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

Abstract

In the last 60 years coding theory has been studied a lot over finite fields�Fq�or commutative rings�R�with unity. Although in�1993, a study on the classification of the rings (not necessarily commutative or ring with unity) of order�p2�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,a2=a,b2=b,ab=a,ba=b?�by presenting the classification of optimal and nice codes of length�n?7�over�E, along with respective weight enumerators and complete weight enumerators.