  Motivated by the application in geometric orthogonal codes (GOCs), Wang et
al. introduced the concept of generalized perfect difference families (PDFs),
and established the equivalence between GOCs and a certain type of generalized
PDFs recently. Based on the relationship, we discuss the existence problem of
generalized $(n\times m,K,1)$-PDFs in this paper. By using some auxiliary
designs such as semi-perfect group divisible designs and several recursive
constructions, we prove that a generalized $(n\times m, \{3,4\}, 1)$-PDF exists
if and only if $nm\equiv1\pmod{6}$. The existence of a generalized $(n\times m,
\{3,4,5\}, 1)$-PDF is also completely solved possibly except for a few values.
As a consequence, some variable-weight perfect $(n\times m,K,1)$-GOCs are
obtained.\vspace{0.2cm}
  {\bf Keywords}: generalized perfect difference family, generalized perfect
difference packing, geometric orthogonal code, semi-perfect group divisible
design
