  In this paper, we will study finite multiple $T$-values (MTVs) and their
alternating versions, which are level two and level four variations of finite
multiple zeta values, respectively. We will first provide some structural
results for level two finite multiple zeta values (i.e., finite Euler sums) for
small weights, guided by the author's previous conjecture that the finite Euler
sum space of weight, $w$, is isomorphic to a quotient Euler sum space of
weight, $w$. Then, by utilizing some well-known properties of the classical
alternating MTVs, we will derive a few important $\Q$-linear relations among
the finite alternating MTVs, including the reversal, linear shuffle, and sum
relations. We then compute the upper bound for the dimension of the $\Q$-span
of finite (alternating) MTVs for some small weights by rigorously using the
newly discovered relations, numerically aided by computers.
