On constacyclic codes of length 4p^sover F<inf>p^m</inf>+uF<inf>p^m</inf>

Abstract

© 2016 Elsevier B.V. For any odd prime p such that pm≡1(mod4), the structures of all λ-constacyclic codes of length 4psover the finite commutative chain ring Fpm+uFpm(u2=0) are established in terms of their generator polynomials. If the unit λ is a square, each λ-constacyclic code of length 4psis expressed as a direct sum of an −α-constacyclic code and an α-constacyclic code of length 2ps. In the main case that the unit λ is not a square, it is shown that any nonzero polynomial of degree <4 over Fpmis invertible in the ambient ring (F p m + uF p m )[ x]〈 x 4 p s− λ〉. When the unit λ is of the form λ=α+uβ for nonzero elements α,β of Fpm, it is obtained that the ambient ring (F p m + uF p m )[ x]〈 x 4 p s−( α+ u β)〉 is a chain ring with maximal ideal 〈x4−α0〉, and so the (α+uβ)-constacyclic codes are 〈(x4−α0)i〉, for 0≤i≤2ps. For the remaining case, that the unit λ is not a square, and λ=γ for a nonzero element γ of Fpm, it is proven that the ambient ring (F p m + uF p m )[ x]〈 x 4 p s− γ〉 is a local ring with the unique maximal ideal 〈x4−γ0,u〉. Such λ-constacyclic codes are then classified into 4 distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each λ-constacyclic code are provided.