Hamming codes and the binary Golay codes
Oliver Pretzel · 1992
Abstract In the previous chapter we saw that a binary linear code C can correct all single errors if and only if its check matrix H has distinct columns. In Chapter 3 it was shown that the rank of C is bounded by the number of columns of H minus the rank of H. If H has a sufficiently large set of columns, then its rank is just the length of the columns. So to make the rank of the code maximal we must choose as many distinct columns for H as possible. This leads us to define a class of single error-correcting binary codes by choosing the check matrix to have all possible non-zero columns of length k. These are the celebrated Hamming codes, which we shall denote by Ham(k).