二元Hamming空间中的鉴别码
Discriminating and Identifying Codes in the Binary Hamming Space
摘要：
令$F^n$是二进制$n$ -立方体，或维度为$n$的二进制汉明空间，赋予汉明距离，并且${\cal E}^n$（分别为${\cal O}^n$）是具有偶（分别为奇）权重的向量的集合。对于$r\geq 1$和$x\in F^n$，我们用$B_r(x)$表示半径为$r$和中心为$x$的球。代码$C\subseteq F^n$被称为$r$ -标识集合$B_r(x) \cap C$、$x\in F^n$是否都是非空且不同的。如果集合$B_r(x) \cap C$、$x\in {\cal O}^n$都是非空且不同的，则称代码$C\subseteq {\cal E}^n$是$r$ -判别的。我们证明了这两个定义，这是给一般的图，在汉明空间的情况下是等价的，在以下意义上：对于任何奇数$r$，有一个双射之间的一组$r$ -识别码在$F^n$和一组$r$ -歧视码在$F^{n+1}$。然后，我们扩展了以前的研究在汉明空间中的识别码的最小基数的建设性上界。
英文摘要：
Let $F^n$ be the binary $n$-cube, or binary Hamming space of dimension $n$, endowed with the Hamming distance, and ${\cal E}^n$ (respectively, ${\cal O}^n$) the set of vectors with even (respectively, odd) weight. For $r\geq 1$ and $x\in F^n$, we denote by $B_r(x)$ the ball of radius $r$ and centre $x$. A code $C\subseteq F^n$ is said to be $r$-identifying if the sets $B_r(x) \cap C$, $x\in F^n$, are all nonempty and distinct. A code $C\subseteq {\cal E}^n$ is said to be $r$-discriminating if the sets $B_r(x) \cap C$, $x\in {\cal O}^n$, are all nonempty and distinct. We show that the two definitions, which were given for general graphs, are equivalent in the case of the Hamming space, in the following sense: for any odd $r$, there is a bijection between the set of $r$-identifying codes in $F^n$ and the set of $r$-discriminating codes in $F^{n+1}$. We then extend previous studies on constructive upper bounds for the minimum cardinalities of identifying codes in the Hamming space.
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