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RT distance and weight distributions of Type 1 constacyclic codes of length 4ps over 𝔽pm[u] / <ua>

Authors: 

Hai Q. Dinh, Bac T. Nguyen, Songsak Sriboonchitta

Source title: 
Turkish Journal of Mathematics, 43: 561-578, 2019 (ISI)
Academic year of acceptance: 
2019-2020
Abstract: 

For any odd prime p such that pm ≡ 1 (mod 4), the class of Λ -constacyclic codes of length 4ps over the finite commutative chain ring a, for all units Λ of ℛa that have the form Λ = Λ0+uΛ1+ · · ·+ua-1Λa−1, where Λ01, ...,Λa−1 ∈ Fpm, Λ0 ≠ 0, Λ1 ≠ 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 2ps. In the main case that the unit Λ is not a square, we show that any nonzero polynomial of degree < 4 over Fpm is invertible in the ambient ring a and use it to prove that the ambient ring a is a chain ring with maximal ideal ⟨x4 − λ0⟩, where b. 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.