On the Classification of Codes over Non-Unital Ring of Order 4
(2208.08710)Abstract
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 $p2$ 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 $E= \langle 2a=2b=0, a2=a, b2=b, ab=a, ba=b \rangle$ by presenting the classification of optimal and nice codes of length $n\leq7$ over $E$, along-with respective weight enumerators and complete weight enumerators.
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