Codes over the non-unital non-commutative ring E using simplicial complexes
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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 linear left-E-codes C^L_D={(v· d)_d∈ D : v∈ E^m} and right-E-codes C^R_D={(d· v)_d∈ D : v∈ 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.
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