SELF-DUAL PERMUTATION CODES OVER FORMAL POWER SERIES RINGS AND FINITE PRINCIPAL IDEAL RINGS

In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.     

形式幂级数环上的自对偶码

形式幂级数环R_∞=F[[γ]]={sum from l=0 to a_lγ~l|a_l∈F}与有限链环R_i={a_0+a_1γ+…+a_(i-1)γ~(i-1)|a_i∈F}的码的投影与提升有密切关系.利用形式幂级数环R_∞上码C在有限链环R_i的投影码的自正交性与自对偶性来研究码C的自正交性与自对偶性,得到了两个有意义的结果.     

有限域上的重根自对偶负循环码

Let F_q be a finite field with q = p~m, where p is an odd prime. In this paper, we study the repeated-root self-dual negacyclic codes over Fq. The enumeration of such codes is investigated. We obtain all the self-dual negacyclic codes of length 2~ap~r over F_q, a ≥ 1.The construction of self-dual negacyclic codes of length 2~abp~r over F_q is also provided, where gcd(2, b) = gcd(b, p) = 1 and a ≥ 1.     

Negacyclic codes over Galois rings of characteristic 2 a

We investigate negacyclic codes over the Galois ring GR(2 a ,m) of length N = 2 k n,where n is odd and k 0.We first determine the structure of u-constacyclic codes of length n over the finite chain ring GR(2 a ,m)[u]/ u 2 k + 1 .Then using a ring isomorphism we obtain the structure of negacyclic codes over GR(2 a ,m) of length N = 2 k n (n odd) and explore the existence of self-dual negacyclic codes over GR(2 a ,m).A bound for the homogeneous distance of such negacyclic codes is also given.     

CYCLIC CODES OVER FORMAL POWER SERIES RINGS

In this article, cyclic codes and negacyclic codes over formal power series rings are studied. The structure of cyclic codes over this class of rings is given, and the relationship between these codes and cyclic codes over finite chain rings is obtained. Using an isomorphism between cyclic and negacyclic codes over formal power series rings, the structure of negacyclic codes over the formal power series rings is obtained.     

Type Ⅱ codes over finite rings

In this paper,we generalize the concept of Type Ⅱ codes to arbitrary finite rings. We focus on Type Ⅱ codes over finite chain rings and use the Chinese Remainder Theorem on these codes to study Type Ⅱ codes over principal ideal rings.     

Notes on linear codes over finite commutative chain rings

The properties of the generator matrix are given for linear codes over finite commutative chain rings,and the so-called almost-MDS (AMDS) codes are studied.     

弱$r$-Clean环

As generalization of r-clean rings and weakly clean rings, we define a ring R is weakly r-clean if for any a∈R there exist an idempotent e and a regular element r such that a = r + e or a = r-e. Some properties and examples of weakly r-clean rings are given. Furthermore, we prove the weakly clean rings and weakly r-clean rings are equivalent for abelian rings.     

Some Sharpening and Generalizations of a Result of T. J. Rivlin

Let p(z)=a_0+a_1z+a_2z~2+a_3z~3+···+a_nz~n be a polynomial of degree n.Rivlin[12]proved that if p(z)≠0 in the unit disk,then for 0r≤1,max|z|=r|p(z)|≥((r+1)/2)~nmax|p(z)||z|=1.In this paper,we prove a sharpening and generalization of this result and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin’s Theorem.     

有限域上一类方程组解数的直接公式

本文给出有限域F=F_q(q=p~f,f≥1,p是一个奇素数)上一类方程组∑_(i=s_(r-1)+1~(s_r)∑_(j=1)~(m_i-m_(i-1))a_(m_(i-1)+j)x_1~(d_m(i-1)+j,1)…x_(n_i)~d_(m_(i-1)+j,n_i)=b_r,r=1,…,k当指数满足一定条件时,在F~(n_s_k)上解数的一个直接公式,这里d_(ij)＞0,a_i∈F~*,b_i∈F,0= s_0＜s_1＜…＜s_k,0=m_0＜m_1＜…＜m_(s_k),0=n_0＜n_1＜…＜n_(s_k), m_1≤n_1,…,m_(s_k)≤n_(s_k)．     

On Constacyclic Codes over Z_(p_1p_2…p_t)

Let t ≥ 2 be an integer, and let _(p_1, ···, p_t)be distinct primes. By using algebraic properties, the present paper gives a sufficient and necessary condition for the existence of non-trivial self-orthogonal cyclic codes over the ring Z_(p_1p_2···p_t)and the corresponding explicit enumerating formula. And it proves that there does not exist any self-dual cyclic code over Z_(p_1p_2···p_t).     

