The Nonexistence of Ternary [38, 6, 23] Codes

It has been shown by Bogdanova and Boukliev [1] that there exist a ternary [38,5,24] code and a ternary [37,5,23] code. But it is unknown whether or not there exist a ternary [39,6,24] code and a ternary [38,6,23] code. The purpose of this paper is to prove that (1) there is no ternary [39,6,24] code and (2) there is no ternary [38,6,23] code using the nonexistence of ternary [39,6,24] codes. Since it is known (cf. Brouwer and Sloane [2] and Hamada and Watamori [14]) that (i) n₃(6,23) = 38> or 39 and d₃(38,6) = 22 or 23 and (ii) n₃(6,24) = 39 or 40 and d₃(39,6) = 23 or 24, this implies that n₃(6,23) = 39, d₃(38,6) = 22, n₃(6,24) = 40 and d₃(39,6) = 23, where n3<>(k,d) and d<>3(n,k) denote the smallest value of n and the largest value of d, respectively, for which there exists an [n,k,d] code over the Galois field GF(3).     

A Necessary and Sufficient Condition for the Existence of Some Ternary [n, k, d] Codes Meeting the Greismer Bound

Let k and d be any integers such that k 4 and . Then there exist two integers and in {0,1,2} such that . The purpose of this paper is to prove that (1) in the case k 5 and (,) = (0,1), there exists a ternary code meeting the Griesmer bound if and only if and (2) in the case k 4 and (,) = (0,2) or (1,1), there is no ternary code meeting the Griesmer bound for any integers k and d and (3) in the case k 5 and , there is no projective ternary code for any integers k and such that 1k-3, where and for any integer i 0. In the special case k=6, it follows from (1) that there is no ternary linear code with parameters [233,6,154] , [234,6,155] or [237,6,157] which are new results.     

On the nonexistence of ternary linear codes attaining the Griesmer bound

Designs, Codes and Cryptography - An $$[n,k,d]_q$$ code is a linear code of length n, dimension k and minimum weight d over the field of order q. It is known that the Griesmer bound is attained for...     

Niveau hermitien de Certaines algèbres de quaternions

Let D [d] =(a,b/F) a quaternion divisior algebra over a field F of characteristic ? 2. Denote 1, i, j , k the basis of D, such that i²[d] n, j²[d] b, ij [d] -ji [d] k and A :D → D the involution given by i [d] -i, j [d] j (and k [d] k). In [LE] D. LEWIS asks the following question :Does there exist a quadratic Pfister form [S p. 721 [d] such that the hermitian form [d] [d] D is isotropic over (D, [d]) but not hyperbolic &; In this note, we show that the answer of this question is negative, so that the hermitien level [§I], when it is finite, of (D, A) is a power of two. This result holds for quaternion algebras with standard involution [LE].     

二元叠加码Mq(i:n,k,d)的检纠错性质

对于自然数i,d,k,n,0q(i:n,k,d)是一个基于有限域Fq上n维向量空间中子空间的相交关系的二元叠加码,研究了二元叠加码M_q(i:n,k,d)任意列之间的汉明距离,给出了它的检错性和纠错性.     

Elements in one-sided unit regular rings

Let R be regular. We show that the following are equivalent:(1) R is a one sided unit regular ring. (2) For every x [euro] R, there exist an idempotente and a right or left invertible u such that x [d] eu or x [d] ue. (3) For every x [euro] R,there exists a right or left invertible u such that xu or ux is an idempotent. Moreover, we give some characterizations of one-sided unit regular rings by group inverses.     

循环逆M-矩阵的判定

给出了循环逆M-矩阵的判定方法:如果一个n×n非负循环矩阵非正且不等于c0I,若存在一个正整数K是n的真因子,使得cjk>0,j=0,1[,…,n-k]k,其余的ci等于0且Circ[c0,ck,…,cn-k]是一个逆M-矩阵,则A是一个逆M-矩阵.     

仿射合成代数的三角分解

设是一个仿射箭图,它的极小虚单根为ｎ．设ｋ是一个有限域,记Ａ＝ｋ为ｋ上关于箭图的路代数,而记Ｃ(Ａ)为关于Ａ的合成代数．由Ｃ．Ｒｉｎｇｅｌ和Ｊ．Ｇｒｅｅｎ的工作,Ｃ(Ａ)揭示了Ａ的表示与量子群有密切的关系．文[１１]证明了对应于Ａ的不可分解表示可以分成预投射,正则,和预内射三个部分,Ｃ(Ａ)具有一个三角分解．[１１]中的证明需要假设维数向量为ｎ的拟单模存在,而对于|ｋ|＝２,是ｎ型和ｍ型(ｍ＝６,７,８)的情形,此假设不满足,本文的目的是给出一个简化的,而且不需要前面所提假设的证明．由此,得到一个与域ｋ无关的Ｃ(Ａ)的三角分解．     

A construction of mutually disjoint Steiner systems from isomorphic Golay codes

It is well known that the extended binary Golay [24,12,8] code yields 5-designs. In particular, the supports of all the weight 8 codewords in the code form a Steiner system S(5,8,24). In this paper, we give a construction of mutually disjoint Steiner systems S(5,8,24) by constructing isomorphic Golay codes. As a consequence, we show that there exists at least 22 mutually disjoint Steiner systems S(5,8,24). Finally, we prove that there exists at least 46 mutually disjoint 5-(48,12,8) designs from the extended binary quadratic residue [48,24,12] code.     

