RT distance and weight distributions of Type 1 constacyclic codes of length 4ps over Fpm[u

Abstract

© Tubitak. For any odd prime p such that p m ≡ 1 (mod 4), the class of Λ-constacyclic codes of length 4p s over the finite commutative chain ring R a =F p m[u]/〈u a 〉 =F p m + uF p m + . . . + ua -1 F p m, for all units Λ of R a that have the form Λ = Λ 0 +uΛ 1 +. . .+u a-1 Λ a-1 , where Λ 0 ;Λ 1 ;: :: ;Λ a-1 ∈ F p m, Λ 0 ≠0; Λ 1 ≠0, is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a -λ-constacyclic code and a λ-constacyclic code of length 2p s . In the main case that the unit Λ is not a square, we show that any nonzero polynomial of degree < 4 over F p m is invertible in the ambient ring R a [x] /〈x 4ps -Λ〉 and use it to prove that the ambient ring 〈 a [x] /x 4ps -Λ〉 is a chain ring with maximal ideal 〈x 4 - λ 0 〉, where λ 0ps = λ 0 : As an application, the number of codewords and the dual of each λ-constacyclic code are provided. Furthermore, we get the Rosenbloom-Tsfasman (RT) distance and weight distributions of such codes. Using these results, the unique MDS code with respect to the RT distance is identified.