Pseudop olarity of Generalized Matrix Rings over a Lo cal Ring

Pseudopolar rings are closely related to strongly π-regular rings, uniquely strongly clean rings and semiregular rings. In this paper, we investigate pseudopolarity of generalized matrix rings K s(R) over a local ring R. We determine the conditions under which elements of K s(R) are pseudopolar. Assume that R is a local ring. It is shown that A ∈ K s(R) is pseudopolar if and only if A is invertible or A2∈ J(K s(R)) or A is similar to a diagonal matrix[u 00 j], where l u-r j and l j-r u are injective and u ∈ U(R) and j ∈ J(R). Furthermore, several equivalent conditions for K s(R)over a local ring R to be pseudopolar are obtained.     

环$F_2 uF_2 u^2F_2$上的循环码及自对偶码

We give the structures of a cyclic code over ring
R = F2 ＋ uF2 ＋ u^2F2 = {0, 1,u, u^2,v, v^2,uv, v^3},
where u^3 = 0, of odd length and its dual code. For the cyclic code, necessary and sufficient conditions for the existence of self-dual code are provided.     

一个递推公式及其应用

本文介绍一个递推公式及其在解题中的广泛应用。1 递推公式设F(n)=a_1x_1~n+a_2x_2~n+…+a_kx_k~n(n≥0,n∈Z),构造以x_1,x_2,…,x_k为根的方程: x~k+m_1x~(k-1)+m_2x~(k-2)+…+m_k=0 我们称这个方程为F(n)的特征方程,则F(n)=a_1x_1~n+a_2x_2~n+…+a_kx_k~n(n≥k,x∈Z)满足下列递推公式:     

Non-symmetric association schemes of symmetric matrices

Let X_n be the set of n×n symmetric matrices over a finite field F_q,where q is a power of an odd prime.For S_1,S_2 ∈ X_n,we define (S_1,S_2)∈ R_0 iff S_1=S_2;(S_1,S_2)∈R_(r,ε)iff S_1-S_2 is congruent to where?=1 or z,z being afixed non-square element of F_q.Then X_n=(X_n,{R_0,R_(r,ε)|1≤r≤n,?=1 or z}) is a non-symmetric association scheme of class 2n on X_n.The parameters of X_n have been computed.And we also prove that X_n is commutative.     

On the Number of Conjugacy Classes in Semi-Products of Certain Finite Groups

Let V be a vector space of n-dimension over the field GF(p) of p elements,where p is a prime. V is also an elementary abelian p-group.Let G be a p'-group oflinear transformations on V.Theorem 1 Let π_v(a_1,a_2) be the number of the common fixed points of a_1and a_2 on V, a_1, a_2 ∈G. Let k(GV) be the number of conjugacy classes in thesemi-product GV (We also denote it by GV) of G and V. Then     

On the Characterization of Cyclic Co des over Ring F2＋uF2＋v F2

In this work, we investigate the cyclic codes over the ring F2＋uF2＋vF2. We first study the relationship between linear codes over F2＋uF2＋vF2 and that over F2. Then we give a characterization of the cyclic codes over F2＋uF2＋vF2. Finally, we obtain the number of the cyclic code over F2＋uF2＋vF2 of length n.     

EXISTENCE PROBLEMS OF ADDITIVE SELECTION MAPS FOR ANOTHER TYPE SUBADDITIVE SET-VALUED MAP

In this paper,we consider the following subadditive set-valued map F:X → P 0 (Y):F r ∑ i=1 x i + s ∑ j=1 x r+ j rF r ∑ i=1 x i r + sF s ∑ j=1 x r+ j s, x i ∈ X,i=1,2,,r + s,where r and s are two natural numbers.And we discuss the existence and unique problem of additive selection maps for the above set-valued map.     

A common property of R(E,F) and B(R~n,R~m) and a new method for seeking a path to connect two operators

Given two Banach spaces E,F,let B(E,F) be the set of all bounded linear operators from E into F,and R(E,F) the set of all operators in B(E,F) with finite rank.It is well-known that B(Rn) is a Banach space as well as an algebra,while B(Rn,Rm) for m = n,is a Banach space but not an algebra;meanwhile,it is clear that R(E,F) is neither a Banach space nor an algebra.However,in this paper,it is proved that all of them have a common property in geometry and topology,i.e.,they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces).Let Σr be the set of all operators of finite rank r in B(E,F) (or B(Rn,Rm)).In fact,we have that 1) suppose Σr∈ B(Rn,Rm),and then Σr is a smooth and path-connected submanifold of B(Rn,Rm) and dimΣr = (n + m)r-r2,for each r ∈ [0,min{n,m});if m = n,the same conclusion for Σr and its dimension is valid for each r ∈ [0,min{n,m}];2) suppose Σr∈ B(E,F),and dimF = ∞,and then Σr is a smooth and path-connected submanifold of B(E,F) with the tangent space TAΣr = {B ∈ B(E,F) : BN(A)-R(A)} at each A ∈Σr for 0 r ∞.The routine methods for seeking a path to connect two operators can hardly apply here.A new method and some fundamental theorems are introduced in this paper,which is development of elementary transformation of matrices in B(Rn),and more adapted and simple than the elementary transformation method.In addition to tensor analysis and application of Thom’s famous result for transversility,these will benefit the study of infinite geometry.