Symplectic Quasi Self-dual, Self-dual, and LCD Codes over Non-unital Rings of Order Six

We consider codes over the two semi-local non-unital rings of order six, $$H_{23}=⟨a,b\mid 2a=0,3b=0,a^2=a,b^2=0,ab=0=ba⟩,$$ and $$H_{32}=⟨a,b\mid 2a=0,3b=0,a^2=0,b^2=b,ab=0=ba⟩\mathrm{.}$$ with respect to a symplectic inner product. Via the decomposition $\mathcal{C}=a{\mathcal{C}}_a\oplus b{\mathcal{C}}_b$, any $H_z$-code splits into a binary component ${\mathcal{C}}_a$ and a ternary component ${\mathcal{C}}_b$; this yields characterizations of symplectic self-orthogonal, self-dual, and quasi self-dual (QSD) codes, and allows the introduction of symplectic nice and symplectic linear complementary dual (LCD) codes. Using the automorphism groups of ${\mathcal{C}}_a$ and ${\mathcal{C}}_b$ and double-coset enumeration, we classify, up to permutation equivalence, all symplectic self-orthogonal and QSD $H_z$-codes of short even lengths.