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We consider the structures of the plane-line and point-line incidence matrices of the projective space PG (3 , q) connected with orbits of planes, points, and lines under the stabilizer group of the twisted cubic. In the literature, lines are partitioned into classes, each of which is a union of line orbits. In this paper, for all q, even and odd, we determine the incidence matrices connected with a family of orbits of the class named O 6 . This class contains lines external to the twisted cubic. The considered family includes an essential part of all O 6 orbits, whose complete classification is an open problem. [ABSTRACT FROM AUTHOR]

The point-plane, the point-line, and the plane-line incidence matrices of PG (3 , q) are of interest in combinatorics, finite geometry, graph theory and group theory. Some of the properties of these matrices and their submatrices are related with the interplay of orbits of points, lines and planes under the action of subgroups of PGL (4 , q) . A remarkable particular case is the subgroup G ≅ PGL (2 , q) , viewed as the stabilizer group of the twisted cubic C . For this case, the study of the point-plane incidence matrix, initiated by D. Bartoli and the present authors, has attracted attention as being related to submatrices with useful applications in coding theory for the construction of multiple covering codes. In this paper, we extend our investigation to the plane-line incidence matrix apart from just one class of the line orbits, named O 6 in the literature. For all q ≥ 2 , in each submatrix, the numbers of lines in any plane and planes through any line are obtained. As a tool for the present investigation, we use submatrices of incidences arising from orbits of planes and unions of line orbits, including O 6 . For each such submatrix we determine the total number of lines from the union in any plane and the average number of planes from the orbit through any line of the union. [ABSTRACT FROM AUTHOR]

We consider the $[\text {q}+1,\text {q}-3,5]_{\text {q}}3$ generalized doubly-extended Reed-Solomon code of codimension 4 as the code associated with the twisted cubic in the projective space $\mathrm {PG}(3,\text {q})$. Basing on the point-plane incidence matrix of $\mathrm {PG}(3,\text {q})$ , we obtain the number of weight 3 vectors in all the cosets of the considered code. This allows us to classify the cosets by their weight distributions and to obtain these distributions. The weight of a coset is the smallest Hamming weight of any vector in the coset. For the cosets of equal weight having distinct weight distributions, we prove that the difference between the w-th components, $3 < \text {w}\le \text {q}+1$ , of the distributions is uniquely determined by the difference between the 3-rd components. This implies an interesting (and in some sense unexpected) symmetry of the obtained distributions. [ABSTRACT FROM AUTHOR]

In PG (3 , q) , q = 2 n , n ≥ 3 , let A h , q = { (1 , t , t 2 h , t 2 h + 1) | t ∈ F q } ∪ { (0 , 0 , 0 , 1) } , with gcd ⁡ (n , h) = 1 , be a (q + 1) -arc and let G h ≃ PGL (2 , q) be the stabilizer of A h , q in PGL (4 , q). The G h -orbits on points, lines and planes of PG (3 , q) , together with the point-plane incidence matrix with respect to the G h -orbits on points and planes of PG (3 , q) are determined. The point-line incidence matrix with respect to the G 1 -orbits on points and lines of PG (3 , q) is also considered. In particular, for a line ℓ belonging to a given line G 1 -orbit, say L , the point G 1 -orbit distribution of ℓ is either explicitly computed or it is shown to depend on the number of elements x in F q (or in a subset of F q) such that Tr q | 2 (g (x)) = 0 , where g is an F q -map determined by L. [ABSTRACT FROM AUTHOR]

Fundamental matrix encapsulates all the geometric information in two views, and it plays an important role in many applications of three-dimensional computer vision. However, some configurations have an inherent ambiguity, namely, degeneracy, and no matter how many matched image points are used, the fundamental matrix could not be determined uniquely. It is well-known that degeneracies occur when all the space points and both the camera centers belong to a ruled quadric. But it is not so well-known how great the degenerate degrees in different configurations are. In this paper, we discuss the degeneracies caused by a quadric cone and give the corresponding degenerate degrees. We parameterize all the points on a quadric cone by a twisted cubic lying on the cone, and obtain a parametric coefficient matrix of the equations for estimating the fundamental matrix. By analyzing the coefficient matrix, we get the rank of it, i.e. the degenerate degree of the given configuration. It gives a more intuitive degeneracy and the degenerate degrees of the configurations, and reveals the full picture of the degeneracies on a cone. [ABSTRACT FROM AUTHOR]

In this paper we study combinatorial invariants of the equivalence classes of pencils of cubics on PG (1 , q) , for q odd and q not divisible by 3. These equivalence classes are considered as orbits of lines in PG (3 , q) , under the action of the subgroup G ≅ PGL (2 , q) of PGL (4 , q) which preserves the twisted cubic C in PG (3 , q). In particular we determine the point orbit distributions and plane orbit distributions of all G -orbits of lines which are contained in an osculating plane of C , have non-empty intersection with C , or are imaginary chords or imaginary axes of C. [ABSTRACT FROM AUTHOR]

Via the XJC-correspondence proved in [L. Pirio and F. Russo, Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras, submitted] we provide some structure theorems for quadro-quadric Cremona transformations and for extremal varieties 3-covered by twisted cubics by reinterpreting for these objects the algebraic results on the solvability of the radical of Jordan algebras. In this way, we can define the semi-simple part and the radical part of a quadro-quadric Cremona transformation, respectively of an extremal variety 3-covered by twisted cubics, and then describe how general objects are constructed from the semi-simple ones, which are completely classified modulo certain equivalences, via suitable null radical extensions. [ABSTRACT FROM AUTHOR]

Abstract: Three-level secret sharing schemes arising from the vector space construction over a finite field are presented. Compared to the previously known schemes, they allow a larger number of participants with respect to the size of . The key tool is the twisted cubic of . [ABSTRACT FROM AUTHOR]

Abstract: In order to understand the triple action of PGL n+1 on the projective space of nonzero (n +1)×(n +1) matrices of linear forms on , we associate a quadratic rational map to any such matrix A. The properties of the dynamical system obtained by iteration of ϕ, some of which are of a geometric nature, generate invariants and a canonical form for the orbit of A. We study a family of matrices parametrized by , whose associated geometry is given by the rational normal curve for each dimension n =2,3,4. Our analysis involves the osculating flags to the curves; and we calculate the stabilizers of our rational maps and matrices. [Copyright &y& Elsevier]

Abstract: Using the connection between translation spreads of the classical generalized hexagon and the twisted cubic of , established in [European J. Combin. 23 (2002) 367–376], we prove that if , q odd, and , then does not admit an -translation spread. [Copyright &y& Elsevier]

Cayley’s (ruled cubic) surface carries a three-parameter family of twisted cubics. We describe the contact of higher order and the dual contact of higher order for these curves and show that there are three exceptional cases. [ABSTRACT FROM AUTHOR]

Let PG (3 , q) be the projective space of dimension three over the finite field with q elements. Consider a twisted cubic in PG (3 , q). The structure of the point-plane incidence matrix in PG (3 , q) with respect to the orbits of points and planes under the action of the stabilizer group of the twisted cubic is described. This information is used to view generalized doubly-extended Reed-Solomon codes of codimension four as asymptotically optimal multiple covering codes. [ABSTRACT FROM AUTHOR]