Example
The index groups in the catalogue follow the numbering of library of small groups used
in GAP3, GAP4, and MAGMA. For instance, the Kleinian four-group is given by G(4,2),
a group of order 4 and the second in the list of small groups of this order.
The catalogue lists the following information for this group:
Generators of nice error basis 1:
The group G(4,2) can be generated by two elements. The catalogue shows the representing
matrices of these generators.
A nice error basis is obtained as follows.
Construct the group H generated by these matrices. This is a w-covering group of G.
The group
H of order 8 and has the following
elements:
Note that the center Z(H) of H is given by the identity matrix I and the matrix -I.
If we take a transversal of Z(H) in H, then we obtain four matrices which span the whole ring of 2x2 matrices. For instance, the last four matrices are such a transversal of Z(H) in H.
Therefore, we obtain the nice error basis
All other nice error bases with index group G(4,2) are projectively equivalent to this basis.
The catalogue lists generators for the w-covering groups H such
that all nice error bases with index group G=H/Z(H)
are projectively equivalent to a transversal of Z(H) in H for one of the
w-covering groups H of G.
An entry from the Catalogue of Nice Error Bases, a joint project of
Andreas Klappenecker and
Martin Rötteler.
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