极大线性无关组的构造

Construction of Maximal Linearly Independent Arrays

记Vr(F2)是有限域F2上的r维向量空间,令S是Vr(F2)的含有n个向量的子集合,如果S中任意t个向量在有限域F2上都线性无关,则称S是n元t无关组。称S为极大n元t无关组,是指在所有的n元t无关组中,S的向量个数达到最大值,把这个极大值记为M(r,t)。n元t无关组在密码、纠错码理论以及区组设计等方面有着重要的应用。利用若干线性纠错码的结果,给出了关于M(r,t)的若干下界。

Let Vr（F2） be the vector space of dimension r over finite field F2,and S be a subset of Vr（F2）,consisting of n nonzero vectors,such that the arbitrary t vectors of S are linearly independent over F2,then M is called a（n,t）-linearly independent array of length n over Vr（F2）.The（n,t）-linearly independent array that has the maximal number of elements is called the maximal（r,t）-linearly independent array,and the maximal number is denoted by M（r,t）.The（n,t）-linearly independent array has many applications in cryptography,coding theory and block design and so on.In this paper,some low bounds on M（r,t） by linear coding theory is given.