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Mathematics Department
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Recent Publications
Trilinear alternating forms and related CMLs and GECs
of the base space. This classification seems to be a difficult problem (unlike in the bilinear case).
For n ≤ 8 there exist finitely many trivector classes under the action of the general linear group
GL(n). The methods of Galois cohomology can be used to determine the classes of nondegenerate
trivectors which split into multiple classes when going from K¯ (the algebraic closure of K ) to K.
In this paper, we are interested in the classification of trivectors of an eight dimensional vector
space over a finite field of characteristic 3, K = F3m . We obtain a 31 inequivalent trivectors,
20 of which are full rank. Having its motivation in the theory of the generalized elliptic curves
and commutative moufang loop, this research studies the case of the forms over the 3 elements
field. We use a transfer theorem providing a one-to-one correspondence between the classes of
trilinear alternating forms of rank 8 over a finite field with 3 elements F3 and the rank 9 class 2
Hall generalized elliptic curves (GECs) of 3-order 9 and commutative moufang loop (CMLs). We
derive a classification and explicit descriptions of the 31 Hall GECs whose rank and 3-order both
equal 9 and the number of order 39 -CMLs.
Citation
M. MIDOUNE Noureddine, (2023), "Trilinear alternating forms and related CMLs and GECs", [national] International Journal of Group Theory , http://ijgt.ui.ac.ir/
Some Weights of the Fq-forms Of 2-step Splitting Trivector
Citation
M. MIDOUNE Noureddine, (2019), "Some Weights of the Fq-forms Of 2-step Splitting Trivector", [international] RAMA 11 , Université de Sidi Bel Abbès
Weights of the Fq-forms of 2-step splitting trivectors of rank 8 over a finite field
These codes are conveniently described using the correspondence between non-degenerate [n, k]q
linear codes on one hand and non-degenerate [n, k] projective systems on the other hand. A nondegenerate [n, k] projective system is simply a collection of n points in projective space Pk−1satisfying the condition that no hyperplane of Pk−1
contains all the n points under consideration. Inthis paper we will determine the weight of linear codes C(3, 8) associated with Grassmann varieties G(3, 8) over an arbitrary finite field Fq. We use a formula for the weight of a codeword of C(3, 8),in terms of the cardinalities certain varieties associated with alternating trilinear forms on F8q . Form = 6 and 7, the weight spectrum of C(3, m) associated with G(3, m), have been fully determined by Kaipa K.V, Pillai H.K and Nogin Y. A classification of trivectors depends essentially on the dimension n of the base space. For n ≤ 8 there exist only finitely many trivector classes under the action of the general linear group GL(n). The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from F¯ to F.
This program is partially determined by Noui L and Midoune N and the classification of trilinearalternating forms on a vector space of dimension 8 over a finite field Fq of characteristic other than 2 and 3 was solved by Noui L and Midoune N. We describe the Fq-forms of 2-step splitting trivectors of rank 8, where char Fq 6= 3. This fact we use to determine the weight of the Fq-forms.
Citation
M. MIDOUNE Noureddine, (2019), "Weights of the Fq-forms of 2-step splitting trivectors of rank 8 over a finite field", [national] Carpathian Mathematical Publications , 10.15330/cmp.11.2.422-430
Weight of Codes Associated With the Grassmanniann
sional subspaces of an m dimensional vector space Fmq . They were studied by Nogin for general q.
These codes are conveniently described using the correspondence between non-degenerate [n, k]q
linear codes on one hand and non-degenerate [n, k] projective systems on the other hand. A non-
degenerate [n, k] projective system is simply a collection of n points in projective space Pk−1 satis-
fying the condition that no hyperplane of Pk−1 contains all the n points under consideration
Citation
M. MIDOUNE Noureddine, (2018), "Weight of Codes Associated With the Grassmanniann", [national] Workshops on Pure and Applied Mathematics (WPAM 2018) , Université de M’sila
Une classe de cohomologie affine
Citation
M. MIDOUNE Noureddine, (2015), "Une classe de cohomologie affine", [international] The 3rd International Conference On Applied Algebra (Icaa’2015) , Université de M’sila
Algèbres de Lie 2-Nilpotentes et Structures Symplectiques
We study symplectic structures on 2-step nilpotent Lie algebras and we show that they are rarely symplectic algebras. Finally, symplectic Lie algebras play a role in superconformal field theories [see S. E. Parkhomenko, Quasi-Frobenius Lie algebra constructions of N = 4 superconformal field theories, Mod. Phys. Lett. A 11 (1996) 445--461] and have been studied in connection with rational solutions of the classical Yang-Baxter equation [see A. Stolin, Rational solutions of the classical Yang-Baxter equation and quasi Frobenius Lie algebras, J. Pure Appl. Algebra 137 (1999) 285--293].
Citation
M. MIDOUNE Noureddine, (2013), "Algèbres de Lie 2-Nilpotentes et Structures Symplectiques", [national] journal of Lie theory , Copyright Heldermann Verlag 2013
Trilinear alternating forms on a vector space of dimension 8 over a finite field
Citation
M. MIDOUNE Noureddine, (2013), "Trilinear alternating forms on a vector space of dimension 8 over a finite field", [national] Linear and Multilinear Algebra , Taylor & Francis Online
K-Forms of 2-Step Splitting Trivectors
Citation
M. MIDOUNE Noureddine, (2008), "K-Forms of 2-Step Splitting Trivectors", [national] International journal of algebra 2 , Hikari