  We study parallel algorithms for addition of numbers having finite
representation in a positional numeration system defined by a base $\beta$ in
$\mathbb{C}$ and a finite digit set $\mathcal{A}$ of contiguous integers
containing $0$. For a fixed base $\beta$, we focus on the question of the size
of the alphabet allowing to perform addition in constant time independently of
the length of representation of the summands. We produce lower bounds on the
size of such alphabet $\mathcal{A}$. For several types of well studied bases
(negative integer, complex numbers $ -1 + \imath$, $2 \imath$, and $\imath
\sqrt{2}$, quadratic Pisot unit, and the non-integer rational base), we give
explicit parallel algorithms performing addition in constant time. Moreover we
show that digit sets used by these algorithms are the smallest possible.
