The build up construction for codes over a non-commutative non-unitary ring of order $9$

The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order four to generate quasi self-dual codes. In the present paper we introduce three such propagation rules to generate self-orthogonal, one-sided self-dual, and self-dual codes over a special non-unitary ring of order 9. As an application, we classify the three categories of codes in lengths at most $7,$ up to monomial equivalence. Mass formulas for the three classes of codes considered ensure that the classification is complete.