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.     

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.     

关于Goppa 几何码最小距离的一个新方法

引进一个关于Goppa几何码(代数几何码)最小距离界的一个新方法.应用Maharaj的思想(即用显示基来近似表达Riemann-Roch空间)到Goppa几何码的最小距离的界上去.通过厄米特曲线上的代数几何码的一类例子,来证明标准的几何码的下界在某些情形下可以被显著地改进.进一步地,我们给出了这些码的最小距离上界,并说明了我们的下界非常接近这个上界.     

有限域上一类等维码的下界

等维码凭借其在随机线性网络编码中的良好的差错控制得到广泛研究,对于给定维数和最小距离的等维码所含码字的最大个数目前还没有一般性结果.Tuvi Etzion和Alexander Vardy给出了一定等维码所含码字最大个数的上界和下界,首先利用对偶空间构造等维码C(n,M,2k,k),达到了此类码所含码字的下界,然后具体构造了最优等维码C(7,41,4,2).     

最优循环局部修复码的构造

局部修复码是分布式存储编码领域的一个热门研究方向,具有局部性(r,δ)的局部修复码对丢失节点修复具有更高的效率.文章利用限域F_q上循环码构造了两类具有局部性(r,δ),最小距离为d=δ+4,码长n为q+1倍数的最优局部修复码.     

代数函数域的一些Artin-Schreier型扩张相伴的Riemann-Roch空间

阐明给定代数函数域上一些除子的Riemann-Roch空间是代数几何码构造的基础.给出代数函数域的一些Artin-Schreier型扩张的Riemann-Roch空间的一组基,并应用于编码理论,得到F_(16)上参数分别是[54,43,5],[54,41,7],[54,40,8]的代数几何码.     

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.     

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.     

代数几何码的测试

性质测试是90年代开始由多种研究引发的,GF（q）^n中一个线性码C称为局部可测试的,当且仅当存在一个随机化算法,使得只要输入任一个GF（q）^n中向量的很少一部分坐标（一般而言是常数个坐标）,这个随机化算法就可以很高的概率判定此向量是否是C中码字．Blum,Luby和Rubinfeld由于和概率可验证证明的紧密关系研究了码的局部可测试性,然而怎样刻画局部可测试码是一个复杂且甚具挑战性的问题．对Reed—Solomon（RS）码、Reed．Muller（RM）码、循环码、BCH码的对偶码及代数几何码的迹子码,已经研究了局部可测试问题．在本文中我们给出了代数几何码的线性参数的测试子,并证明了在一个不太强的限制条件下代数几何码不是局部可测试的．     

关于两个特殊环曲面码的参数

扩充了Hansen的关于几个环曲面码的结果,给出两个新的环曲面码,并且用上同调的方法来计算码的维数,用相交理论的方法来估计最小距离,主要目的是为了证明对应两个不同的多边形可能具有相同的参数.最后,证明了这些新的码在某些有限域上是最好的码.     

GOPPA CODES FROMARTIN-SCHREIER FUNCTION FIELDS

ＧＯＰＰＡＣＯＤＥＳＦＲＯＭＡＲＴＩＮ－ＳＣＨＲＥＩＥＲＦＵＮＣＴＩＯＮＦＩＥＬＤＳ￥ＨＡＮＷＥＮＢＡＯ（ＤｅｐａｔｍｅｌｌｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙｔＣｈｅｎｇｄｕ６１００６４，Ｓｉｃｈｕａｎ，Ｃｈｉｎａ．）Ａ...     

A symmetric Roos bound for linear codes

The van Lint-Wilson AB-method yields a short proof of the Roos bound for the minimum distance of a cyclic code. We use the AB-method to obtain a different bound for the weights of a linear code. In contrast to the Roos bound, the role of the codes A and B in our bound is symmetric. We use the bound to prove the actual minimum distance for a class of dual BCH codes of length q²−1 over Fq. We give cyclic codes [63,38,16] and [65,40,16] over F₈ that are better than the known [63,38,15] and [65,40,15] codes.     

