代数函数域上的局部恢复码

假设C是有限域Fq上的[n,k]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)线性码,这里r是较小的数.本文在代数函数域上构造出了局部恢复码,它的码长不受字符集大小的限制,实际上,它的码长可以远远大于字符集的大小;并将此方法应用于广义Hermite函数域,得到了一类广义Hermite函数域上的局部恢复码.进一步地,通过构造子码的方式改进了广义Hermite函数域上的局部恢复码的最小距离的下界.     

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,k]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双恢复集,并且码长可以突破字符集的大小的限制.结果表明,此构造方法得出的最小距离下界明显地改进了Alexander Barg的最小距离的下界.     

关于mr(2,q)的一个新值

编码理论中关于寻找某一线性码的最大长度涉及到有限射影空间中关于t-blockingsets,(k,r-ares和caps集所含元素的个数的问题,本文研究了(k,r)-arc集的元素的个数,找到了使得(k,r)-arc集存在的最大k值,即mr(2,q)的一个新值,丰富了编码理论的相关内容。     

Sidon空间和循环子空间码的构造

本文首先给出Sidon空间和Sidon集的构造,用这些Sidon空间我们构造一些码字个数是τ·(q~n-1)/(q-1)并且最小距离是2k-2的循环子空间码,其中τ是一个正整数.进一步,我们给出码字个数是2τ·(q~n-1)/(q-1)并且最小距离2k-2的循环子空间码.     

一类BSC码的汉明距离界

根据二元叠加码(Binary Superimposed Code)M_q(n,k,d)的定义研究了这个BSC码任意两个码字的汉明(Hamming)距离上下界,并由此给出了它的平均汉明距离和均方差的界.     

几类最优避免冲突码

长度为n重量为w的避免冲突码C是群Z_n的w元子集族,满足对任意的x,y∈C,x≠y有d*(x)∩d*(y)=Φ,其中d*(x)={a-b(mod n):a,b∈x,a≠b}.避免冲突码适用于无反馈时隙同步多址冲突信道.C中的元素称为码字,C中所包含的码字的个数称为码的容量,它是系统中所支持的潜在用户的个数.利用已有的3种构造方法给出了重量在4到10之间的一些最优CAC(p,w)码类.     

一种OLS/OOC二维光正交码的构造与性能分析

利用不同的序列作为波长跳频序列和时间扩频序列可以构造出不同的二维光正交码在众多文献中已有所报道.在经过正交拉丁方(OLS)与跳频序列的相关性研究之后.做了以下主要工作:首先,将正交拉丁方(OLS)序列作为波长跳频序列,结合一维时间扩频序列(OOC),构造了一种OLS/OOC二维光正交码.然后,本文对构造的OLS/OOC进行了多种性能仿真和分析.相对于PC/OOC、OCFHC/OOC等二维光正交码而言,OLS/OOC的波长数并不局限于素数,更能充分利用MWOCDMA系统中的有效波长数.仿真和分析表明:码字具有很好的相关性能,码字容量直逼理论极限,为一种渐近最优二维光正交码.     

4维3元断链码的重量谱

GF(q)上[n,k;q]线性码C的重量谱为序列(d1,d2,…,dk),这里dr是C的r维子码的最小支持重量(1≤r≤k).用有限射影几何方法确定了满足含有2个邻接断点的断链条件的4维3元线性码的重量谱.     

具有多恢复集的局部恢复码的构造

局部恢复码(LRC)是指码字的任意一个坐标位置的值都可以通过较少的r个其它位置的值来恢复.构造具有多恢复集的LRC码是为了解决通信中节点访问的拥堵问题.基于代数函数域上的自同构群,利用其子群的内直积构造多恢复集,进而构造出具有多恢复集的局部恢复码.此外,在恢复码的构造中,赋值空间的生成集是显式表达的,这使得码的维数、最小距离等参数计算非常方便.     

重量为3的2q周期强避免冲突中心码的容量

强避免冲突码适用于无反馈异步多址冲突信道.码中所包含的码字的个数称为码的容量,它是系统中所支持的潜在用户的个数.给出了重量ω=3,长度为2q的最优码的构造方法及其容量.     

