On the size of identifying codes in binary hypercubes

TL;DR

This paper investigates the minimal size of identifying codes in binary hypercubes, providing asymptotic estimates, existence results, and explicit constructions for certain parameters relevant to applications like sensor networks.

Contribution

It offers new bounds and explicit constructions for (r,<= l)-identifying codes in binary hypercubes, advancing understanding of their minimal sizes.

Findings

Existence of codes of size O(n^{3/2}) for fixed l and r< n/2

Asymptotic estimates for minimal code sizes as n grows

Explicit constructions for specific parameters, e.g., l=2, r≈ n/2-1

Abstract

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).