Correction to combinatiorial properties of some classes of matrices over GF(2)

Due to my oversight in checking the galley proofs, an error has occured in Tran [1]. The wording of proposition 4.2 on page 162 should read.     

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.     

Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries

Letq be an odd prime power not divisible by 3. In Part I of this series, it was shown that the number of points in a rank-n combinatorial geometry (or simple matroid) representable over GF(3) and GF(q) is at mostn ². In this paper, we show that, with the exception ofn = 3, a rank-n geometry that is representable over GF(3) and GF(q) and contains exactlyn ² points is isomorphic to the rank-n Dowling geometry based on the multiplicative group of GF(3).This research was partially supported by the National Science Foundation under Grants DMS-8521826 and DMS-8500494.     

Combinatorial geometries representable over GF(3) and GF(q). I. The number of points

Letq be a prime power not divisible by 3. We show that the number of points (or rank-1 flats) in a combinatorial geometry (or simple matroid) of rankn representable over GF(3) and GF(q) is at mostn ². Whenq is odd, this bound is sharp and is attained by the Dowling geometries over the cyclic group of order 2.This research was partially supported by National Science Foundation Grant DMS-8521826 and a North Texas State University Faculty Research Grant.     

On the properties of matrices defining some classes of BVMs

Summary It is known that the matrices defining the discrete problem generated by a k-step Boundary Value Method (BVM) have a quasi-Toeplitz band structure [7]. In particular, when the boundary conditions are skipped, they become Toeplitz matrices. In this paper, by introducing a characterization of positive definiteness for such matrices, we shall prove that the Toeplitz matrices which arise when using the methods in the classes of BVMs known as Generalized BDF and Top Order Methods have such property. Mathematics Subject Classification (2000):65L06, 47B35, 15A48Work supported by G.N.C.S.     

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.     

Equivalence classes of matrices inGL 2 (Q) andSL 2 (Q)

LetG denote either of the groupsGL ₂ (q) orSL ₂ (q). The mapping θ sending a matrix to its transpose-inverse is an automophism ofG and therefore we can form the groupG ⁺ =G. <θ>. In this paper conjugacy classes of elements inG ⁺ -G are found. These classes are closely related to the congruence classes of invertible matrices inG.     

Binary images of some self-dual codes over GF(2m) with respect to trace-orthogonal basis

The note deals with a class of self-dual binary codes containing external doubly-even codes for lengths 8, 16, 24, 32, 40 and 64. For most of them, our construction also provides a simple method for calculating the distance of the code. To our knowledge, this is the first example of a (64, 32, 12) extermal doubly-even self-dual code.     

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.     

Equivalence classes of matrices over finite fields

Let F=GF(q) denote the finite field of order q, and F_mn the ring of m×n matrices over F. Let Ω be a group of permutations of F. If A,B∈F_mn, then A is equivalent to B relative to Ω if there exists ?∈Ω such that ?(a_ij) = b_ij. Formulas are given for the number of equivalence classes of a given order and for the total number of classes induced by various permutation groups. In particular, formulas are given if Ω is the symmetric group on q letters, a cyclic group, or a direct sum of cyclic groups.     

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.     

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].     

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.     

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.