Part 2 to 6 of the paper explains the elements of notation I used.
The algorithm I wrote had to do with both the usual matrix product and the kronecker product (in short it lets you compute the Kronecker product of several matrices together time a given matrix much more efficiently and with a significantly lower memory usage).
The notation was useful to me as it surfaces the property Wikipedia describes in the "Matrix Equation" [0] section on the Kronecker product and makes it intuitive.
[0]: https://en.wikipedia.org/wiki/Kronecker_product#Matrix_equat...
(could you add a bit more detail about any notation in particular that was useful? I see the matrix direct product symbol and have a vague initial understanding of that and block matrices)