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Abstract
Let Z = F p m [u]/(u 3 ) be the finite commutative chain ring, where p is a prime, m is a positive integer and Fpm is the finite field with pm elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over Z and their dual codes. We also determine the number of codewords in each repeated-root constacyclic code over Z. We also obtain Hamming distances, RT distances, RT weight distributions and ranks (i.e., cardinalities of minimal generating sets) of some repeated-root constacyclic codes over Z. Using these results, we also identify some isodual and maximum distance separable (MDS) constacyclic codes over Z with respect to the Hamming and RT metrics.
Highlights
Constructing codes that are easy to encode and decode, can detect and correct many errors and have a sufficiently large number of codewords is the primary aim of coding theory
We show that there does not exist any non-trivial maximum distance separable (MDS) (α + βu + γ u2)-constacyclic code of length nps over R when β = 0
We show that there does not exist any non-trivial MDS (α + βu + γ u2)-constacyclic code of length nps over R with respect to the RT metric when β = 0
Summary
Constructing codes that are easy to encode and decode, can detect and correct many errors and have a sufficiently large number of codewords is the primary aim of coding theory. Dinh et al [18] studied repeated-root (α + aw)-constacyclic codes of length ps over a finite commutative chain ring R with the maximal ideal as w , where p is a prime number, s ≥ 1 is an integer and α, a are units in R. The main goal of this paper is to determine all repeated-root constacyclic codes of arbitrary lengths over the finite commutative chain ring Fpm [u]/ u3 , their sizes and their dual codes, where p is a prime and m is a positive integer. We list all MDS repeated-root constacyclic codes of length 2ps over Fpm [u]/ u3 with respect to the Hamming metric (Theorem 35).
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