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In this paper we study the non-existence problem of (nearly) perfect (almost)$m$-ary sequences via their connection to (near) Butson-Hadamard (BH) matricesand (near) conference matrices. Firstly, we apply a result on vanishing sums ofroots of unity and a result of Brock on the unsolvability of certain equationsover a cyclotomic number field to derive non-existence results for near BHmatrices and near conference matrices. Secondly, we refine the idea of Brock inthe case of cyclotomic number fields whose ring of integers is not a principalideal domains and get many new non-existence results.

Non-Existence of Some Nearly Perfect Sequences, Near Butson-Hadamard Matrices, and Near Conference Matrices