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

There are exactly two non-commutative rings of size 4, namely, <tex-math notation="LaTeX">E = a, b 2a = 2b = 0, a² = a, b² = b, ab= a, ba = b </tex-math> and its opposite ring <tex-math notation="LaTeX">F </tex-math> . These rings are non-unital. A subset <tex-math notation="LaTeX">D </tex-math> of <tex-math notation="LaTeX">Eᵐ </tex-math> is defined with the help of simplicial complexes, and utilized to construct the linear left- <tex-math notation="LaTeX">E </tex-math> -code <tex-math notation="LaTeX">CLD=\(v· d)d∈ D: v∈ Eᵐ\ </tex-math> and the right- <tex-math notation="LaTeX">E </tex-math> -code <tex-math notation="LaTeX">CRD=\(d· v)d∈ D: v∈ Eᵐ\ </tex-math> . We study a certain binary subfield-like code corresponding to <tex-math notation="LaTeX">CDL </tex-math> . By using a Gray map, we also obtain the binary Gray images of <tex-math notation="LaTeX">CDL </tex-math> and <tex-math notation="LaTeX">CDR </tex-math> . 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.