Cocyclic butson Hadamard matrices and codes over Zn via the trace map

Over the past couple of years, trace maps over Galois fields and Galois rings have been used very succesfully o construct cocyclic Hadamard, complex Hadamard and Butson Hadamard matrices and subsequently to generate simplex codes over Z4, Z2 and ZP and new linear codes over ZP. Here we define a new map, the trace-like map and more generally the weighted map and extend these techniques to construct cocyclic Budson Hadamard matrices of order (nm) for all n and m and linear and non-linear codes over Zn.