  Affine finite automata (AfA) can be more succinct than probabilistic and
quantum finite automata when recognizing some regular languages with
bounded-error. In this paper, we improve previously known constructions given
for the succinctness of AfAs in three ways. First, we replace some of fixed
error bounds with arbitrarily small error bounds. Second, we present new
constructions by using less states than the previous constructions. Third, we
show that any language recognized by a nondeterministic finite automaton (NFA)
is also recognized by bounded-error AfAs having one more state, and so, AfAs
inherit all succinct results by NFAs. As a special case, we also show that any
language recognized by a NFA is recognized by AfAs with zero error if the
number of accepting path(s) for each member is exactly the same number.
