(untitled) (Journal of Nonlinear Modeling and Analysis, 2026)
In this paper, we determine the algebraic structure of all λ-constacyclic codes of length 4p^s over the ring R_u^2,v^2,p^m = F_p^m[u,v]/u^2, v^2, uv-vu and u^2 = 0, v^2 = 0 where λ = (α+β u+γ v+δ uv) with β,γ,δ F_p^m, α F_p^m^* and β, γ are not both zero. If λ is a square, each λ-constacyclic codes of length 4p^s is expressed as a direct sum of an -α-constacyclic code and α-constacyclic code of length 2p^s. In the primary case where the unit λ is not square, it is shown that any non-zero polynomial of degree ≤ 4 over F_p^m is invertible in the ring R_α,β,γ,δ = R_u^2,v^2,p^m[x]/x^4p^s-λ when λ = (α+β u+γ v +δ uv) for non-zero elements α, β,γ F_p^m^*, δ F_p^m, it follows that the ring R_α,β,γ,δ is a chain ring with maximal ideal (x^4-α_0) and (α+β u+γ v+δ uv)-constacyclic codes are (x^4-α_0)^i , for 0 ≤ i ≤ 2p^s. In the second case where λ is not square and λ =γ for γ F_p^m, it is obtain that R_u^2,v^2,p^m[x]/x^4p^s-γ is a local ring with maximal ideal x^4-γ_0,u but not a chain ring, such λ-constacyclic codes are classified into four distinct types. We provide the number of codewords for each type, and give the details of their dual codes.