Galois环上本原序列压缩映射的单性

设Ω是 Ｇａｌｏｉｓ环 ＧＲ（２~ｄ,ｒ）的 Ｔｅｉｃｈｍｕｌｌｅｒ代表集,则 ＧＲ（２~ｄ,ｒ）上每条序列ａ有唯一的权位分解,　其中ａ-ｉ是Ω上序列,同时也可自然视为有限域Ｆ-（２~ｒ）,上序列．设ｆ（ｘ）是环 ＧＲ（２~ｄ,ｒ）上强本原多项式,Ｇ（ｆ（ｘ））表示 ＧＲ（２~ｄ,ｒ）上以ｆ（ｘ）为特征多项式的序列的全体,是Ｆ-（２~ｒ）上一类ｄ－１元多项式, 　本文证明了压缩映射是单射,即对 ａ＝ ｂ当且仅当对所有 ａ,ｂ ∈ Ｇ（ｆ（ｘ））．     

环Z／（p^d）上序列的迹表示及前馈序列空间结构分析

本文利用ｐ－ａｄｉｃ数域理论，给出了乘余类环Ｚ／（ｐ＾ｄ）上线性递归序列的迹表示。并通过应用迹表示，刻划了前馈序列空间Ｇ（ｆ（ｘ））＾ｍ的结构。     

Z／（2∧e）上本原序列不同压缩映射的导出序列

设f(x）是Z/（2∧e）上n次强本原多项式,对形如xe-1 η（x0,…,xe-2）的二个e元布尔函数φ（xo,…,xe-1）和ψ（x0,…,xe-1）及二条序列a,b∈G（f（x)）e,若φ（a0,…,ae-1）=ψ（b0,…,be-1）,给出了函数φ（x0,…,xe-1）和ψ（x0,…,xe-1）之间的关系与序列a和b之间的关系,所给出的结论进一步说明了导出的二元序列具有良好的密码性质。     

Z2e上本原序列最高权位序列的0, 1分布

研究了由 Z_2^e 上n次本原多项式生成的本原序列的最高权位序列的0, 1分布. 首先, 利用Galois环上的指数和估计, 得到了0, 1分布的一个界, 该界当e相对n较小时有效. 同时, 还得到了另一个估计, 该估计当e相对n较大时比较适用. 综合两者, 得到0, 1分布的一个只依赖于n的估计, 该估计说明, n越大, 1在最高权位序列中所占的比率越接近1/2.     

Galois环上的本原多项式的一个判别准则

本文给出Ｇａｌｏｉｓ环Ｒ上的基本不可约多项式ｆ（ｘ）的根的具体表达式和其阶的联系；由此，对本原多项式和次本原多项式分别推导出代数判别式，其主要部分分别由ｆ（ｘ）ｍｏｄｐ和ｆ（ｘ）ｍｏｄｐ２的系数所确定．     

GR(4,r)上本原序列的元素分布

本文利用GR（4，r)上本原序列的迹表示及二次型的有关结论，给出了本原序列的第一权位序列的元素分布，同时求得本原序列的元素分布。     

Galois环上极大周期序列的平移等价

本文给出了Galois环上两个具有相同特征多项式的极大周期序列是否平移等价的—个判定方法,以及在两个序列平移等价的情况下,利用模p方幂提升技术,给出了—个计算它们的平移距离的方法.     

Galois环上的连分式与线性递归序列综合

本文讨论了 Galois环上连分式的性质 ,并将其用于 Galois环上线性递归序列综合问题 .     

Z/(2e)上本原序列不同压缩映射的导出序列

设 f( x)是 Z/ ( 2 e)上 n次强本原多项式 ,对形如 xe- 1 +η( x0 ,… ,xe- 2 )的二个 e元布尔函数 Φ( x0 ,… ,xe- 1 )和 Ψ( x0 ,… ,xe- 1 )及二条序列 a,b∈G( f( x) ) e,若Φ( a0 ,… ,ae- 1 ) =Ψ ( b0 ,… ,be- 1 ) ,给出了函数Φ ( x0 ,… ,xe- 1 )和Ψ ( x0 ,… ,xe- 1 )之间的关系与序列 a和 b之间的关系 .所给出的结论进一步说明了导出的二元序列具有良好的密码性质     

环Z／（2^e）上压缩序列的0,1分布

本文研究环Ｚ／（２＾ｅ）上本原序列最高权位的０,１分布,证明了当ｅ≥８,次数ｎ≥２０时,本的序列ａ的最高权位序列ａｅ－１在一个周期中０（或１）所占的比例λ（ａｅ－１）满足４３．６７６８％〈λ（ａｅ－１）〈５６．３２３２％。     

Primitive polynomial with three coefficients prescribed

The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given (a₁,a₂,a₃)F_q³, there exists a primitive polynomial f(x)=xⁿ−σ₁xⁿ⁻¹++(−1)ⁿσ_n over F_q of degree n with the first three coefficients σ₁,σ₂,σ₃ prescribed as a₁,a₂,a₃ when n8. But the methods in Fan and Han (in press) are not effective for the case of n=7. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the n=7 case for finite fields of characteristic at least 5. In this paper, we deal with the remaining cases and prove that there exists a primitive polynomial of degree 7 over F_q with the first three coefficient prescribed where the characteristic of F_q is 2 or 3.     

