Erasure codes are a critical component in reliable storage systems today, and many blockchain systems use consensus protocols that involve erasure codes to reduce their communication cost. Existing erasure codes rely on a threshold failure assumption, but recent blockchain systems have departed from this simple model and use generalized failure assumptions. This paper introduces monotone erasure codes that respect arbitrary trust assumptions on a set of nodes. The paper first describes a method for constructing a monotone erasure code from any access structure given by a monotone Boolean formula. Next, the relevant notion of a linear monotone erasure code is introduced, which works on vectors over a finite field and where the encoding is a linear operation. We then focus on constructing linear monotone erasure codes: We give an efficient algorithm to construct linear monotone erasure codes for any access structure, and we show how to efficiently construct linear monotone erasure codes for the special case of partitioned access structures with minimal storage overhead. Last but not least, this work also shows how to use monotone erasure codes to obtain a communication-efficient, generalized version of the well-known asynchronous verifiable information dispersal (AVID) primitive, which is a key building block for developing efficient reliable broadcast and consensus protocols.