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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 commutativeor ring with unity) of order $p^2$ had been presented, the construction ofcodes over non-commutative rings or non-commutative non-unital rings surfacedmerely two years ago. In this letter, we extend the diverse research onexploring the codes over the non-commutative and non-unital ring $E= \langle2a=2b=0, a^2=a, b^2=b, ab=a, ba=b \rangle$ by presenting the classification ofoptimal and nice codes of length $n\leq7$ over $E$, along-with respectiveweight enumerators and complete weight enumerators.

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