### Weights of the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$ over a finite field

#### Abstract

Grassmann codes are linear codes associated with the Grassmann variety

$G(\ell,m)$ of $\ell$-dimensional subspaces of an $m$ dimensional vector space $\mathbb{F}_{q}^{m}.$

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 $\mathbb{P}^{k-1}$

satisfying the condition that no hyperplane of $\mathbb{P}^{k-1}$ 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 $\mathbb{F}_{q}$. 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 $\mathbb{F}_{q}^{8}.$ 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\leq 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 $\mathbb{\bar{F}}$ to $\mathbb{F}.$

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 $\mathbb{F}_{q}$ of characteristic other than $2$ and $3$ was solved by Noui L and Midoune N. We describe

the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$, where char $\mathbb{F}_{q}\neq 3.$ This fact we use

to determine the weight of the $\mathbb{F}_{q}$-forms.

$G(\ell,m)$ of $\ell$-dimensional subspaces of an $m$ dimensional vector space $\mathbb{F}_{q}^{m}.$

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 $\mathbb{P}^{k-1}$

satisfying the condition that no hyperplane of $\mathbb{P}^{k-1}$ 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 $\mathbb{F}_{q}$. 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 $\mathbb{F}_{q}^{8}.$ 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\leq 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 $\mathbb{\bar{F}}$ to $\mathbb{F}.$

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 $\mathbb{F}_{q}$ of characteristic other than $2$ and $3$ was solved by Noui L and Midoune N. We describe

the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$, where char $\mathbb{F}_{q}\neq 3.$ This fact we use

to determine the weight of the $\mathbb{F}_{q}$-forms.

#### Keywords

trivector, Grassmannian, weight

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