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

TL;DR

This paper improves the lower bounds for locating-dominating codes in binary Hamming spaces, narrowing the gap between known bounds and advancing understanding of code size constraints.

Contribution

The authors provide a new, tighter lower bound for locating-dominating codes in binary Hamming spaces for all n ≥ 10, notably improving the bound for n=11.

Findings

Lower bound for n=11 increased from 309 to 317

Bounds are now closer to the known upper bound of 320

Improved bounds for all n ≥ 10

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 $n\geq10$. 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 $320$.