  In his 1992 Ph.D. thesis Chang identified an efficient way to dominate $m
\times n$ grid graphs and conjectured that his construction gives the most
efficient dominating sets for relatively large grids. In 2011 Gon\c{c}alves,
Pinlou, Rao, and Thomass\'e proved Chang's conjecture, establishing a closed
formula for the domination number of a grid. In March 2013 Fata, Smith and
Sundaram established upper bounds for the $k$-distance domination numbers of
grid graphs by generalizing Chang's construction of dominating sets to
$k$-distance dominating sets. In this paper we improve the upper bounds
established by Fata, Smith, and Sundaram for the $k$-distance domination
numbers of grids.
