特征2的有限域中一些特殊元素的存在性(II)

设q=2s.s,n为正整数,Fqn为qn元素的有限域.在本文中,我们考虑Fqn中一些特殊元素的存在性.主要结果是:当下面的条件之一成立时,在Fqn中存在ξ使得ξ和ξ+ξ-1都是本原元并且ξ+ξ-1还是一个正规元:1.当n|(q-1)时,n37,s>6,或者2.当n|■(q-1)时,n≥34,s>6.进一步,如果n是奇数,则当下列条件之一成立时,存在ξ∈Fqn使得ξ和ξ+ξ-1都是Fqn的本原正规元:1.当n|(q-1)时,n≥257,s>9,或者2.当n■(q-1)时,n≥43,s≥9.     

有限域上幂剩余正规元的存在性

GF(q)是q个元的有限域,q是素数的方幂,n是正整数,GF(qn)为GF(q)的n次扩张.用指数和估计的方法给出了3种情形下幂剩余正规元存在的充分条件,即(1)GF(qn)中存在元ξ为GF(q)上的幂剩余正规元;(2)GF(qn)中存在元ξ与ξ-1同时为GF(q)上幂剩余正规元;(3)对GF(qn)*中任意给定的非零元a和b,GF(qn)中存在元ξ与ξ-1同时为GF(q)上d次幂剩余正规元,且满足Tr(ξ)=a,Tr(ξ-1)=b.     

有限域上幂剩余正规元的存在性

GF(q)是q个元的有限域,q是素数的方幂,n是正整数,GF(q~n)为GF(q)的n次扩张.用指数和估计的方法给出了3种情形下幂剩余正规元存在的充分条件,即(1)GF(q~n)中存在元ξ为GF(q)上的幂剩余正规元;(2)GF(q~n)中存在元ξ与ξ~(-1)同时为GF(q)上幂剩余正规元;(3)对GF(q~n)~*中任意给定的非零元a和b,GF(q~n)中存在元ξ与ξ~(-1)同时为GF(q)上d次幂剩余正规元,且满足Tr(ξ)=a,Tr(ξ~(-1))=b.     

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妥条件:设q为素数幂,E=Fqn为q元域F=Fq的n次扩张,N={αi=αqi|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.     

有限域上的弱自对偶正规基

对有限域上的弱自对偶正规基的乘法表的特征进行了刻画,并对其复杂度进行了研究,得到了在几种不同类型的有限域扩张时此类正规基的下界描述.例如,若q为素数幂,E=Fqn为q元域F=Fq的n次扩张,N={αi=αqi|I=0,1,…,n-1}为E在F上的一组弱自对偶正规基,其对偶基由β=cαr生成,其中c∈F*,0≤r≤n-1,则当r≠0,n/2时,N的复杂度CN为偶数且CN≥4n-2.     

关于有限域上的置换二项式

设l为一奇正整数,q是某素数的方幂,二者满足l|q-1,记s=(q-1)/l;又设Fq是一个q元有限域,r,e为正整数,(e,l)=1.本文应用序列{an=∑(l-1)/2t=1(2(-1)t-1cos(tπ/l))n}∞n=-∞的性质给出了当l=9时Fq上的二项式f(x)=xr(1+xes)成为Fq上的置换多项式的充要条件.     

Existence of Some Specific Elements in Finite Fields北大核心CSCD

Let q = pk and Fqn be the extension field of Fq of degree n, where p is an odd prime and n, k are positive integers. The main contribution of this paper is as follows: If n | (q − 1), k ≥ 11, n ≥ 14 or n (q − 1), k ≥ 10, n ≥ 8, then there exists a primitive element α in Fqn such that α + α−1 is a normal element, and 1 + α2 is a square element, and there exists a normal element β, such that β + β−1 is a primitive element, and 1 + β2 is a square element. © 2022 Chinese Academy of Sciences. All rights reserved.     

有限域上本原多项式系数的研究

设Fq表示有q个元素的有限域,q为素数的方幂,f(x)=xn+a1xn-1+…+an-1x+an∈Fq[x].当n≥7时,文[8]指出存在Fq上可预先指定a1,a2的n次本原多项式.本文讨论了剩余的n=5,6两种情形,利用有限域上的两类特征和估计及Cohen筛法(见[4,6]),改进了文[8]中关于本原解个数的下界,并得到当n=5,6时,在特征为奇的有限域上存在可预先指定前两项系数的n次本原多项式.     

有限域上本原多项式系数的研究

设Fq表示有q个元素的有限域,q为素数的方幂,f(x)=xn+a1xn-1+…+an-1x+an∈Fq[x].当n(≥)7时,文[8]指出存在Fq上可预先指定a1,a2的n次本原多项式.本文讨论了剩余的n=5,6两种情形,利用有限域上的两类特征和估计及Cohen筛法(见[4,6]),改进了文[8]中关于本原解个数的下界,并得到当n=5,6时,在特征为奇的有限域上存在可预先指定前两项系数的n次本原多项式.     

