Duality of Codes over Non-unital Rings of Order Six

We present some basic theory on the duality of codes over two non-unital rings of order $6$, namely $H_{23}$ and $H_{32}$. For a code $\mathcal{C}$ over these rings, we associate a binary code ${\mathcal{C}}_a$ and a ternary code ${\mathcal{C}}_b$. We characterize self-orthogonal, self-dual and quasi self-dual (QSD) codes over these rings using the codes ${\mathcal{C}}_a$ and ${\mathcal{C}}_b$. In addition, we present a building-up construction for self-orthogonal codes, introduce cyclic codes and linear complementary dual (LCD) codes. We also gave a classification of self-orthogonal codes for short lengths.