  We refine a uniform algebraic approach for deriving upper bounds on reset
thresholds of synchronizing automata. We express the condition that an
automaton is synchronizing in terms of linear algebra, and obtain upper bounds
for the reset thresholds of automata with a short word of a small rank. The
results are applied to make several improvements in the area.
  We improve the best general upper bound for reset thresholds of finite prefix
codes (Huffman codes): we show that an $n$-state synchronizing decoder has a
reset word of length at most $O(n \log^3 n)$. In addition to that, we prove
that the expected reset threshold of a uniformly random synchronizing binary
$n$-state decoder is at most $O(n \log n)$. We also show that for any non-unary
alphabet there exist decoders whose reset threshold is in $\varTheta(n)$.
  We prove the \v{C}ern\'{y} conjecture for $n$-state automata with a letter of
rank at most $\sqrt[3]{6n-6}$. In another corollary, based on the recent
results of Nicaud, we show that the probability that the \v{C}ern\'y conjecture
does not hold for a random synchronizing binary automaton is exponentially
small in terms of the number of states, and also that the expected value of the
reset threshold of an $n$-state random synchronizing binary automaton is at
most $n^{3/2+o(1)}$.
  Moreover, reset words of lengths within all of our bounds are computable in
polynomial time. We present suitable algorithms for this task for various
classes of automata, such as (quasi-)one-cluster and (quasi-)Eulerian automata,
for which our results can be applied.
