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  

A method of constructing linear mapping families to achieve the asymptotic GV bound

作　　者：王晓京[1,2] 彭行一 Xiaojing Wang;Xingyi Peng

机构地区：[1]中国科学院成都计算机应用研究所,成都610041 [2]中国科学院大学计算机科学与技术学院,北京100049

出　　处：《中国科学：数学》2021年第1期225-238,共14页Scientia Sinica：Mathematica

摘　　要：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_(q)^(n)(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.Given a finite field F_(q)(q≥2)and arbitrary positive integers n and k(n>k),the construction methods of linear mapping sequences(algebraic families){σ_(i)}(σ_(i):F_(q)^(k)→F_(q)^(n)(i→∞))over F_(q)have been a central problem of coding theory in information science,which is generally called the algebraic family approach to the Shannon ideal.Until now,finding a better structure of the algebraic family{σ_(i)},and using it to derive asymptotic sequences{[n_(i),k_(i),d_(i)]}of the Shannon good codes,is still an unresolved challenge and ongoing goal.To measure the algebraic family,in addition to seeing whether the construction{σ_(i)}is convenient for implementation by communication engineering(simple construction,low complexity),a vital standard is to see if the asymptotic parameters of sequences{[n_(i),k_(i),d_(i)]}derived by{σ_i}do not decrease under the asymptotic Gilbert-Varshamov bound(asymptotic GV bound),which has attracted the attention of many mathematicians.In this paper,we give a new method by the matrix mapping to generate the algebraic family{σ_i}over any finite field F_(q),and we indicate the asymptotic code sequences{[n_(i),k_(i),d_(i)]}derived by{σ_(i)}are above the asymptotic GV bound.The new approach to construct algebraic families is of great significance to information coding theory and its engineering application.

关 键 词：线性映射族 代数构造 渐近Gilbert-Varshamov界 

分 类 号：O157.4[理学—数学]