  In this article, we study the possibility of producing a Calabi-Yau threefold
in positive characteristic which is a counter-example to Kodaira vanishing. The
only known method to construct the counter-example is so called inductive
method such as the Raynaud-Mukai construction or Russel construction. We
consider Mukai's method and its modification. Finally, as an application of
Shepherd-Barron vanishing theorem of Fano threefolds, we compute $H^1(X,
H^{-1})$ for any ample line bundle $H$ on a Calabi-Yau threefold $X$ on which
Kodaira vanishing fails.
