Let
be a finite abelian
group and
be a positive
integer. A subset
of
is called a
perfect -basisof if each
element of
can be written uniquely as the sum of at most
(not necessarily
distinct) elements of
;
similarly, we say that
is
a
perfect restricted -basisof if each
element of
can be written uniquely as the sum of at most
distinct elements of
. We prove that perfect
-bases exist only in
the trivial cases of
or
. The situation
is different with restricted addition where perfection is more frequent; here we treat the case of
and prove that
has a perfect restricted
-basis if, and only if,
it is isomorphic to
,
,
,
,
, or
.
Keywords
Abelian group, sumset, restricted addition, basis, $B_h$
set