Certain binary minimal codes constructed using simplicial complexes
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In this article, we work over the non-chain ring ℛ = ℤ_2[u]/⟨ u^3 - u⟩. Let m∈ℕ and let L, M, N ⊆ [m]:={1, 2, …, m}. For X⊆ [m], define Δ_X := {v ∈ℤ_2^m : Supp(v)⊆ X} and D:= (1+u^2)D_1 + u^2(D_2 + (u+u^2)D_3), a subset of ℛ^m, where D_1∈{Δ_L, Δ_L^c}, D_2∈{Δ_M, Δ_M^c}, D_3∈{Δ_N, Δ_N^c}. The linear code C_D over ℛ defined by {(v· d)_d∈ D : v ∈ℛ^m } is studied for each D. For instance, we obtain the Lee weight distribution of C_D. The Gray map Φ: ℛ⟶ℤ_2^3 given by Φ(a+ub+u^2d) = (a+b, b+d, d) is utilized to derive a binary linear code, namely, Φ(C_D) for each D. Sufficient conditions for each of these binary linear codes to be minimal are obtained. In fact, sufficient conditions for minimality are mild in nature, for example, | L| , | M| , | N| < m-2 is a set of conditions for minimality of Φ(C_D) for each D. Moreover, these binary codes are self-orthogonal if each of L, M and N is nonempty.
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