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Let <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> be any prime, <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> be positive integers. In this paper, we completely determine the Hamming distance of all constacyclic codes of length <inline-formula> <tex-math notation="LaTeX">$p^s$ </tex-math></inline-formula> over the finite commutative chain ring <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{p^m}+ u\mathbb {F}_{p^m} + u^{2}\mathbb {F}_{p^m}\,\,\, (u^3=0)$ </tex-math></inline-formula>. As applications, we identify all maximum distance saparable codes (i.e., optimal codes with respect to the Singleton bound) among them.

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Hamming Distance of Constacyclic Codes of Length <italic>ps</italic> Over F<italic>pm</italic>&#x002B;<italic>u</italic>F<italic>pm</italic>&#x002B;<italic>u</italic>&#x00B2;F<italic>pm</italic>

Hamming Distance of Constacyclic Codes of Length ps Over Fpm+uFpm+u²Fpm

Hamming Distance of Constacyclic Codes of Length p^s Over F_p^m+uF_p^m+u²F_p^m