Does this technique apply if the dimension of the null space of the linear transformation is greater than 1? In other words, if there is more than 1 free variable, can you still use the Smith normal form of the matrix to find a bijection from the integers to the solution set?

Edit: related follow up: any chance this technique is a good fit for enumeration of [Magic Squares][0] of a given order?

[0]: https://en.wikipedia.org/wiki/Magic_square