Compute the minimum hamming distance

Question

to compute the min hammning distance of the 4 codewords:

1110001110010111

1001011010001110

0010111101101111

1100000000011111

Do I have to compute the hamming distance between every two codewords? is there a faster way?

Answer 1

There is no faster way in general, after all the pair of codewords you do not compute the distance for could have the minimum distance.

In specific cases, there might be faster ways if the codewords are clustered. Say you check the Hamming weights of the codewords and a subset of codewords of length $n$ have Hamming weights $>2n/3$ while another subset have Hamming weights $<n/3.$ Then, clearly the minimum distance is achieved within those clusters. This gives some improvement, especially if roughly half of the codewords are in one cluster and the rest in the other, though not necessarily significant unless you have a huge number of codewords.