Repeated-Root Constacyclic Codes of Prime Power Length Over Fpm [u] / 〈ua〉 and Their Duals
constacyclic codes, dual codes, chain rings, polynomial residue rings, hamming distance, homogeneous distance
Algebra | Discrete Mathematics and Combinatorics | Mathematics
The units of the chain ring ℛa = Fpm [u]/〈ua〉 = Fpm + uFpm + ⋯ + ua−1Fpm are partitioned into a distinct types. It is shown that for any unit Λ of Type k, a unit λ of Type k∗ can be constructed, such that the class of λ-constacyclic of length ps of Type k∗ codes is one-to-one correspondent to the class of Λ-constacyclic codes of the same length of Type k via a ring isomorphism. The units of ℛa of the form Λ = Λ0 + u Λ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, … , Λa−1 ∈ Fpm, Λ0 ≠ 0, Λ1 ≠ 0, are considered in detail. The structure, duals, Hamming and homogeneous distances of Λ-constacyclic codes of length ps over ℛa are established. It is shown that self-dual Λ-constacyclic codes of length ps over ℛa exist if and only if a is even, and in such case, it is unique. Among other results, we discuss some conditions when a code is both α- and β-constacyclic over ℛa for different units α, β.
Dinh, Hai Q.; Dhompongsa, Sompong; and Sriboonchitta, Songsak (2016). Repeated-Root Constacyclic Codes of Prime Power Length Over Fpm [u] / 〈ua〉 and Their Duals. Discrete Mathematics 339(6), 1706-1715. doi: 10.1016/J.DISC.2016.01.020 Retrieved from https://digitalcommons.kent.edu/mathpubs/22