0018-9448 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIT.2018.2864293, IEEE Transactions on Information Theory 1 On the structure and distances of repeated-root constacyclic codes of prime power lengths over finite commutative chain rings Anuradha Sharma and Tania Sidana Abstract—Let p be a prime, s be a positive integer, and let R be a finite commutative chain ring with the characteristic as a power of p. For a unit λ ∈R,λ-constacyclic codes of length p s over R are ideals of the quotient ring R[x]/〈x p s − λ〉. In this paper, we derive necessary and sufficient conditions under which the quotient ring R[x]/〈x p s − λ〉 is a chain ring. When R[x]/〈x p s − λ〉 is a chain ring, all λ-constacyclic codes of length p s over R are known. In this paper, we establish algebraic structures of all λ-constacyclic codes of length p s over R when R[x]/〈x p s − λ〉 is a non-chain ring. We also determine the number of codewords in each of these codes. Using their algebraic structures, we obtain symbol-pair distances, Rosenbloom-Tsfasman (RT) distances, and Rosenbloom-Tsfasman (RT) weight distributions of all constacyclic codes of length p s over R. Apart from this, we derive necessary and sufficient conditions under which a constacyclic code of length p s over R is maximum- distance separable (MDS) with respect to the (i) Hamming metric, (ii) symbol-pair metric, and (iii) Rosenbloom-Tsfasman (RT) metric. We also provide an algorithm to decode constacyclic codes of length p s over R using the known decoding algorithms of linear codes over finite fields with respect to the Hamming, symbol-pair and RT metrics. Index Terms—Cyclic codes; Negacyclic codes; Local rings; Optimal codes. 2010 Mathematics Subject Classification: 94B15. I. I NTRODUCTION The main aim of coding theory is to construct codes that are easier to encode and decode, can detect and correct many errors, and contain a sufficiently large number of codewords. In other words, the goal is to find codes with efficient encoding and decoding procedures, and with the largest possible value of distance for given code length, code size and cardinality of the code alphabet. To study error-detecting and error-correcting properties of a code with respect to various communication This work is supported by Science and Engineering Research Board (SERB), India, under grant no. EMR/2017/000662. The authors are with the Department of Mathematics, IIIT-Delhi, New Delhi 110020, India (e-mail: anuradha@iiitd.ac.in; taniai@iiitd.ac.in). Copyright (c) 2017 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs- permissions@ieee.org. channels, several metrics (e.g. Hamming metric, Lee metric, RT metric, symbol-pair metric, etc.) have been introduced and studied in coding theory. The most studied metric in coding theory is the Hamming metric, which is suitable for orthogonal modulated channels. Singleton [31] derived the following upper bound (called the Singleton bound) on the size M of an arbitrary block code with respect to the Hamming metric: M ≤ q n-d+1 , (I.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 attaining the Singleton bound are called MDS codes with respect to the Hamming metric. Later, Rosenbloom and Tsfasman [29] introduced a non-Hamming metric, called the Rosenbloom-Tsfasman metric (or RT metric), which is motivated by a problem related to transmission over several parallel communication channels with some channels not available for the transmission. This metric is also useful in case of interference between several consecutive communication channels. They also derived Singleton bound for the RT metric. Linear codes attaining the Singleton bound for the RT metric are called MDS codes with respect to the RT metric. On the other hand, Cassuto and Blaum [4], [5] introduced a new metric, called the symbol- pair metric, which is suitable for a channel whose outputs are overlapping pairs of symbols. Such channels are motivated by reading limitations in high-density data storage systems or the storage systems in which the spatial resolution of the reader may be insufficient to isolate adjacent symbols. Chee et al. [6] derived a Singleton type bound for codes with respect to the symbol- pair metric and the codes attaining this bound are called MDS codes with respect to the symbol-pair metric. In the same work, they constructed many MDS codes with respect to this metric. MDS codes are optimal in the sense that they have the highest possible error-detecting and error-correcting capability for given code length and code size. This motivates many researchers to further study and construct MDS codes with respect to various metrics (see [12], [13], [17], [18], [20], [26], [28], [33]). In this paper, we shall also study and find MDS codes with respect