  We study the properties of secret sharing schemes, where a random secret
value is transformed into shares distributed among several participants in such
a way that only the qualified groups of participants can recover the secret
value. We improve the lower bounds on the sizes of shares for several specific
problems of secret sharing. To this end, we use the method of non-Shannon-type
information inequalities going back to Z. Zhang and R.W. Yeung. We employ and
extend the linear programming technique that allows to apply new information
inequalities indirectly, without even writing them down explicitly. To reduce
the complexity of the problems of linear programming involved in the bounds we
extensively use symmetry considerations.
