Codes Over the Non-Unital Non-Commutative Ring E Using Simplicial Complexes

There are exactly two non-commutative rings of size 4, namely, E = a, b 2a = 2b = 0, a^2 = a, b^2 = b, ab= a, ba = b 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^L₃=\ (v d) ₃ ₃: v E^{m\} and the right- E -code C^R₃=\ (d v) ₃ ₃: v E^{m\}. We study a certain binary subfield-like code corresponding to C₃^L. By using a Gray map, we also obtain the binary Gray images of C₃^L and C₃^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.