Codes Over the Non-Unital Non-Commutative Ring E Using Simplicial Complexes
Vidya Sagar, Ritumoni Sarma · IEEE Transactions on Information Theory · 2024
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}$is defined with the help of simplicial complexes, and utilized to construct the linear left-$E$-code$C^{L}_{D}=\{(v\cdot d)_{d\in D}: v\in E^{m}\}$and the right-$E$-code$C^{R}_{D}=\{(d\cdot v)_{d\in D}: v\in E^{m}\}$. We study a certain binary subfield-like code corresponding to$C_{D}^{L}$. By using a Gray map, we also obtain the binary Gray images of$C_{D}^{L}$and$C_{D}^{R}$. 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 self-orthogonal and few-weight codes under certain mild conditions. This is the first attempt to study the structure of linear codes over a non-unital non-commutative ring using simplicial complexes.