RT distance and weight distributions of Type 1 constacyclic codes of length $4p^s$ over $frac{F_{p^{mleft[uright]}}}{leftlangle u^arightrangle}$

Abstract

For any odd prime p such that pm ≡ 1 (mod 4) , the class of Λ -constacyclic codes of length 4ps over thefinite commutative chain ring R = Fpm [u] = F m + uF m + · · · + ua−1F m , for all units Λ of Rthat have the forma⟨ua⟩pppaΛ = Λ0 + uΛ1 +· · ·+ ua−1Λa−1 , where Λ0, Λ1, . . . , Λa−1 ∈ Fpm , Λ0 ̸= 0, Λ10 , 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 2ps . In the main case that the unit Λ is not a square, we show that any nonzero polynomial of degree < 4 over F m is invertible in the ambient ring Ra[x] and use it to prove that the ambient ring Ra[x] is a chain p⟨x4ps −Λ⟩⟨x4ps −Λ⟩ring with maximal ideal ⟨x4 − λ ⟩ , where λps = Λ . As an application, the number of codewords and the dual of eachλ -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.