Finite and Symmetric Euler Sums and Finite and Symmetric (Alternating) Multiple $T$-Values

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.