  This paper deals with three technical ingredients of geometry for quantum
information. Firstly, we give an algorithm to obtain diagonal basis matrices
for submodules of the Z_{d}-module Z_{d}^{n} and we describe the suitable
computational basis. This algorithm is set along with the mathematical
properties and tools that are needed for symplectic diagonalisation. Secondly,
with only symplectic computational bases allowed, we get an explicit
description of the Lagrangian submodules of Z_{d}^{2n}. Thirdly, we introduce
the notion of a fringe of a Gram matrix and provide an explicit algorithm using
it in order to obtain a diagonal basis matrix with respect to a symplectic
computational basis whenever possible. If it is possible, we call the
corresponding submodule nearly symplectic. We also give an algebraic property
in order to single out symplectic submodules from nearly symplectic ones.
