On the Structure and Distances of Repeated-Root Constacyclic Codes of Prime Power Lengths Over Finite Commutative Chain Rings.

Let $p$ be a prime, $s$ be a positive integer, and $\mathcal {R}$ be a finite commutative chain ring with the characteristic as a power of $p$. For a unit $\lambda \in \mathcal {R},~\lambda $ -constacyclic codes of length $p^{s}$ over $\mathcal {R}$ are ideals of the quotient ring $\mathcal {R}[x]/\langle x^{p^{s}}-\lambda \rangle $. In this paper, we derive necessary and sufficient conditions under which the quotient ring $\mathcal {R}[x]/\langle x^{p^{s}}-\lambda \rangle $ is a chain ring. When $\mathcal {R}[x]/\langle x^{p^{s}}-\lambda \rangle $ is a chain ring, all $\lambda $ -constacyclic codes of length $p^{s}$ over $\mathcal {R}$ are known. In this paper, we establish the algebraic structures of all $\lambda $ -constacyclic codes of length $p^{s}$ over $\mathcal {R}$ when $\mathcal {R}[x]/\langle x^{p^{s}}-\lambda \rangle $ 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 RT weight distributions of all constacyclic codes of length $p^{s}$ over $\mathcal {R}$. Apart from this, we derive necessary and sufficient conditions under which a constacyclic code of length $p^{s}$ over $\mathcal {R}$ is maximum-distance separable with respect to the: 1) Hamming metric; 2) symbol-pair metric; and 3) RT metric. We also provide an algorithm to decode the constacyclic codes of length $p^{s}$ over $\mathcal {R}$ using the known decoding algorithms of linear codes over finite fields with respect to the Hamming, symbol-pair, and RT metrics. [ABSTRACT FROM AUTHOR]