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.
翻译:在过去60年中,对编码理论进行了大量的研究,研究范围超过了限定的字段$\mathbb{F ⁇ q$或通货环$\mathcal{R}$\mathcal{R}$。虽然在1993美元中,提出了关于圆环分类(不一定具有通货性或具有统一性的环)$p ⁇ 2美元的研究,对非混合环或非混合非单一环的代码的构建仅在两年前才浮出水面。在本信中,我们扩展了关于在非混合和非整体环$=\langle 2a=2b=0, a ⁇ 2=a, b ⁇ 2=b, ab=a,ba=b=b\rangle$的代码的分类,对长度超过$nleq7美元的最佳和漂亮的代码进行了分类,同时列出了相应的重量计数器和完整重量计数器。