Providing increased capacity in heterogeneous storage elements including a method for reading from memory. The method includes receiving a read word from a block of memory cells, where physical characteristics of the memory cells support different sets of data levels. The read word is separated into two or more virtual read vectors. For each of the virtual read vectors, the codebook that was utilized to generate the virtual read vector is identified and a partial read data vector is generated. The generating includes multiplying the virtual read vector by a matrix that represents the codebook. The partial read data vectors are combined into a read message and the read message is output.