Codes and Grafts as Codes

Codes and grafts have been central to the understanding of codes ever since first clever codes were created by richard hamming. Hamming's work set the stage for the ubiquitous error correcting codes of the 19 eighties. To understand why testability is so hard to obtain, we need to think of a message not just as a string of bits, but as a collection of vertices or dots connected by edges or lines.