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 [Formula: see text] or commutative rings [Formula: see text] with unity. Although in [Formula: see text], a study on the classification of the rings (not necessarily commutative or ring with unity) of order [Formula: see text] 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 [Formula: see text] by presenting the classification of optimal and nice codes of length [Formula: see text] over [Formula: see text], along with respective weight enumerators and complete weight enumerators.