  For a digraph $G$ without multisquares and a field $\mathbb{F}$, we construct
a basis of the vector space of path $n$-chains $\Omega_n(G;\mathbb{F})$ for
$n\geq 0$, generalising the basis of $\Omega_3(G;\mathbb{F})$ constructed by
Grigory'an. For a field $\mathbb{F},$ we consider the $\mathbb{F}$-path Euler
characteristic $\chi^\mathbb{F}(G)$ of a digraph $G$ defined as the alternating
sum of dimensions of path homology groups with coefficients in $\mathbb{F}.$ If
$\Omega_\bullet(G;\mathbb{F})$ is a bounded chain complex, the constructed
bases can be applied to compute $\chi^\mathbb{F}(G)$. We provide an explicit
example of a digraph $\mathcal{G}$ whose $\mathbb{F}$-path Euler characteristic
depends on whether the characteristic of $\mathbb{F}$ is two, revealing the
differences between GLMY theory and the homology theory of spaces. This allows
us to prove that there is no topological space $X$ whose homology is isomorphic
to path homology of the digraph $H_*(X;\mathbb{K})\cong {\rm
PH}_*(\mathcal{G};\mathbb{K})$ simultaneously for $\mathbb{K}=\mathbb{Z}$ and
$\mathbb{K}=\mathbb{Z}/2\mathbb{Z}.$