The 0 ,1 - Distribution of the First CoordinateSequences Derived from ML- Sequences over Z/(2~e)

§1.　IntroductionThereseachindetailonlinearrecurringsequencesovertheresideclassringZ/(pe)start-edfromWardworkin30s(see[1]).Inthepastfewyears,fromthepointofviewofcrypto-graphandcodetheory,peopleisinterestedparticularlyinstudyingthecoordinatesequencesde-rivedfromML-sequencesoverZ/(pe),asnon-linearrecurringsequencesoverFp(see[2],[3]).SomeanalogousresultsextendedtoGaliosringcanbefoundin[4],[5].Butthereisnobetterresultontheresearchofthe0,1-distributionproperties.Fromavastamountofcalcula-tion,t…     

本原环为除环的若干条件(英)

本文推广文［1－3］的结果，给出了本原环为除环的几个条件．     

A new proof of Fitzgerald's characterization of primitive polynomials

We give a new proof of Fitzgerald's criterion for primitive polynomials over a finite field. Existing proofs essentially use the theory of linear recurrences over finite fields. Here, we give a much shorter and self-contained proof which does not use the theory of linear recurrences.     

Compressing Mappings on Primitive Sequences over Z/(2) and Its Galois Extension

Let f(x) be a strongly primitive polynomial of degree n over Z/(2^e), η(x₀,x₁,…,x_e−2) a Boolean function of e−1 variables and (x₀,x₁,…,x_e−1)=x_e−1+η(x₀,x₁,…,x_e−2)G (f(x),Z/(2^e)) denotes the set of all sequences over Z/(2^e) generated by f(x), F₂^∞ the set of all sequences over the binary field F₂, then the compressing mapping

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is injective, that is, for , G(f(x),Z/(2^e)), = if and only if Φ( )=Φ( ), i.e., ( ₀,…, _e−1)=( ₀,…, _e−1) mod 2. In the second part of the paper, we generalize the above result over the Galois rings.     

On the Splitting Ring of a Polynomial

Let f(Z) = Zⁿ ? a₁Z^n?1 + … + (?1)^n?1a_n?1Z + (?1)ⁿa_n be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal I_f of R[X₁,…, X_n] generated by σ_i(X₁,…, X_n) ? a_i, where σ₁,…, σ_n are the elementary symmetric polynomials, as well as the quotient ring R[X₁,…, X_n]/I_f.     

Noncommutative Valuation Rings of the Quotient Artinian Ring of a Skew Polynomial Ring

Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let Q[X,] be the skew polynomial ring over Q in an indeterminate X, where is an automorphism of Q. Consider the natural map from Q[X,]_XQ[X,] to Q, where Q[X,]_XQ[X,] is the localization of Q[X,] at the maximal ideal XQ[X,] and set , the complete inverse image of R by . It is shown that is a Dubrovin valuation ring of Q(X,) (the quotient ring of Q[X,]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism is classified into five types, in order to study the structure of (the value group of ). It is shown that there is a commutative valuation ring R with automorphism which belongs to each type and which makes Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional. Presented by A. VerschorenMathematics Subject Classifications (2000) 16L99, 16S36, 16W60.     

Modular Species and Prime Ideals for the Ring of Monomial Representations of a Finite Group#

The ring of monomial representations of a finite group has been investigated by Dress (1971 Dress , A. W. M. ( 1971 ). The ring of monomial representations I: structure theory . J. Algebra 18 : 137 – 157 . [CSA] [CROSSREF]  [Google Scholar]) and Boltje (1990 Boltje , R. ( 1990 ). A canonical Brauer induction formula . Astérisque 181–182 : 31 – 59 . [Google Scholar]), among others. It is of interest in connection with induction theorems in representation theory. Its species have recently been determined by Boltje. In this article, we will analyze the block distribution of species. As an application, we will determine the prime ideals of the ring of monomial representations. The results here constitute a slightly modified version of part of the first author's Diplomarbeit (Fotsing, 2003 Fotsing , B. ( 2003 ). Zum Ring der monomialen Darstellungen einer endlichen Gruppe . Diplomarbeit . Jena . [Google Scholar]), written under the direction of the second author.