We provide a complete enumeration of all complex Golay pairs of length up to 25, verifying that complex Golay pairs do not exist in lengths 23 and 25 but do exist in length 24. This independently verifies work done by F. Fiedler in 2013 that confirms the 2002 conjecture of Craigen, Holzmann, and Kharaghani that complex Golay pairs of length 23 don't exist. Our enumeration method relies on the recently proposed SAT+CAS paradigm of combining computer algebra systems with SAT solvers to take advantage of the advances made in the fields of symbolic computation and satisfiability checking. The enumeration proceeds in two stages: First, we use a fine-tuned computer program and functionality from computer algebra systems to construct a list containing all sequences which could appear as the first sequence in a complex Golay pair (up to equivalence). Second, we use a programmatic SAT solver to construct all sequences (if any) that pair off with the sequences constructed in the first stage to form a complex Golay pair.

It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous G-algebras, are finite factorization domains (FFD for short).

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial nth Weyl algebra, the polynomial nth shift algebra, and Zn-graded polynomials in the nth q-Weyl algebra.