Non-Existence of Some Nearly Perfect Sequences, Near Butson-Hadamard   Matrices, and Near Conference Matrices
  In this paper we study the non-existence problem of (nearly) perfect (almost) $m$-ary sequences via their connection to (near) Butson-Hadamard (BH) matrices and (near) conference matrices. Firstly, we apply a result on vanishing sums of roots of unity and a result of Brock on the unsolvability of certain equations over a cyclotomic number field to derive non-existence results for near BH matrices and near conference matrices. Secondly, we refine the idea of Brock in the case of cyclotomic number fields whose ring of integers is not a principal ideal domains and get many new non-existence results. 
Arne Winterhof and O\u{g}uz Yayla and Volker Ziegler
math.NT math.CO