Discriminating and Identifying Codes in the Binary Hamming Space

Abstract

Let $F^n$ be the binary $n$-cube, or binary Hamming space of dimension $n$, endowed with the Hamming distance, and ${\mathcal{E}}^n$ (respectively, ${\mathcal{O}}^n$) the set of vectors with even (respectively, odd) weight. For $r\ge 1$ 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 $C\subseteq {\mathcal{E}}^n$ is said to be $r$-discriminating if the sets $B_r(x)\cap C$, $x\in {\mathcal{O}}^n$, 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.