Gauss Sum of Index Four

Let p be a prime number, q = pf and m| q - 1. For a multiplicative character χ of Fq with order m, the Gauss sum G(χ) over Fq is defined by G(χ)=∑(X()Fq)χ(x)ζTP(x),ζp=e2πi/p,(1)where T is the mapping for Fq/Fp.     

The Fundamental Units ofAlgebraic Real Quadratic Function Fields

A general theorem on the fundamental unit E of any algebraic quadratic function field K isgiven here; and then the fundamental unit E is exhibited explicitly for sixteen types of four seriesof quadratic function fields K.Suppose that k = Fq.(T) is the rational function field in the indeterminate (variable) T overF,, the finite field with q elements (q is a power of odd prime number p). Let R = Fq[T] bethe polynomial ring of T over F., which is also said to be the ring (domain) of integers…     

AN ANZAHL THEOREM IN SINGULAR PSEUDO-SYMPLECTIC GEOMETRY OVER FINITE FIELDS

1 IntroductionLet F, be a fiuite field witl1 q elelllents. wl1ere q is a power of 2. We will follow tlie notationsand terntillologies in [11. LetThe singular pseudthsymplectic group with respect to Se,I over Fq denoted by PS=.+.+..=.+, (F,)is defiued to be tlie group fOrnled by all (2P + 6 + l) x (2v + b + l) nonsingUlar n1atrices T suchthat TSb.,T' == Sb,i. wl1ere TT denotes the transpose of T. If T 6 Ps=.+,+,.=.+, (Fq), then T isof the forlllwhere T1,S,T1,' = Se and T22 is nousin…     

GEOMETRIC CONSTRUCTION OF ASSOCIATION SCHEMES FROM DEGENERATE QUADRICS INPROJECTIVE SPACES PG(2v+δ+l-1,Fq) FOR q ODD

Let Fq be a finite field with q elements,where q is a power of an odd prime.In this paper,the authors consider a projective space PG(2v+δ+l,Fq) with dimension 2v+δ+l,partitioned into an affine space AG(2v+δ+l,Fq) of dimension 2v+δ+l and a hyperplane H=PG(2v+δ+l-1,Fq) of dimension 2v+δ+l-1 at infinity,where l≠0.The points of the hyperplane Hare next partitioned into four subsets.A pair of points a and b of the affine space is defined to belong to class i if the line ab meets the subset i of H.Finally,a family of four-class association schemes are constructed,and parameters are also computed.     

有限域上存在弱自对偶正规基的一个充要条件

对于将有限域上的自对偶基概念推广到了更一般的弱自对偶的情形,给出了有限域上存在这类正规基的一个充要条件设q为素数幂,E=Fqn为q元域F=Fq的n次扩张,N={αi=αq2| i=0,1,…,n-1}为E在F上的一组正规基.则存在c∈F*及r,0≤r≤n-1,使得β=cαr生成N的对偶基的充要条件是以下三者之一成立(1)q为偶数且n≠0(mod 4);(2)n与q均为奇数;(3)q为奇数,n为偶数,(-1)为F中的非平方元且r为奇数.     

有限域上仿射m面诱导的图Ⅱ

Let AG(n,Fq) be the n-dimensional affine space over Fq,where Fq is a finite field with q elements. Denote by Г(m) the graph induced by m-flats of AG(n,Fq). For any two adjacent vertices E and F of Гr(m) ,Г(m) (E)∩ Г(m) (F) is studied. In particular,sizes of maximal cliques in Г(m) are determined and it is shown that Г(m) is not edge-regular when m＜n-1.     

Full automorphism group of the generalized symplectic graph

Let Fq be a finite field of odd characteristic, m, ν the integers with 1≤m≤ν and Ka 2ν× 2ν nonsingular alternate matrix over Fq. In this paper, the generalized symplectic graph GSp2ν (q, m) relative to K over Fq is introduced. It is the graph with m-dimensional totally isotropic subspaces of the 2ν-dimensional symplectic space F(2ν)q as its vertices and two vertices P and Q are adjacent if and only if the rank of PKQT is 1 and the dimension of P ∩ Q is m-1. It is proved that the full automorphism group of the graph GSp2ν(q, m) is the projective semilinear symplectic group PΣp(2ν, q).     

On enumeration of polynomial equivalence classes

The isomorphism of polynomials(IP),one of the hard problems in multivariate public key cryptography induces an equivalence relation on a set of systems of polynomials.Then the enumeration problem of IP consists of counting the numbers of different classes and counting the cardinality of each class that is highly related to the scale of key space for a multivariate public key cryptosystem.In this paper we show the enumeration of the equivalence classes containing ∑n-1 i=0 aiX2qi when char(Fq) = 2,which implies that these polynomials are all weak IP instances.Moreover,we study the cardinality of an equivalence class containing the binomial aX 2q i + bX 2qj(i=j) over Fqn without the restriction that char(Fq) = 2,which gives us a deeper understanding of finite geometry as a tool to investigate the enumeration problem of IP.     

On Enumeration of the Special Permutations

Let n be an odd number and P_n be the set of all permutations on the set{0,1,…,n-1}.A permutation a=(a_0a_1…a_(n-1))∈P_n is called a special permutation ofdegree n,if a_i-i a_j-j(mod n),0≤i相似文献   

TO SOLVE MATRIX EQUATION sum (A~iXBi=c) BY THE SMITH NORMAL FORM

§ 1　IntroductionLet F be a field,F[λ] be the polynomial ring over F,Fm× n( or Fm× n[λ] ) be the setofall m×n matrices over F( or F[λ] ) .Let M(i) be the ith column of M∈Fm× m[λ] ,i=1 ,...,n.A g-inverse of M∈Fm× n will be denoted by M- and understood as a matrix for whichMM- M=M.In this paper,we discuss the linear matrix equation ki=0Ai XBi =C, ( 1 )where A∈Fm× m,Bi∈Fn× q,i=0 ,1 ,...,k,and C∈Fm× q.Equation( 1 ) is called universally solvable if ithas a solution f…