k-XORSAT is the problem of deciding whether a Boolean formula $$\bigwedge_{i \in I} \oplus_{j=1}^k l_{s_{ij}}$$ is satisfiable. Here $\oplus$ denotes the binary XOR operation, $I$ is some index set, and each clause has $k$ distinct literals $l_{s_{ij}}$ each of which is either a variable $x_{s_{ij}}$ or its negation.

Equivalently, $k$-XORSAT requires deciding whether a set of linear equations, each of the form $\sum_{j=1}^k x_{s_{ij}}\equiv 1\; (\mod 2)$, has a solution over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$.

Every decision problem has an associated counting problem, which (informally speaking) requires establishing the number of distinct solutions. Such counting problems form the complexity class #P. The "hardest" problems in #P are #P-complete, as any problem in #P can be reduced to a #P-complete problem.

The counting problem associated with any NP-complete decision problem is #P-complete. However, many "easy" problems (some even solvable in linear time) also lead to #P-complete counting problems: Leslie Valiant's original 1979 paper The Complexity of Computing the Permanent shows that computing the permanent of a 0-1 matrix is #P-complete. There is a list of #P-complete problems in the companion paper The Complexity of Enumeration and Reliability Problems. These include #MONOTONE 2-SAT, which requires counting the number of solutions to Boolean formulas in conjunctive normal form, where each clause has two variables (no negated variables are allowed).

Andrea Montanari has written about the partition function of $k$-XORSAT in some lecture notes, and his book with Marc Mézard apparently discusses this (unfortunately I do not have a copy available to hand, and the relevant Chapter 17 is not included in Montanari's online draft).

Some counting problems are not #P-complete, but instead can be approximated in a certain sense, by means of a scheme called an FPRAS. Jerrum, Sinclair, and collaborators have linked the existence of an FPRAS for some #P problems to the question of whether $NP = RP$.

These considerations lead to the following question:

Is $k$-XORSAT #P-complete? If not, does it have an FPRAS?

Note that the formula in Montanari's notes does not obviously appear to answer this question. Just because there is a nice closed form solution, doesn't mean we can evaluate it efficiently: consider the Tutte polynomial.