form u⊗v⊗w for some u in U , v in V and w in W . We prove that if U , V and W have dimension three, then the rank of a tensor in U ⊗ V ⊗ W is at most six, and such a bound cannot be improved in general. Moreover we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U ⊗ V ⊗ W when the dimensions of U , V and W are higher.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the rank of 3×3×3-tensors

Lavrauw

Pavan

Zanella

2013

Linear and Multilinear Algebra

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“…Both the second and the third contraction spaces of the canonical form A for o 14 give the plane PG(A 2 ), defined by A 2 = e 1 ⊗ e 1 , (e 1 + e 2 ) ⊗ e 2 , e 2 ⊗ e 3 , which is not contained in an S 2,2 and intersects S 2,3 in exactly three points.…”

Section: Finite Fieldsmentioning

confidence: 99%

“…However, tensors over finite fields are also of great interest, due for example to their connections to complexity theory, and finite semifields [14].…”

Section: Introductionmentioning

confidence: 99%

Classification of subspaces in $${{\mathbb{F}}^2\otimes {\mathbb{F}}^3}$$ F 2 ⊗ F 3 and orbits in $${{\mathbb{F}}^2\otimes{\mathbb{F}}^3 \otimes {\mathbb{F}}^r}$$ F 2 ⊗ F 3 ⊗ F r

Lavrauw

Sheekey

2016

J. Geom.

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“…Semifields play a key role in the study of projective planes, as they correspond to translation planes which are also dual translation planes, and they are also related to various other structures from finite geometry and the theory of finite fields. For more background, history, and known classifications, see for example [10], [8], [15] and [14]. Most of the remainder of this section is standard knowledge in the subject, but we reproduce it here for clarity of exposition and to establish notation and conventions.…”

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confidence: 99%

On BEL-configurations and finite semifields

Lavrauw

Sheekey

2014

Des. Codes Cryptogr.

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The BEL-construction for finite semifields was introduced in [3]; a geometric method for constructing semifield spreads, using so-called BELconfigurations in V (rn, q). In this paper we investigate this construction in greater detail, and determine an explicit multiplication for the semifield associated with a BEL-configuration in V (rn, q), extending the results from [3], where this was obtained only for r = n. Given a BELconfiguration with associated semifields spread S, we also show how to find a BEL-configuration corresponding to the dual spread S d . Furthermore, we study the effect of polarities in V (rn, q) on BEL-configurations, leading to a characterisation of BEL-configurations associated to symplectic semifields.We give precise conditions for when two BEL-configurations in V (n 2 , q) define isotopic semifields. We define operations which preserve the BEL property, and show how non-isotopic semifields can be equivalent under this operation. We also define an extension of the "'switching" operation on BEL-configurations in V (2n, q) introduced in [3], which, together with the transpose operation, leads to a group of order 8 acting on BELconfigurations.

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Finite semifields and nonsingular tensors

Cited by 23 publications

References 62 publications

On the rank of 3×3×3-tensors

On the rank of 3×3×3-tensors

Classification of subspaces in $${{\mathbb{F}}^2\otimes {\mathbb{F}}^3}$$ F 2 ⊗ F 3 and orbits in $${{\mathbb{F}}^2\otimes{\mathbb{F}}^3 \otimes {\mathbb{F}}^r}$$ F 2 ⊗ F 3 ⊗ F r

On BEL-configurations and finite semifields

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