Constacyclic codes of length 4p^s over the Galois ring GR(p^a,m)
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For prime p, GR(p^a,m) represents the Galois ring of order p^am and characterise p, where a is any positive integer. In this article, we study the Type (1) λ-constacyclic codes of length 4p^s over the ring GR(p^a,m), where λ=ξ_0+pξ_1+p^2z, ξ_0,ξ_1∈ T(p,m) are nonzero elements and z∈ GR(p^a,m). In first case, when λ is a square, we show that any ideal of R_p(a,m,λ)=GR(p^a,m)[x]/〈 x^4p^s-λ〉 is the direct sum of the ideals of GR(p^a,m)[x]/〈 x^2p^s-δ〉 and GR(p^a,m)[x]/〈 x^2p^s+δ〉. In second, when λ is not a square, we show that R_p(a,m,λ) is a chain ring whose ideals are 〈 (x^4-α)^i〉⊆R_p(a,m,λ), for 0≤ i≤ ap^s where α^p^s=ξ_0. Also, we prove the dual of the above code is 〈 (x^4-α^-1)^ap^s-i〉⊆R_p(a,m,λ^-1) and present the necessary and sufficient condition for these codes to be self-orthogonal and self-dual, respectively. Moreover, the Rosenbloom-Tsfasman (RT) distance, Hamming distance and weight distribution of Type (1) λ-constacyclic codes of length 4p^s are obtained when λ is not a square.
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