arXiv:1911.03089v1 [cs.IT] 8 Nov 2019 Constacyclic codes of length 4p s over the Galois ring GR(p a ,m) Om Prakash, Habibul Islam and Ram Krishna Verma Department of Mathematics Indian Institute of Technology Patna Patna- 801 106, India E-mail: om@iitp.ac.in, habibul.pma17@iitp.ac.in, ram.pma15@iitp.ac.in Abstract For prime p, GR(p a ,m) represents the Galois ring of order p am and charac- terise p, where a is any positive integer. In this article, we study the Type (1) λ- constacyclic codes of length 4p s over the ring GR(p a ,m), where λ = ξ 0 + pξ 1 + p 2 z, ξ 0 ,ξ 1 ∈ T (p, m) are nonzero elements and z ∈ GR(p a ,m). In first case, when λ is a square, we show that any ideal of R p (a, m, λ)= GR(p a ,m)[x] 〈x 4p s −λ〉 is the direct sum of the ideals of GR(p a ,m)[x] 〈x 2p s −δ〉 and GR(p a ,m)[x] 〈x 2p s +δ〉 . In second, when λ is not a square, we show that R p (a, m, λ) is a chain ring whose ideals are 〈(x 4 − α) i 〉⊆R p (a, m, λ), for 0 ≤ i ≤ ap s where α p s = ξ 0 . Also, we prove the dual of the above code is 〈(x 4 − α −1 ) ap s −i 〉⊆R p (a, m, λ −1 ) and present the necessary and sufficient condi- tion for these codes to be self-orthogonal and self-dual, respectively. Moreover, the Rosenbloom-Tsfasman (RT) distance, Hamming distance and weight distribution of Type (1) λ-constacyclic codes of length 4p s are obtained when λ is not a square. Key Words : Constacyclic code, Chain rings, Galois rings, RT distance. 2010 MSC : 94B15, 94B05, 94B60. 1. Introduction The constacyclic codes over finite rings have been extensively studied for the last six decades due to their theoretical and practical importance. Recall that when the charac- teristic of a ring is relatively prime to the length n of the code over the ring is known as simple-root constacyclic code. It is called the repeated-root constacyclic code if the length n is not relatively prime to the characteristic of the ring. Due to vide application, the repeated-root constacyclic codes are the center of attention of the present research in coding theory. After 1970, several researchers such as Massey et al. [21], Falkner et al. [16], Roth and Seroussi [24], Van Lint [31] and Castagnoli et al. [3] have been worked on 1