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...     

On Linear Programming Bounds for Spherical Codes and Designs

We investigate universal bounds on spherical codes and spherical designs that could be obtained using Delsartes linear programming methods. We give a lower estimate for the LP upper bound on codes, and an upper estimate for the LP lower bound on designs. Specifically, when the distance of the code is fixed and the dimension goes to infinity, the LP upper bound on codes is at least as large as the average of the best known upper and lower bounds. When the dimension n of the design is fixed, and the strength k goes to infinity, the LP bound on designs turns out, in conjunction with known lower bounds, to be proportional to k^n-1.     

R~n中构形和单纯复形的一个最优度量不等式

对于R~n中一般位置的点构形,定义了第r个极小凸包距离的概念,证明了极小凸包距离和极小点-超平面距离之间的一个最优不等式.该不等式的一个直接推论是:对于R~n中一个k-维单纯复形K,我们能用其顶点集的极小点-超平面距离下估计K的Gromov-Guth厚度.进一步,在每一个维数k,构造了例子说明该下界几乎是最优的.     

一类循环码的极小距离

循环码的极小距离大于或等于BCH界。本文考虑的是极小距离等于BCH界的特殊情形。利用一类自反循环码的事实。证明了使循环码的极小距离等于其BCH界的两个充分条件；并指出极小距离等于任意给定值，维数任意大的循环码可以构造。     

Proof of Conjectures on the True Dimension of Some Binary Goppa Codes

There is a classical lower bound on the dimension of a binary Goppa code. We survey results on some specific codes whose dimension exceeds this bound, and prove two conjectures on the true dimension of two classes of such codes.Part of this work has been presented at the Sixth International Conference on Finite Fields and Applications, Oaxaca, Mexico, May 2001.AMS classification: 94B65     

On the Geometry of Metric Measure Spaces with Variable Curvature Bounds

Motivated by a classical comparison result of J. C. F. Sturm, we introduce a curvature-dimension condition CD(k, N) for general metric measure spaces, variable lower curvature bound \(k\) and upper dimension bound \(N\ge 1\). In the case of non-zero constant lower curvature, our approach coincides with the celebrated condition that was proposed by Sturm (Acta Math 196(1):133–177, 2006). We prove several geometric properties as sharp Bishop–Gromov volume growth comparison or a sharp generalized Bonnet–Myers theorem (Schneider’s Theorem). In addition, the curvature-dimension condition is stable with respect to measured Gromov–Hausdorff convergence, and it is stable with respect to tensorization of finitely many metric measure spaces provided a non-branching condition is assumed. We also briefly describe possible extensions for variable dimension bounds.     

Gårding inequality in large dimension

We give a lower bound for a pseudodifferential operator in a large dimensional setting, with a nonnegative real-valued symbol. The lower bound is explicitly written as a function of the semiclassical parameter, and of parameters used for the estimations of some derivatives of the symbol, without any constant depending on the dimension. This is an analog, in large dimension, of the sharp Gårding inequality.     

Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain

In this paper we introduce a weighted Cheeger constant and show that the gap between the first two eigenvalues of a Riemannian manifold given Dirichlet conditions can be bounded from below in terms of this constant. When the Riemannian manifold is a bounded Euclidean domain satisfying an interior rolling sphere condition we give an estimate on the weighted Cheeger constant in terms of the rolling sphere radius, volume, a bound on the principal curvatures of the boundary and the dimension. This yields a lower bound on the nontrivial gap for Euclidean domains. S-Y. Cheng’s research partially supported by the CUHK direct grant A/C # 220600260. K. Oden’s research partially supported by the Department of Education Graduate Fellowship     

A lower bound for Hausdorff dimensions of harmonic measures on negatively curved manifolds

Employing the methods of [KL], a lower bound for Hausdorff dimension of harmonic measures on negatively curved manifolds is derived yielding, in particular, that if the curvature tends to a constant then the above Hausdorff dimension tends to the dimension of the sphere at infinity. Supported by U.S.-Israel BSF.