有限域上的本原最优正规元

Let q be a prime or prime power and Fqn the extension of q elements finite field Fq with degree n(n1).Davenport,Lenstra and Schoof proved that there exists a primitive element α∈ Fqn such that α generates a normal basis of Fqn over Fq.Later,Mullin,Gao and Lenstra,etc.,raised the definition of optimal normal bases and constructed such bases.In this paper,we determine all primitive type I optimal normal bases and all finite fields in which there exists a pair of reciprocal elements α and α-1 such that both of them generate optimal normal bases of Fqn over Fq.Furthermore,we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other.     

Estimate of a multidimensional sum with the Legendre symbol for a polynomial of odd degree

The estimate $\left| {\sum\nolimits_{x_1 ,...,x_n \in F_q } {x(f(x_1 ,...,x_n ))} } \right| \leqslant (d - 1)^n q^{n/2} $ is derived for the quadratic character Λ of a field F_q of q elements and a polynomial f of odd degree d over F_q under certain natural conditions.     

Some remarks concerning the individual ergodic theorem of information theory

Let (X,μ, T) be an ergodic dynamic system and let ξ = (C₁, C₂, ...) be a discrete decomposition of X. Conditions are considered for the existence almost everywhere of $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left| {\log \mu (C_{\xi ^n } (x))} \right|,$$ whereC _ξn(x) is the element of the decomposition ξⁿ = ξ V T ξ V ... < T^n-1ξ containing x. It is proved that the condition H(ξ) < ∞ is close to being necessary. If T is a Markov automorphism and ξ is the decomposition into states, then the limit exists, even if H(ξ) = ∞, and is equal to the entropy of the chain.     

On the Existence for Some Special Primitive Elements in Finite Fields

Let F_q be a finite field of characteristic p. In this paper, by using the index sum method the authors obtain a sufficient condition for the existence of a primitive elementα∈ F_(q~n) such that α + α~(-1)is also primitive or α + α~(-1)is primitive and α is a normal element of F_(q~n) over F_q.     

An example of a function which is Denjoy integrable but not khinchin summable

The following result is proven: if ξ is an irrational number “anomalously badly“ approximable by rationals, then there are functions which are not Khinchin ξ-summable but which are Denjoy integrable. Let I be the interval 0 ≤ x ≤ 1, and let ξ be an irrational, 0 < ξ< 1. Let T _ξ denote the transformation of I into itself defined as follows: $$T_\xi x = \left\{ {\begin{array}{*{20}c} {x + \xi ,ifx + \xi \in I;} \\ {x + \xi - 1} \\ \end{array} } \right.$$ otherwise.     

Geometric Properties of Generalized Hypergeometric Functions

Let denote the generalized hypergeometric function where no denominator parameter can be zero or a negative integer and (a,n) denotes the ascending factorial notation. Ponnusamy and Vuorinen raised the problem of finding conditions on the parameters a_j > 0, b_j > 0 so that the function is univalent in . The main aim of this paper is to discuss this problem in detail for the case q = 2.     

Asymptotic number of solutions of some systems of diophantine inequalities

The problem of finding the asymptotic number of solutions of the system of inequalities $$\begin{gathered} \left\| {\alpha _i q} \right\|< q^{ - \sigma _i } (i = 1,...,n), \sigma _i > 0, \hfill \\ \sigma = \sum\nolimits_{i = 1}^n {\sigma _i< c(\alpha _1 ,...,\alpha _n ), q = 1,...,N,} \hfill \\ \end{gathered}$$ is solved under the assumption that for real numbers α₁,..., α_n, starting from some Q=max(q₁...,q_n) the inequality holds for any real λ≥0.     

有限域上长为2mpn 的负循环码

设F_q 是奇数阶有限域. 本文主要借助X^2mpn+1 在F_q 上的不可约因式分解来确定有限域F_q上所有长为2^mpⁿ 的负循环码和自对偶的负循环码的生成多项式, 这里p 是q-1 的奇素因子, m 和n是正整数.     

On global relations for hypergeometric series

Let [K:Q]=k, f_i εK[[z]], ξ εK, Q εK[y₁,...,y_m]. A relation Q(f₁(ξ),..., f_m(ξ))=0 is called global if it holds in any local field where all f_i(ξ) exist. The paper establishes that for series of the form $$\sum\limits_{n = 0}^\infty {\frac{{(\mu _1 )_n \ldots (\mu _p )_n }}{{(\lambda _1 )_n \ldots (\lambda _{q - 1} )_n n!}}\left( {\frac{{z^{p - q} }}{{q - p}}} \right)^n , p > q,} $$ with some natural hypotheses on parameters global relations do not exist. Bibliography: 9 titles.