Characteristic-Dependent Linear Rank Inequalities with Applications to   Network Coding
  Two characteristic-dependent linear rank inequalities are given for eight variables. Specifically, the first inequality holds for all finite fields whose characteristic is not three and does not in general hold over characteristic three. The second inequality holds for all finite fields whose characteristic is three and does not in general hold over characteristics other than three. Applications of these inequalities to the computation of capacity upper bounds in network coding are demonstrated. 
Randall Dougherty, Eric Freiling, and Kenneth Zeger
cs.IT math.IT