Some Repeated-Root Constacyclic Codes Over Galois Rings

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over GR(2
^a
, m) of length 2
^s
have been characterized: the ring R
₂
(a, m, -1) = GR(2
^a
, m)[x]/(x
^2s
t 1) is a chain ring. Furthermore, these results have been generalized to λ-constacyclic codes for any unit λ of the form 4z - 1, z ∈ GR(2
^a
, m). In this paper, we study more general cases and investigate all cases, where R
_p
(a, m, y) = GR(pa, m)[x]/(xps - y) is a chain ring. In particular, the necessary and sufficient conditions for the ring R
_p
(a, m, y) to be a chain ring are obtained. In addition, by using this structure we investigate all y-constacyclic codes over GR(p
^a
, m) when R
_p
(a, m, y) is a chain ring. The necessary and sufficient conditions for the existence of self-orthogonal and self-dual y-constacyclic codes are also provided. Among others, for any prime p, the structure of R
_p
(a, m, y) = GR(p
^a
, m)[x]/(x
^p
s
- y) is used to establish the Hamming and homogeneous distances of y-constacyclic codes.