On neighbor distances and symmetry in group codes (Corresp.)

This correspondence provides counterexamples to two frequently proposed conjectures concerning group codes for the Gaussian channel. It is shown 1) that a code for which the list of distances from codeword
i
to all other words is independent of
i
is not necessarily a group code, and 2) a group code of
M
words need not possess a transitive symmetry group of order
M
.