There are exactly two non-commutative rings of size $4$, namely, $E = \langle a, b ~\vert ~ 2a = 2b = 0, a^2 = a, b^2 = b, ab= a, ba = b\rangle$ and its opposite ring $F$. These rings are non-unital. A subset $D$ of $E^m$ is defined with the help of simplicial complexes, and utilized to construct linear left-$E$-codes $C^L_D=\{(v\cdot d)_{d\in D} : v\in E^m\}$ and right-$E$-codes $C^R_D=\{(d\cdot v)_{d\in D} : v\in E^m\}$. We study their corresponding binary codes obtained via a Gray map. The weight distributions of all these codes are computed. We achieve a couple of infinite families of optimal codes with respect to the Griesmer bound. Ashikhmin-Barg's condition for minimality of a linear code is satisfied by most of the binary codes we constructed here. All the binary codes in this article are few-weight codes, and self-orthogonal codes under certain mild conditions. This is the first attempt to study the structure of linear codes over non-unital non-commutative rings using simplicial complexes.

翻译：大小为 $4$ 的非交换环恰有两个，即 $E = \langle a, b ~\vert ~ 2a = 2b = 0, a^2 = a, b^2 = b, ab= a, ba = b\rangle$ 及其对偶环 $F$。这两个环均为非幺环。借助复形结构定义了 $E^m$ 的子集 $D$，并以此构造线性左$E$-码 $C^L_D=\{(v\cdot d)_{d\in D} : v\in E^m\}$ 与右$E$-码 $C^R_D=\{(d\cdot v)_{d\in D} : v\in E^m\}$。我们研究了通过Gray映射得到的二元码，并计算了所有码的重量分布。基于Griesmer界，我们得到了若干无穷最优码族。本文构造的大多数二元码满足Ashikhmin-Barg线性码极小性条件，且所有二元码均为少重量码，在温和条件下具有自正交性。这是首次尝试利用复形结构研究非幺非交换环上线性码的结构。