Browsing by Author "Mohamed Anouar RAKDI"

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    Classification des trivecteurs de rang 8 sur un corps fini et applications
    (Université de M'sila, 2021) Mohamed Anouar RAKDI
    Soit V un espace vectoriel de dimension finie n sur un corps commutatifK: La classification des trivecteurs est l’étude de l’action du groupe linéaire GL(V) sur l’espace vectoriel ^3V; cette classification joue un rôle important pour résoudre certains problèmes dans la théorie des codes, CCEGs et BMC. Nous étudions les classes des trivecteurs de rang 8 sur un corps fini et nous donnons les cardinaux des groupes d’automorphismes et les orbites des trivecteurs. Nous déterminons les poids des codes des trivecteurs sur un espace de dimension 8 et quelques spectres des codes
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    WEIGHTS OF THE Fq-FORMS OF 2-STEP SPLITTING TRIVECTORS OF RANK 8 OVER A FINITE FIELD
    (Université de M'sila, 2021) Mohamed Anouar RAKDI
    Grassmann codes are linear codes associated with the Grassmann variety G(`;m) of `-dimensional 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 satisfying the condition that no hyperplane of Pk􀀀1 contains all the n points under consideration. In this 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 : For m = 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 trilinear alternating 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