On symbol-pair distance of repeated-root constacyclic codes of length $4p^s$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+u^2{\mathbb{F}}_{p^m}$

Abstract

This paper completely determines the symbol-pair distance distributions of all repeated-root $\Delta$-constacyclic codes of length $4p^s$ over the finite commutative chain ring $R_3={\mathbb{F}}_{p^m}[u]/⟨u^3⟩$, where $p^m\equiv 1(\mathrm{mod}4)$. The distance characterization is explicitly classified according to the quadratic character of the shift unit $\Delta \in R_3^{\ast }$. When $\Delta$ is a non-square unit, the exact symbol-pair distances are established across all eight distinct ideal classifications of the ambient ring. Conversely, when $\Delta$ is a square unit, the distance profiles are derived by evaluating direct sum decompositions and local ring reductions. By evaluating the symbol-pair singleton bound, we prove that only the trivial ideal $\mathcal{C}=⟨1⟩$ achieves maximum distance separability (MDS) , as structural constraints rule out any non-trivial MDS configurations. Finally, computational examples of length 20 over ${\mathbb{F}}_5+u{\mathbb{F}}_5+u^2{\mathbb{F}}_5$ are provided to validate the derived distance formulas.