Note: This problem is on holiday (as of Sage 6.7), while some buggy Sage code for the minimum distance of a Hamming code gets sorted out. The r = 2 case produces an error message and for r > 5 the computation of the minimum distance has become intolerably slow. So it is a bit harder to make a reasonable conjecture from just 3 cases.
The dual of an (n,k) block code is formed as all the set of all binary vectors which are orthogonal to every vector of the original code. Exercise 8.6.25 describes this construction and asks about some of its properties.
You can construct the dual of a code in Sage with the .dual\_code() method. Construct the binary Hamming codes, and their duals, with the parameter r ranging from 2 to 5, inclusive. Build a table with six columns (perhaps employing the html.table() function) that lists r, the length of the codes, the dimensions of the original and the dual, and the minimum distances of the original and the dual.
Conjecture formulas for the dimension and minimum distance of the dual of the Hamming code as expressions in the parameter r.