Linearizing multiple XOR operations

I have never encountered this in a model. but nevertheless this is an interesting question.

How can we handle $$y = x_1 \oplus x_2 \oplus \cdots \oplus x_n$$ in a linear MIP model? Here $\oplus$ indicates the xor operation. 

Well, when we look at $x_1 \oplus x_2 \oplus \cdots \oplus x_n$ we can see that this is identical to $\sum x_i$ being odd. A small example illustrates that:

Checking if an expression is odd can be done by seeing if it can not be expressed as $2z$ where $z$ is an integer variable. So combining this, we can do:

Linearization of $y = x_1 \oplus x_2 \oplus \cdots \oplus x_n$
$$\begin{align} & \color{DarkRed}y = \sum_i \color{DarkRed}x_i - 2 \color{DarkRed}z \\ & \color{DarkRed}y, \color{DarkRed} x_i \in\{0,1\} \\ & \color{DarkRed}z \in \{0,1,2,\dots\} \end{align}$$

If we want, we can relax $y$ to be continuous between 0 and 1. An upper bound on z can be
$$z \le \left \lfloor{ \frac{n}{2} }\right \rfloor$$ .

References

1. Express XOR with multiple inputs in zero-one integer linear programming (ILP), https://cs.stackexchange.com/questions/40737/express-xor-with-multiple-inputs-in-zero-one-integer-linear-programming-ilp
2. XOR as linear inequalities, https://yetanothermathprogrammingconsultant.blogspot.com/2016/02/xor-as-linear-inequalities.html
3. A variant of the Lights Out game, https://yetanothermathprogrammingconsultant.blogspot.com/2016/01/a-variant-of-lights-out-game.html