Symplectic codes over a non-unitary ring

In this paper, we consider the commutative non-unitary ring of order four defined as $I=\{a,b|2a=0,2b=0,a^2=b,ab=0\}.$ Self-orthogonal codes for a symplectic inner product over $I$ are introduced. A mass formula to enumerate them under symplectic equivalence is given. An application is a classification of such codes in short lengths.