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

Abstract

Turkish Journal of Mathematics h t t p : / / j o u r n a l s . t u b i t a k . g o v . t r / m a t h / Research Article RT distance and weight distributions of Type 1 constacyclic codes of length 4p s over F p m [u] ⟨u a ⟩ Abstract: 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 Ra = F p m [u]⟨u a ⟩ = Fpm + uFpm + · · · + u a−1 Fpm , for all units Λ of Ra that have the formIf the unit Λ is a square, each Λ -constacyclic code of length 4p s 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 Fpm is invertible in the ambient ring Ra[x] ⟨x 4p s −Λ⟩ and use it to prove that the ambient ring Ra[x] ⟨x 4p s −Λ⟩ is a chain ring with maximal ideal ⟨x 4 − λ0⟩ , where λ p s 0 = Λ0. 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.⟨x n −λ⟩ , where the generator polynomial g(x) is the unique monic polynomial of minimum degree in the code, which is a divisor of x n − λ. In the case of λ = 1, those λ -constacyclic codes are called cyclic codes, and when λ = −1, such λ -constacyclic codes are called negacyclic codes. Cyclic and negacyclic codes are interesting from both theoretical and practical perspectives, which have been well studied since the late 1960s.When the code of length n is relatively prime to the characteristic of the field F, these codes are said to be simple root codes; otherwise, they are called repeated-root codes, which were first studied in 1967 by Berman [5]. Many authors studied repeated-root codes over finite fields ([28, 33, 44]). However, repeated-root codes were *

Constacyclic codes of length $4p^s$ over the Galois ring $GR(p^a,m)$

“…Recently, Dinh et. al [12,13] studied the the Type (1) constacyclic codes of length 2p s and 4p s over the chain ring F p m [u]/ u a of size p ma . They have determined the RT distance and weight distributions for these codes of length 2p s and 4p s , respectively.…”