# WEIGHTS OF THE Fq-FORMS OF 2-STEP SPLITTING TRIVECTORS OF RANK 8 OVER A FINITE FIELD

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## Date

2021

## Authors

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## Publisher

Université de M'sila

## Abstract

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 Pk1 satisfying the condition
that no hyperplane of Pk1 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

## Description

## Keywords

words and phrases: trivector, Grassmannian, weight