The Quasi-self-dual, Right LCD and ACD codes over a noncommutative non-unital ring

This paper presents the study of QSD (quasi-self-dual), right-LCD (linear complementary dual), and ACD (additive complementary dual) codes over a noncommutative local ring $R= \langle a,b ~|~ 3a =3b=0,~ a^2=a,~ b^2=b,~ ab=b,~ ba=a \rangle$ of order $9$. Initially, over this ring $R$, we introduce QSD codes and characterize their multilevel construction. Then, we delve into the study of right LCD codes over the ring $R$ and demonstrate a method for constructing these codes based on ternary LCD codes. Finally, we introduce the right-ACD codes over this ring and present several criteria for the existence of such codes.