On the Classification of Codes over Non-Unital Ring of Order 4

In the last 60 years coding theory has been studied a lot over finite fields ${\mathbb{F}}_q$ or commutative rings $\mathcal{R}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring with unity) of order $p^2$ had been presented, the construction of codes over non-commutative rings or non-commutative non-unital rings surfaced merely two years ago. In this letter, we extend the diverse research on exploring the codes over the non-commutative and non-unital ring $E=⟨2a=2b=0,a^2=a,b^2=b,ab=a,ba=b⟩$ by presenting the classification of optimal and nice codes of length $n\le 7$ over $E$, along-with respective weight enumerators and complete weight enumerators.