Discriminating and Identifying Codes in the Binary Hamming Space

TL;DR

This paper explores the properties of identifying and discriminating codes in binary Hamming spaces, establishing a bijection between them for odd radii and extending bounds on their minimal sizes.

Contribution

It proves the equivalence of r-identifying and r-discriminating codes in Hamming space for odd r, and extends bounds on their minimal cardinalities.

Findings

Established a bijection between r-identifying and r-discriminating codes for odd r.

Extended previous upper bounds on the minimum sizes of identifying codes.

Unified the understanding of these codes in the context of binary Hamming spaces.

Abstract

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.