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

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.

Product of symplectic groups and its cyclic orbit code

Constant dimension subspace codes are subsets of the finite Grassmann Variety. Orbit codes are constant dimension subspace codes that arise as the orbit of subgroup of general linear group acting on subspaces in an ambient space. In particular, orbit codes of symplectic subgroup of the general linear group have been investigated recently. In this paper, we determine product of symplectic groups and its orbit code, and decoding algorithm of this code is considered.

The special linear version of the projective bundle theorem

A special linear Grassmann variety $\text{SGr}(k,n)$ is the complement to the zero section of the determinant of the tautological vector bundle over $\text{Gr}(k,n)$. For an $SL$-oriented representable ring cohomology theory $A^{\ast }(-)$ with invertible stable Hopf map ${\it\eta}$, including Witt groups and $\text{MSL}_{{\it\eta}}^{\ast ,\ast }$, we have $A^{\ast }(\text{SGr}(2,2n+1))\cong A^{\ast }(pt)[e]/(e^{2n})$, and $A^{\ast }(\text{SGr}(k,n))$ is a truncated polynomial algebra over $A^{\ast }(pt)$ whenever $k(n-k)$ is even. A splitting principle for such theories is established. Using the computations for the special linear Grassmann varieties, we obtain a description of $A^{\ast }(\text{BSL}_{n})$ in terms of homogeneous power series in certain characteristic classes of tautological bundles.

Chow forms of congruences

Let V be a 4-dimensional complex space. A congruence Y is an integral surface of the Grassmann variety G = Gr(2, 4) of 2-dimensional subspaces V2 of V (we denote by Vi a subspace of V of dimension i). They have been extensively studied by both classical and modern geometers. We bring to their study the tool of Chow forms, characterizing them by differential equations, following the program of M. Green and I. Morrison [3]. The first results in this direction are due to Cayley ([1], [2]) and are rederived in [3]. Our results share much of the geometrical flavour of Cayley's.