Secant varieties of Grassmann varieties

We consider the dimensions of the higher secant varieties of the Grassmann varieties. We give new instances where these secant varieties have the expected dimension and also a new example where a higher secant variety does not.

     

Triangularizing matrices over GF(2) by congruence

Necessary and sufficient conditions are presented for a square matrix over GF(2) to be triangulable by congruence.     

Polynomial Basis Multiplication over GF(2 m )

In this paper, we describe, analyze and compare various multipliers. Particularly, we investigate the standard modular multiplication, the Montgomery multiplication, and the matrix–vector multiplication techniques.     

The weight distribution of linear codes over GF(ql) having generator matrix over GF(q>)

We study the weight distribution of the linear codes over GF(q^l) which have generator matrices over GF(q) and their dual codes. As an application we find the weight distribution of the irreducible cyclic (23(2¹≈1), 111) codes over GF(2) for all lnot divisible by 11.     

Determinantal varieties over truncated polynomial rings

We study components and dimensions of higher-order determinantal varieties obtained by considering generic m×n (m?n) matrices over rings of the form F[t]/(t^k), and for some fixed r, setting the coefficients of powers of t of all r×r minors to zero. These varieties can be interpreted as spaces of (k−1)th order jets over the classical determinantal varieties; a special case of these varieties first appeared in a problem in commuting matrices. We show that when r=m, the varieties are irreducible, but when r<m, these varieties are reducible. We show that when r=2<m (any k), there are exactly ⌊k/2⌋+1 components, which we determine explicitly, and for general r<m, we show there are at least ⌊k/2⌋+1 components. We also determine the components explicitly for k=2 and 3 for all values of r (for k=3 for all but finitely many pairs of (m,n)).     

Characterizations of Veronese and Segre varieties

We survey the most important characterizations of quadric Veroneseans and Segre varieties of the last thirty years, including some very recent results.     

The eight dimensional ovoids over GF(5)

In this article we outline a computer assisted classification of the ovoids in an orthogonal space of the type .

     

Irreducible polynomials over GF(2) with prescribed coefficients

For an even positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF(2) in which the coefficients of xⁿ⁻¹,xⁿ⁻² and xⁿ⁻³ are specified in advance. Formulas for the number of elements in GF(2ⁿ) with the first three traces specified are also given.     

Orbits under actions of affine groups over GF (2)

Let G be a group (or vector space) and A a group of transformations of G. A then acts as a group of transformations of P(G), the set of subsets of G. It is meaningful to study the orbit structure of P(G) under the action of A. The question of the existence of elements of P(G) with trivial isotropy subgroup seems to be of interest in studying the action of A on G. In this paper actions of affine groups over GF (2) are considered. It is proved, by an inductive construction, that every vector space over GF (2) of dimension at least six contains a subset with trivial isotropy subgroup.     

利用有限域GF（2m）上的三项式构造二元循环码

循环码是一类特殊的线性码,由于循环码快速的编码和译码算法,它被广泛应用于消费电子,数据存储以及通信系统当中.在本文中,利用特征是偶数的有限域上的三项式构造出了两类二元循环码,我们不仅可以确定出这两类循环码最小距离的下界,而且这两类循环码在参数的选取上非常的灵活.     

Irreducible polynomials over GF(2) with three prescribed coefficients

For an odd positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF(2) in which the coefficients of xⁿ⁻¹, xⁿ⁻² and xⁿ⁻³ are specified in advance. Formulas for the number of elements in GF(2ⁿ) with the first three traces specified are also given.     

Symmetry codes over GF(3)

A lower bound $\text{{}\text{1}\text{4}\text{(4q + 5)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+}\text{3}\text{2}$ is given for the minimum weight of the symmetry code C(q) over GF(3), which is introduced by Pless [3].     

An asymptotic bound for secant varieties of Segre varieties

This paper studies the defectivity of secant varieties of Segre varieties. We prove that there exists an asymptotic lower estimate for the greater non-defective secant variety (without filling the ambient space) of any given Segre variety. In particular, we prove that the ratio between the greater non-defective secant variety of a Segre variety and its expected rank is lower bounded by a value depending just on the number of factors of the Segre variety. Moreover, in the final section, we present some results obtained by explicit computation, proving the non-defectivity of all the secant varieties of Segre varieties of the shape $(\mathbb{P }^{n})^4$ , with $2 \le n\le 10$ , except at most $\sigma _{199}((\mathbb{P }^8)^4)$ and $\sigma _{357}((\mathbb{P }^{10})^4)$ .     

The reflexivity of a Segre product of projective varieties

We study the reflexivity of a Segre product of a projective space and a projective variety Y, and give a criterion for to be reflexive in terms of m, the dimension of Y, the rank of the general Hessian of Y and the characteristic of the ground field. Our study is closely related to a question raised by Kleiman and Piene on the relationship between the conormal map and the Gauss map.     

Combinatorial properties of some classes of matrices over GF(2)

We study several classes of matrices of GF(2) constructed from lists of subsets of finite sets In this paper. We show that all matrices in these classes are representations of connected equicardinal matrix over GF(2). In Matrix terms, these irreducible (defined below) matrices all have the property that every minimal dependent set of column has the same cardinality over GF(2). This fact is shown directly in this paper by elementary matrix considerations. In a subsequent paper, we shall show that these classes of matrices are in fact the classes of canonical forms for all representations of nontrivial binary connected equicardinal matroids.     

Classification of 8-Dimensional Trilinear Alternating Forms over GF(2)

Let V be an n-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms we introduce a new invariant called radical polynomial and investigate its behaviour, in particular in the case of the 2-element field. We show that it is compatible with direct products of forms and how it is related to its values on dimension n ? 1. Moreover, it turns out that it is full up to dimension 7. On the other hand, on higher dimensions it is no more full and it is necessary to generalize it to obtain (using computer) a classification of forms on dimension 8 over the 2-element field. This classification is provided, together with the sizes of stabilizers of the corresponding forms.