Received May 8, 2020, accepted May 28, 2020, date of publication June 1, 2020, date of current version June 10, 2020. Digital Object Identifier 10.1109/ACCESS.2020.2998836 Repeated-Root Constacyclic Codes Over the Chain Ring F p m [u]/〈u 3 〉 TANIA SIDANA AND ANURADHA SHARMA Department of Mathematics, IIIT-Delhi, New Delhi 110020, India Corresponding author: Anuradha Sharma (anuradha@iiitd.ac.in) This work was supported by the Science and Engineering Research Board (SERB), India, under Grant EMR/2017/000662. ABSTRACT Let R = F p m [u]/〈u 3 〉 be the finite commutative chain ring, where p is a prime, m is a positive integer and F p m is the finite field with p m elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over R and their dual codes. We also determine the number of codewords in each repeated-root constacyclic code over R. 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 R. Using these results, we also identify some isodual and maximum distance separable (MDS) constacyclic codes over R with respect to the Hamming and RT metrics. INDEX TERMS Cyclic codes, local rings, negacyclic codes, optimal codes. I. INTRODUCTION 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. Several metrics (e.g. Hamming metric, Lee metric, RT met- ric, etc.) have been introduced to study error-detecting and error-correcting properties of a code with respect to various communication channels. Among the prevalent metrics in coding theory, the Hamming metric is the most studied metric and it is suitable for orthogonal modulated channels. The Singleton bound [31] is an upper bound on the size M of an arbitrary block code with respect to the Hamming metric: M ≤ q n−d +1 , (1) where q is the cardinality of the code alphabet, n is the block length and d is the Hamming distance of the code. Linear codes that attain the Singleton bound are called maximum distance separable (MDS) codes with respect to the Hamming metric. Later, motivated by the problem to transmit messages over several parallel communication channels with some channels not available for transmission, a non-Hamming met- ric, called the Rosenbloom-Tsfasman metric (or RT metric), was introduced by Rosenbloom and Tsfasman [30]; they also derived Singleton bound for the RT metric. Linear codes that attain the Singleton bound for the RT metric are called The associate editor coordinating the review of this manuscript and approving it for publication was Zesong Fei . MDS codes with respect to the RT metric. MDS codes have the highest possible error-detecting and error-correcting capabilities for given code length, code size and alphabet size, hence they are considered optimal codes in that sense. This has encouraged many coding theorists to further study and construct MDS codes with respect to various metrics (see [20], [23], [39]). Recently, Li and Yue [24] determined Hamming distances of all repeated-root cyclic codes of length 5p s over F p m and identified all MDS codes within this class of codes, where p is a prime, s, m are positive integers and F p m is the finite field of order p m . In this paper, we shall also find MDS codes with respect to Hamming and RT metrics within the family of constacyclic codes over F p m [u]/〈u 3 〉. Berlekamp [4] first introduced and studied constacyclic codes over finite fields, which have a rich algebraic structure and are generalizations of cyclic and negacyclic codes. For recent works on constacyclic codes over finite fields, please refer to [32], [33], [37]. Calderbank et al. [6], Hammons et al. [21] and Nechaev [28] related binary non-linear codes (e.g. Kerdock and Preparata codes) to linear codes over the finite commutative chain ring Z 4 of integers modulo 4 with the help of a Gray map. Since then, codes over finite commutative chain rings have received a great deal of attention. However, their algebraic structures are known only in a few cases. Towards this, Dinh and L ´ opez-Permouth [17] studied algebraic structures of simple-root cyclic and nega- cyclic codes over finite commutative chain rings and their dual codes. In the same work, they also determined all 101320 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ VOLUME 8, 2020