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.     

On the polynomial Ramanujan sums over finite fields

The polynomial Ramanujan sum was first introduced by Carlitz (Duke Math J 14:1105–1120, 1947), and a generalized version by Cohen (Duke Math J 16:85–90, 1949). In this paper, we study the arithmetical and analytic properties of these sums, deriving various fundamental identities, such as Hölder formula, reciprocity formula, orthogonality relation, and Davenport–Hasse type formula. In particular, we show that the special Dirichlet series involving the polynomial Ramanujan sums are, indeed, the entire functions on the whole complex plane, and we also give a square mean values estimation. The main results of this paper are new appearance to us, which indicate the particularity of the polynomial Ramanujan sums.     

Direct factors of polynomial rings over finite fields

Let G_q denote the multiplicative semigroup of all monic polynomials in one indeterminate over a finite field F_q with q elements. By a direct factor of G_q is understood a subset B₁ of G_q such that, for some subset B₂ of G_q, every polynomial w G_q has a unique factorization in the form w = b₁b₂ for b_i B_i. An asymptotic formula B₁^#(n) c₁qⁿ as n → ∞ is derived for the total number B₁^#(n) of polynomials of degree n in an arbitrary direct factor B₁ of G_q, c₁ a constant depending on B₁.     

Unimodular quadratic forms over global function fields

The isometry problem is studied for unimodular quadratic forms over the Hasse domains of global function fields. Over the polynomial ring k[x] the problem reduces to classification of forms over k; but examples are provided showing that in general no such reduction occurs, even when the underlying ring is Euclidean. Connections with the structure of the ideal class group are given, and a complete invariant for the isometry class is found in the ternary isotropic case.     

Distribution of matrices with restricted entries over finite fields

For a prime p, we consider some natural classes of matrices over a finite field F_p of p elements, such as matrices of given rank or with characteristic polynomial having irreducible divisors of prescribed degrees. We demonstrate two different techniques which allow us to show that the number of such matrices in each of these classes and also with components in a given subinterval [-H, H] [-(p - 1)/2, (p - 1)/2] is asymptotically close to the expected value.     

The rank of sparse random matrices over finite fields

Let M be a random (n×n)-matrix over GF[q] such that for each entry M_ij in M and for each nonzero field element α the probability Pr[M_ij=α] is p/(q−1), where p=(log n−c)/n and c is an arbitrary but fixed positive constant. The probability for a matrix entry to be zero is 1−p. It is shown that the expected rank of M is n−𝒪(1). Furthermore, there is a constant A such that the probability that the rank is less than n−k is less than A/q^k. It is also shown that if c grows depending on n and is unbounded as n goes to infinity, then the expected difference between the rank of M and n is unbounded. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10 , 407–419, 1997     

Primitive matrices over polynomial semirings

An extension of the definition of primitivity of a real nonnegative matrix to matrices with univariate polynomial entries is presented. Based on a suitably adapted notion of irreducibility an analogue of the classical characterization of real nonnegative primitive matrices by irreducibility and aperiodicity for matrices with univariate polynomial entries is given. In particular, univariate polynomials with nonnegative coefficients which admit a power with strictly positive coefficients are characterized. Moreover, a primitivity criterion based on almost linear periodic matrices over dioids is presented.     

Relatively prime polynomials and nonsingular Hankel matrices over finite fields

The probability for two monic polynomials of a positive degree n with coefficients in the finite field Fq to be relatively prime turns out to be identical with the probability for an n×n Hankel matrix over Fq to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of m-tuples of relatively prime polynomials over Fq of given degrees and for the number of n×n Hankel matrices over Fq of a given rank.     

Degree matrices and divisibility of exponential sums over finite fields

By using the degree matrix, we provide an elementary and algorithmic approach to estimating the divisibility of exponential sums over prime fields, which improves the Adolphson–Sperber theorem obtained by using the Newton polyhedron. Our result also improves the Ax–Katz theorem on estimating the number of rational points on hypersurfaces over prime fields.     

利用有限域上交错矩阵构造Cartesian认证码

设Fq是q元有限域,q是素数的幂.令信源集S为Fq上所有的n×n交错矩阵的合同标准形,编码规则集E为Fq上所有的n×n非奇异矩阵,信息集M为Fq上所有的n×n交错矩阵,构造映射f:S×E→M,(K'(v,n),g)→gK'(v,n)gT.证明了该四元组(S,E,M;f)是一个Cartesian认证码,并计算了它的参数.进而,假定编码规则按照均匀的概率分布所选取,计算出了该码的成功模仿攻击概率PI和替换攻击概率Ps.     

A note on adjacency preservers on hermitian matrices over finite fields

It is shown that any map which preserves adjacency on hermitian matrices over a finite field is necessary bijective and hence of the standard form.     

The number of polynomial functions which permute the matrices over a finite field

Let F = GF(q) denote the finite field of order q, and let F_n×n denote the algebra of n × n matrices over F. A function f:F_n×n → F_n×n is called a scalar polynomial function if there exists a polynomial f(x) ?F[x] which represents f when considered as a matrix function under substitution. In this paper a formula is obtained for the number of permutations of F_n×n which are scalar polynomial functions.     

Rank properties of subspaces of symmetric and hermitian matrices over finite fields

We investigate constant rank subspaces of symmetric and hermitian matrices over finite fields, using a double counting method related to the number of common zeros of the corresponding subspaces of symmetric bilinear and hermitian forms. We obtain optimal bounds for the dimensions of constant rank subspaces of hermitian matrices, and good bounds for the dimensions of subspaces of symmetric and hermitian matrices whose non-zero elements all have odd rank.