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.