  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 $E= \langle
2a=2b=0, a^2=a, b^2=b, ab=a, ba=b \rangle$ by presenting the classification of
optimal and nice codes of length $n\leq7$ over $E$, along-with respective
weight enumerators and complete weight enumerators.
