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.     

达到渐近GV界的一种线性映射族构造方法——生成Shannon好码渐近序列的新进展

给定有限域F_q(q≥2)、任意正整数n和k(n>k),F_q上的线性映射序列(代数族){σ_i}(σ_i:F_q^k→F_qⁿ(i→∞))的构造方法已经成为信息科学中编码理论的一个中心问题,一般称为实现Shannon理想的代数族途径.迄今为止,发现这种代数族{σ_i}的更好结构,并由它导出Shannon好码渐近序列{[n_i,k_i,d_i]}仍是一个尚未彻底解决的挑战性难题和持续不断的努力目标.衡量这种代数族的好坏,除了看{σ_i}的构造是否有利于通信工程实现(构造简明,执行复杂度低)之外,最重要的一个基本标准是看{σ_i}导出的序列{[n_i,k_i,d_i]}诸参数的渐近极限结果是否不至于衰减到渐近GilbertVarshamov(GV)界之下,该问题吸引了许多数学工作者的关注.本文从矩阵映射的观点给出一种生成任意有限域F_q上代数族{σ_i}的新方法,并表明由{σ_i}导出的渐近码序列{[n_i,k_i,d_i]}可达渐近GV界之上.这种新的代数族生成途径对于信息编码理论及其工程应用都具有很重要的意义.     

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.     

渐近好pm-ary量子码的构造

对任意素数P及自然数m,用代数几何码构造了一列渐近好的pm-ary量子纠错码.     

最小距离固定且优于Gilbert—Varshamov界的码

本文构造了一类GF(q)上的码,其中GF(q)为q个元素的有限域.这些码的冗余取到渐进界r(q,n,7) 4 m,此界优于Gilbert-Varshamov存在界r(q,n,7) 5m.     

一类代数几何码的构造

利用有限域Fq^8(s≥1为正整数，q为素数幂)上代数曲线构造了一类q元线性码，这类线性码是q^8元几何Goppa码的子域子码的子码，同时也是Chaoping Xing，SanLing构造的代数几何码[1]的推广。     

基于有限域上奇异线性空间的码本的新构造

码本广泛应用于码分多址系统用于区分不同用户发出的信号.基于有限域上奇异线性空间构造了一类新的码本.运用奇异线性空间的计数定理,得到了码本的参数,计算了码本的最大互相关振幅,并且给出了最大互相关振幅渐近达到Welch界的条件,证明了所构造的码本是渐近最优码本.     

渐近参数优于 Gilbert-Varshamov 和 Xing 界的非线性码

证明了Gilbert-Varshamov 和 Xing界在它们的交点附近,可以被有限域代数曲线上的非线性码所显著改进.     

基于循环码的常重复合码的构造

研究给出了一类基于循环码的常重复合码的构造,并利用指数和计算其参数.与相关的常重复合码相比,该码具有更多的码字,且渐近性较好.     

一类超W-代数上的Poisson超结构

源于Poisson几何的Poisson代数同时具有代数结构和李代数结构,其乘法与李代数乘法满足Leibniz法则.超W-代数是复数域C上的无限维李超代数.主要研究一类超W-代数上的Poisson超结构.     

Constructions and bounds on linear error-block codes

We obtain new bounds on the parameters and we give new constructions of linear error-block codes. We obtain a Gilbert–Varshamov type construction. Using our bounds and constructions we obtain some infinite families of optimal linear error-block codes over . We also study the asymptotic of linear error-block codes. We define the real valued function α _q,m,a (δ), which is an analog of the important real valued function α _q (δ) in the asymptotic theory of classical linear error-correcting codes. We obtain both Gilbert–Varshamov and algebraic geometry type lower bounds on α _q,m,a (δ). We compare these lower bounds in graphs.      

On maximal Spherical codes II

We consider spherical codes attaining the Levenshtein upper bounds on the cardinality of codes with prescribed maximal inner product. We prove that the even Levenshtein bounds can be attained only by codes which are tight spherical designs. For every fixed n ≥ 5, there exist only a finite number of codes attaining the odd bounds. We derive different expressions for the distance distribution of a maximal code. As a by-product, we obtain a result about its inner products. We describe the parameters of those codes meeting the third Levenshtein bound, which have a regular simplex as a derived code. Finally, we discuss a connection between the maximal codes attaining the third bound and strongly regular graphs. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 316–326, 1999     

The triple distribution of codes and ordered codes

We study the distribution of triples of codewords of codes and ordered codes. Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (8) (2005) 2859–2866] used the triple distribution of a code to establish a bound on the number of codewords based on semidefinite programming. In the first part of this work, we generalize this approach for ordered codes. In the second part, we consider linear codes and linear ordered codes and present a MacWilliams-type identity for the triple distribution of their dual code. Based on the non-negativity of this linear transform, we establish a linear programming bound and conclude with a table of parameters for which this bound yields better results than the standard linear programming bound.     

On Upper Bounds for Minimum Distance and Covering Radius of Non-binary Codes

We consider upper bounds on two fundamental parameters of a code; minimum distance and covering radius. New upper bounds on the covering radius of non-binary linear codes are derived by generalizing a method due to S. Litsyn and A. Tietäväinen lt:newu and combining it with a new upper bound on the asymptotic information rate of non-binary codes. The upper bound on the information rate is an application of a shortening method of a code and is an analogue of the Shannon-Gallager-Berlekamp straight line bound on error probability. These results improve on the best presently known asymptotic upper bounds on minimum distance and covering radius of non-binary codes in certain intervals.     

Nonlinear codes with asymptotic parameters better than the Gilbert-Varshamov and the Xing Bounds

In this paper, we show that the Gilbert-Varshamov and the Xing bounds can be improved significantly around two points where these two bounds intersect by nonlinear codes from algebraic curves over finite fields.     

Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences

This paper is concerned with two applications of bases of Riemann-Roch spaces. In the first application, we define the floor of a divisor and obtain improved bounds on the parameters of algebraic geometry codes. These bounds apply to a larger class of codes than that of Homma and Kim (J. Pure Appl. Algebra 162 (2001) 273). Then we determine explicit bases for large classes of Riemann-Roch spaces of the Hermitian function field. These bases give better estimates on the parameters of a large class of m-point Hermitian codes. In the second application, these bases are used for fast implementation of Xing and Niederreiter's method (Acta. Arith. 72 (1995) 281) for the construction of low-discrepancy sequences.     

Exact bound on the number of limit cycles arising from a periodic annulus bounded by a symmetric heteroclinic loop

In this paper, the bound on the number of limit cycles by Poincare bifurcation in a small perturbation of some seventh-degree Hamiltonian system is concerned. The lower and upper bounds on the number of limit cycles have been obtained in two previous works, however, the sharp bound is still unknown. We will employ some new techniques to determine which is the exact bound between $3$ and $4$. The asymptotic expansions are used to determine the four vertexes of a tetrahedron, and the sharp bound can be reached when the parameters belong to this tetrahedron.     

The New Minimum Distance Bounds of Goppa Codes and Their Decoding

A couple of new lower bounds of the minimum distance of Goppa codes is derived, using an extended field code for a Goppa code which contains the Goppa code as its subfield-subcode. Also presented are procedures for both error-only and error-and-erasure decoding for Goppa codes up to the new lower bounds, based on the Berlekamp-Massey algorithm and the Feng-Tzeng multisequence shift-register synthesis algorithms which have been used for decoding cyclic codes up to the BCH and HT(Hartmann-Tzeng) bounds.