(0,1)-vector XOR problem
Let $v_0, v_1, \ldots, v_n \in \mathbb{Z}_2^m$. The problem is to determine whether the following system has a solution:

$$
\begin{pmatrix}
v_1 &amp; \cdots &amp; v_n
\end{pmatrix}
\begin{pmatrix}
x_1 \\
\vdots \\
x_n
\end{pmatrix}
=
v_0 \, (\text{mod } 2)
$$

This problem is known to be $\oplus L$-complete by [Damm90, BDHM92], thus inside $\text{NC}^2 \subseteq \text{P}$.

[Damm90] Carsten Damm: Problems Complete for $\oplus L$. Inf. Process. Lett. 36(5): 247-250 (1990)
[BDHM92] Gerhard Buntrock, Carsten Damm, Ulrich Hertrampf, Christoph Meinel: Structure and Importance of Logspace-MOD Class. Mathematical Systems Theory 25(3): 223-237 (1992)