文摘
For any odd prime pp such that pm≡1(mod4), the structures of all λλ-constacyclic codes of length 4ps4ps over the finite commutative chain ring Fpm+uFpmFpm+uFpm(u2=0)(u2=0) are established in terms of their generator polynomials. If the unit λλ is a square, each λλ-constacyclic code of length 4ps4ps is expressed as a direct sum of an −α−α-constacyclic code and an αα-constacyclic code of length 2ps2ps. In the main case that the unit λλ is not a square, it is shown that any nonzero polynomial of degree <4<4 over FpmFpm is invertible in the ambient ring (Fpm+uFpm)[x]〈x4ps−λ〉. When the unit λλ is of the form λ=α+uβλ=α+uβ for nonzero elements α,βα,β of FpmFpm, it is obtained that the ambient ring (Fpm+uFpm)[x]〈x4ps−(α+uβ)〉 is a chain ring with maximal ideal 〈x4−α0〉〈x4−α0〉, and so the (α+uβ)(α+uβ)-constacyclic codes are 〈(x4−α0)i〉〈(x4−α0)i〉, for 0≤i≤2ps0≤i≤2ps. For the remaining case, that the unit λλ is not a square, and λ=γλ=γ for a nonzero element γγ of FpmFpm, it is proven that the ambient ring (Fpm+uFpm)[x]〈x4ps−γ〉 is a local ring with the unique maximal ideal 〈x4−γ0,u〉〈x4−γ0,u〉. Such λλ-constacyclic codes are then classified into 44 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 λλ-constacyclic code are provided.
