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.