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

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 $R_{3}=\mathbb{F}_{p^{m}}[u]/\langle u^{3}\rangle$, where $p^{m}\equiv1 \pmod 4$.

The distance characterization is explicitly classified according to the quadratic character of the shift unit $\Delta \in R_{3}^{*}$.

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 $\mathcal{C}=\langle1\rangle$ 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.