  Secret sharing schemes with optimal and universal communication overheads
have been obtained independently by Bitar et al. and Huang et al. However,
their constructions require a finite field of size q > n, where n is the number
of shares, and do not provide strong security. In this work, we give a general
framework to construct communication efficient secret sharing schemes based on
sequences of nested linear codes, which allows to use in particular algebraic
geometry codes and allows to obtain strongly secure and communication efficient
schemes. Using this framework, we obtain: 1) schemes with universal and close
to optimal communication overheads for arbitrarily large lengths n and a fixed
finite field, 2) the first construction of schemes with universal and optimal
communication overheads and optimal strong security (for restricted lengths),
having in particular the component-wise security advantages of perfect schemes
and the security and storage efficiency of ramp schemes, and 3) schemes with
universal and close to optimal communication overheads and close to optimal
strong security defined for arbitrarily large lengths n and a fixed finite
field.
