In my discrete math class this week, did an example of a code that encodes three bits to eleven bits and can correct all two-bit errors.   This particular code was not meant to be particularly optimal, but for one particular example I ran with 3,000,000 bits, the number of errors was roughly 1/4 of what was predicted by the probabilities, given that two bit errors can be corrected.  There were 38 failures instead of a predicted 155.   This is no mystery since the packing of spheres in {0,1}^11 wasn't expected to be all that tight, and three bit errors can often be corrected.  

The encoding function is 

To get an visualize the situation, I create the graph below.   The vertices are some of the points in the code space: the sphere of radius three centered around one of the code words.   The color coding is

• Yellow:   the code word - I used the zero word, but the space is symmetric everywhere.
• Blue:   points one unit from the code word
• Green:  points two units from the code word
• Brown:  points three units from the code word and further that three units from all other code words.
• Red:  points three units from the code word but also three units from another code word.