Improved constructions of secret sharing schemes by applying (λ,ω)-decompositions

By using threshold schemes, λ-decompositions were introduced by Stinson [D.R. Stinson, Decomposition constructions for secret sharing schemes, IEEE Trans. Inform. Theory IT-40 (1994) 118–125] and used to achieve often optimal worst-case information rates of secret sharing schemes based on graphs. By using the broader class of ramp schemes, (λ,ω)-decompositions were introduced in [M. van Dijk, W.-A. Jackson, K.M. Martin, A general decomposition construction for incomplete secret sharing schemes, Des. Codes Cryptogr. 15 (1998) 301–321] together with a general theory of decompositions. However, no improvements of existing schemes have been found by using this general theory. In this contribution we show for the first time how to successfully use (λ,ω)-decompositions. We give examples of improved constructions of secret sharing schemes based on connected graphs on six vertices.