关于二元超立方体中识别码的大小
On the size of identifying codes in binary hypercubes
摘要：
我们考虑在二进制汉明空间F^n中识别码，即，在二元超立方体中。识别码的概念是由Karpovsky，Chakrabarty和Levitin在1998年提出的。目前，该主题形成了自己的主题，具有几种可能的应用，例如传感器网络。设C是F^n的一个子集。对于F^n的任意子集X，用I_r（X）=I_r（C;X）表示C中距离X中至少一个x为r的元素的集合。如果集合I_r（X）对于所有大小不超过l的子集X都是不同的，则称C为（r，<= l）-标识码。我们估计了在固定l和r/n收敛到（0，1）中某个数ρ的情况下这类码的最小尺寸。我们进一步证明了对于任意固定的l和略小于n/2的r，存在一个大小为O（n^{3/2}）的码，并且对于l=2，给出了一个小码的显式构造，其中r是n/2-1的整数部分（最大可能值）.
英文摘要：
We consider identifying codes in binary Hamming spaces F^n, i.e., in binary hypercubes. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks. Let C be a subset of F^n. For any subset X of F^n, denote by I_r(X)=I_r(C;X) the set of elements of C within distance r from at least one x in X. Now C is called an (r,<= l)-identifying code if the sets I_r(X) are distinct for all subsets X of size at most l. We estimate the smallest size of such codes with fixed l and r/n converging to some number rho in (0,1). We further show the existence of such a code of size O(n^{3/2}) for every fixed l and r slightly less than n/2, and give for l=2 an explicit construction of small such codes for r the integer part of n/2-1 (the largest possible value).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
