On the Symbol-Pair Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths

Abstract

Let $p$ be a prime, and $\lambda$ be a nonzero element of the finite field $\mathbb F_{p^{m}}$ . The $\lambda$ -constacyclic codes of length $p^{s}$ over $\mathbb F_{p^{m}}$ are linearly ordered under set-theoretic inclusion, i.e., they are the ideals $\langle (x-\lambda _{0})^{i} \rangle$ , $0 \leq i \leq p^{s}$ of the chain ring $[({\mathbb F_{p^{m}}[x]})/({\langle x^{p^{s}}-\lambda \rangle })]$ . This structure is used to establish the symbol-pair distances of all such $\lambda$ -constacyclic codes. Among others, all maximum distance separable symbol-pair constacyclic codes of length $p^{s}$ are obtained.