  A concise theoretical framework, the $\textit{partial}$ Gauss-Hermite
quadrature (pGHQ), is established for constructing on-node lattices of the
lattice Boltzmann (LB) method under a Cartesian coordinate system. Comparing
with existing approaches, the pGHQ scheme has the following advantages:
$\textbf{a).}$ extremely concise algorithm, $\textbf{b).}$ unifying the
constructing procedure of symmetric and asymmetric on-node lattices,
$\textbf{c).}$ covering full-range quadrature degree of a given discrete
velocity set. We employ it to search the local optimal and asymmetric lattices
for $\left\{ {n = 3,4,5,6,7} \right\}$ moment degree equilibrium distribution
discretization on range $\left[ { - 10,10} \right]$. The search reveals a
surprising abundance of available lattices. Through a brief analysis, the
discrete velocity set shows a significant influence on the positivity of
equilibrium distributions, which is considered as one major impact to the
numerical stability of the LB method. Hence the results of the pGHQ scheme lay
a foundation for further investigations on improving the numerical stability of
the LB method by modifying the discrete velocity set. It also worths noting
that pGHQ can be extended into the entropic LB model though it was proposed for
the Hermite polynomial expansion LB theory.
