Enumeration of nilpotent associative algebras of class 2 over arbitrary finite fields

Higman's PORC theory implies that the number Nd,r(q) of isomorphism types of nilpotent associative algebras of dimension d, rank r and class 2 over a finite field with q elements, considered as a function in q, can be described by a polynomial on residue classes in q. We describe an algorithm that, given a rank r, determines such polynomials for Nd,r(q) for all dimensions d. Using this, we determine Nd,r(q) for r∈{1,…,5} and arbitrary d.