  A word $w$ is called synchronizing (recurrent, reset, magic, directable) word
of deterministic finite automaton (DFA) if $w$ sends all states of the
automaton to a unique state. In 1964 Jan \v{C}erny found a sequence of n-state
complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. He
conjectured that it is an upper bound on the length of such words for complete
DFA. Nevertheless, the best upper bound $(n^3-n)/6$ was found almost 30 years
ago. We reduce the upper bound on the length of the minimal synchronizing word
to $n(7n^2+6n-16)/48$. An implemented algorithm for finding synchronizing word
with restricted upper bound is described. The work presents the distribution of
all synchronizing automata of small size according to the length of an almost
minimal synchronizing word.
