Linear rank inequalities in 4 subspaces are characterized by Shannon-type inequalities and the Ingleton inequality in 4 random variables. Examples of random variables violating these inequalities have been found using finite groups, and are of interest for their applications in nonlinear network coding [1]. In particular, it is known that the symmetric group S5 provides the first instance of a group, which gives rise to random variables that violate the Ingleton inequality. In the present paper, we use group theoretic methods to construct random variables which violate linear rank inequalities in 5 random variables. In this case, linear rank inequalities are fully characterized [8] using Shannon-type inequalities together with 4 Ingleton inequalities and 24 additional new inequalities. We show that finite groups which do not produce violators of the Ingleton inequality in 4 random variables will also not violate the Ingleton inequalities for 5 random variables. We then focus on 2 of the 24 additional inequalities in 5 random variables and formulate conditions for finite groups which help us eliminate those groups that obey the 2 inequalities. In particular, we show that groups of order pq, where p; q are prime, always satisfy them, and exhibit the first violator, which is the symmetric group S4.
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