17. A computer implemented method for transmitting data, the method comprising: receiving data to be encoded into a word for transmission across a transmission medium; receiving constraints on symbol values associated with the word; encoding the data into the word, the encoding comprising: representing the data and the constraints as a first linear system in a first field of a first size; embedding the first linear system into a second linear system in a second field of a second size, the second size larger than the first size; solving the second linear system in the second field resulting in a solution; and collapsing the solution into the first field resulting in the word, the word satisfying the constraints on symbol values associated with the word; and outputting the word on the transmission medium.
17. A computer implemented method for transmitting data, the method comprising: receiving data to be encoded into a word for transmission across a transmission medium; receiving constraints on symbol values associated with the word; encoding the data into the word, the encoding comprising: representing the data and the constraints as a first linear system in a first field of a first size; embedding the first linear system into a second linear system in a second field of a second size, the second size larger than the first size; solving the second linear system in the second field resulting in a solution; and collapsing the solution into the first field resulting in the word, the word satisfying the constraints on symbol values associated with the word; and outputting the word on the transmission medium. 21. The method of claim 17 , wherein the second linear system includes an underspecified system of equations.