  We describe two new algorithms for the generation of all non-isomorphic cubic
graphs with girth at least $k\ge 5$ which are very efficient for $5\le k \le 7$
and show how these algorithms can be efficiently restricted to generate snarks
with girth at least $k$.
  Our implementation of these algorithms is more than 30, respectively 40 times
faster than the previously fastest generator for cubic graphs with girth at
least 6 and 7, respectively.
  Using these generators we have also generated all non-isomorphic snarks with
girth at least 6 up to 38 vertices and show that there are no snarks with girth
at least 7 up to 42 vertices. We present and analyse the new list of snarks
with girth 6.