An optimal ternary [69, 5, 45] code and related codes

A ternary [69, 5, 45] code is constructed, thus solving the problem of finding the minimum length of a ternary code of dimension 5 and minimum distance 45. Furthermore, this code is shown to be a unique two-weight code with weight enumerator 1+210Z⁴⁵+32Z⁵⁴. It is also shown that a ternary [70, 6, 45] code, which would have been a projective two-weight code giving rise to a new strongly regular graph, does not exist. In order to prove the main results, the uniqueness of some other optimal ternary codes with specified weight enumerators is also established.     

带弦圈的最小2宽直径

设k为正整数,G是简单k连通图.图G的k宽直径,dk(G),是指最小的整数ι使得对任意两不同顶点x,y∈V(G),都存在k条长至多为ι的内部不交的连接x和y的路.用C(n,t)表示在圈Gn上增加t条边所得的图.定义h(n,t):min{d2(C(n,t))}.本文给出了h(n,2)=[n/2].而且,给出了当t较大时h(n,t)的界.     

Separable polynomials over finite dimensional algebras

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality n^d or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly n^d distinct zeros in [Ktilde] ?_k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ?_k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.     

A New Table of Binary/Ternary Mixed Covering Codes

A table of upper bounds for K3,2(n1,n2;R), the minimum number of codewords in a covering code with n1 ternary coordinates, n2 binary coordinates, and covering radius R, in the range n = n1 + n2 13, R 3, is presented. Explicit constructions of codes are given to prove the new bounds and verify old bounds. These binary/ternary covering codes can be used as systems for the football pool game. The results include a new binary code with covering radius 1 proving K2(13,1) 736, and the following upper bound for the football pool problem for 9 matches: K3(9,1) 1356.     

A NOTE ON LEFT STROUGLY NIL AND LEFT STROUGLY NILPOTENT

Abstract

In [2] van der Walt called a left ideal L of a ring A, left strongly nil, if given 1 ε L and k ε K, K a left ideal. there is an n such that (1+k)ⁿ ε K. L is called left strongly nilpotent if for any left ideal K there exists an m such that (L+K)^m ? K. In this paper we will prove that if A is a left artinian ring (not necessarily with unity) then every left strongly nil left ideal is left strongly nilpotent. This result is a generalization of the main theorem of [2].     

Smoothing the Moment Estimator of the Extreme Value Parameter

Let {X _n be a sequence of i.i.d. random variables whose common distribution F belongs to the domain of attraction of an extreme value law. A semi-parametric estimator of the extreme value parameter is the Dekkers, Einmahl and de Haan [8] moment estimator. Practical use of this estimator requires the problematic choice of a number k=k(n) of upper order statistics and there are few reliable guidelines for this choice. An averaging or smoothing technique is proposed for this estimator yielding a less volatile function of k which in practice aids estimation.     

Hamiltonian[k,k+1]-因子

本文考虑n/2-临界图中Hamiltonian[k,k+1]-因子的存在性。Hamiltonian[k,k+1]-因子是指包含Hamiltonian圈的[k,k+1]-因子;给定阶数为n的简单图G,若δ(G)≥n/2而δ(G\e)相似文献   

一类具有双恢复集的局部恢复码的构造

假设C是有限域Fq上的[n,κ]线性码,如果码字的每个坐标是其它至多r个坐标的函数,称C是(n,k,r)局部恢复码,这里r是较小的数.在分布式存储系统中,具有多个恢复集的局部恢复码使得数据在系统中更具实际意义,因为它可以避免热数据的频繁访问.引入代数函数域、特别是Hermite函数域去构造局部恢复码,这类局部恢复码具有...     

二元叠加码M_q(n,k,d)码的平均汉明距离和均方差

根据二元叠加码(Binary Superimposed Code)M_q(n,k,d)的定义及有限域F_q上n维向量空间的k维子空间的维数性质定义了一个高斯组合函数,利用这个组合函数研究了M_q(n,k,d)码的平均汉明(Hamming)距离和它的均方差问题,给出了计算公式.     

有限个互素因子链上幂GCD矩阵与幂LCM矩阵的行列式的整除性

设S={x1,x2,...,xn}是由n个不同的正整数组成的集合,并设a为正整数.如果一个n阶矩阵的第i行j列元素是S中元素xi和xj的最大公因子的a次幂(xi,xj)a,则称该矩阵为定义在S上的a次幂最大公因子(GCD)矩阵,用(Sa)表示;类似定义a次幂LCM矩阵[Sa].如果存在{1,2,...,n}上的一个置换σ使得xσ(1)|xσ(2)|···|xσ(n),则称S为一个因子链.如果存在正整数k,使得S=S1∪S2∪···∪Sk,其中每一个Si(1ik)均为一个因子链,并且对所有的1i=jk,Si中的每个元素与Sj中的每个元素互素,则称S由有限个互素因子链构成.本文中,设S由有限个互素的因子链构成,并且1∈S.我们首先给出幂GCD矩阵与幂LCM矩阵的行列式的公式,然后证明:如果a|b,则det(Sa)|det(Sb),det[Sa]|det[Sb],det(Sa)|det[Sb].最后我们指出:如果构成S的有限个因子链不互素,则此结论一般不成立.     

关于Erdos-Jacobson-Lehel 问题的门槛

设σ(k,n）表示最小的正整数m，使得对于每个n项正可图序列，当其项和至少为m时，有一个实现含k 1个顶点的团作为其子图。Erdos等人猜想：σ(k,n）＝（k-1)(2n-k) 2.Li等人证明了这个猜想对于k≥5，n≥（^k2））＋3是对的，并且提出如下问题：确定最小的整数N（k），使得这个猜想对于n≥N(k）成立。他们同时指出：当k≥5时，[5k-1/2]≤N(k）≤（^k2) 3.Mubayi猜想：当k≥5时，N(k）＝[5k-1/2]。在本文中，我们证明了N（8）＝20，即Mubayi猜想对于k=8是成立的。