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Let <inline-formula> <tex-math notation="LaTeX">$p\not =5$ </tex-math></inline-formula> be any odd prime. Using the algebraic structures of all cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$5p^{s}$ </tex-math></inline-formula> over the finite commutative chain ring <inline-formula> <tex-math notation="LaTeX">${\mathcal{ R}}=\mathbb F_{p^{m}}+u\mathbb F_{p^{m}}$ </tex-math></inline-formula>, in this paper, the exact values of Hamming distances of all cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$5p^{s}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\cal R$ </tex-math></inline-formula> are established. As an application, we identify all maximum distance separable cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$5p^{s}$ </tex-math></inline-formula>.

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On Hamming Distance Distributions of Repeated-Root Cyclic Codes of Length 5ps Over Fp m &#x002B; uFp m