  We present a construction which shows that there is an infinite set of
cyclically 4-edge connected cubic graphs on $n$ vertices with no cycle longer
than $c_4 n$ for $c_4=\frac{12}{13}$, and at the same time prove that a certain
natural family of cubic graphs cannot be used to lower the shortness
coefficient $c_4$ to 0.
  The graphs we construct are snarks so we get the same upper bound for the
shortness coefficient of snarks, and we prove that the constructed graphs have
an oddness growing linearly with the number of vertices.
