  It is well-known that the dimension of optimal anticodes in the rank-metric
is divisible by the maximum m between the number of rows and columns of the
matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes
are the codes with least maximum rank, among those of dimension k. In this
paper, we study the family of rank-metric codes whose dimension is not
divisible by m and whose maximum rank is the least possible for codes of that
dimension, according to the Anticode bound. As these are not optimal anticodes,
we call them quasi optimal anticodes (qOACs). In addition, we call dually qOAC
a qOAC whose dual is also a qOAC. We describe explicitly the structure of
dually qOACs and compute their weight distributions, generalized weights, and
associated q-polymatroids.
