Improved Lower Bound for Locating-Dominating Codes in Binary Hamming Spaces

Abstract

In this article, we study locating-dominating codes in binary Hamming spaces ${\mathbb{F}}^n$. Locating-dominating codes have been widely studied since their introduction in 1980s by Slater and Rall. They are dominating sets suitable for distinguishing vertices in graphs. Dominating sets as well as locating-dominating codes have been studied in Hamming spaces in multiple articles. Previously, Honkala et al. (2004) have presented a lower bound for locating-dominating codes in binary Hamming spaces. In this article, we improve the lower bound for all values $n\ge 10$. In particular, when $n=11$, we manage to improve the previous lower bound from $309$ to $317$. This value is very close to the current best known upper bound of $320$.