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。     

Failed disk recovery in double erasure RAID arrays

Reliability is a major concern in the design of large disk arrays. In this paper, we examine the effect of encountering more failures than that for which the RAID array was initially designed. Erasure codes are incorporated to enable system recovery from a specified number of disk erasures, and strive beyond that threshold to recover the system as frequently, and as thoroughly, as is possible. Erasure codes for tolerating two disk failures are examined. For these double erasure codes, we establish a correspondence between system operation and acyclicity of its graph model. For the most compact double erasure code, the full 2-code, this underlies an efficient algorithm for the computation of system operation probability (all disks operating or recoverable).When the system has failed, some disks are nonetheless recoverable. We extend the graph model to determine the probability that d disks have failed, a of which are recoverable by solving one linear equation, b of which are further recoverable by solving systems of linear equations, and d−a−b of which cannot be recovered. These statistics are efficiently calculated for the full 2-code by developing a three variable ordinary generating function whose coefficients give the specified values. Finally, examples are given to illustrate the probability that an individual disk can be recovered, even when the system is in a failed state.     

A sequence of one-point codes from a tower of function fields

We construct a sequence of one-point codes from a tower of function fields whose relative minimum distances have a positive limit. Our tower is characterized by principal divisors. We determine completely the minimum distance of the codes from the first field of our tower. These results extend those of Stichtenoth [IEEE Trans Inform Theory (1988), 34(15):1345–1348], Yang and Kumar [Lecture Notes in Mathematics, 1518, (1991), Springer-Verlag, Berlin Hidelberg New York, pp. 99–107], and Garcia [Comm. Algebra, 20(12): 3683–3689]. As an application, we show that the minimum distance corresponds to the Feng–Rao bound.     

Codes of Small Defect

The parameters of a linear code C over GF(q) are given by [n,k,d], where n denotes the length, k the dimension and d the minimum distance of C. The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n-k+1 Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which have minimum distance close to the Singleton bound. Of particular interest is the class of almost MDS codes, i.e. codes for which d=n-k. We will present a condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code. This extends a result of Dodunekov and Landgev Dodunekov. Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost MDS codes up to one parameter. As a consequence we obtain surprising combinatorial relations in such codes. This leads, among other things, to an answer to a question of Assmus and Mattson 5 on the existence of self-dual [2d,d,d]-codes which have no code words of weight d+1. Actually there are more codes than Assmus and Mattson expected, but the examples which we know are related to the expected ones.     

On Ordinary Words of Standard Reed-Solomon Codes over Finite Fields

Reed-Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm.Usually we use the maximum likelihood decoding(MLD) algorithm in the decoding process of Reed-Solomon codes.MLD algorithm relies on determining the error distance of received word.Dür,Guruswami,Wan,Li,Hong,Wu,Yue and Zhu et al.got some results on the error distance.For the Reed-Solomon code C,the received word u is called an ordinary word of C if the error distance d(u,C) =n-deg u(x) with u(x) being the Lagrange interpolation polynomial of u.We introduce a new method of studying the ordinary words.In fact,we make use of the result obtained by Y.C.Xu and S.F.Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed-Solomon codes over the finite field of q elements.This completely answers an open problem raised by Li and Wan in[On the subset sum problem over finite fields,Finite Fields Appl.14 (2008) 911-929].     

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.     

The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes

One of the most important problems of coding theory is to construct codes with best possible minimum distances. Recently, quasi-cyclic (QC) codes have been proven to contain many such codes. In this paper, we consider quasi-twisted (QT) codes, which are generalizations of QC codes, and their structural properties and obtain new codes which improve minimum distances of best known linear codes over the finite fields GF(3) and GF(5). Moreover, we give a BCH-type bound on minimum distance for QT codes and give a sufficient condition for a QT code to be equivalent to a QC code.