Hamming distance of linear block code(7,2)

Question

How would I go about calculating the hamming distance of linear block code(7,2)?

I'm given nothing else, I know that the code will be 7 bits long, have 2 data bits and 5 error bits.

Answer 1

Edit: This answer describes the limits on the possible minimum distance values. Too long for a comment.

Since the code has $2^2=4$ codewords and there are $2^7=128$ vectors in the space, the maximum possible volume $V$ of disjoint equal radius hamming spheres must obey $$ V\leq 32. $$ Since $$ 1+ \binom{7}{1}+ \binom{7}{2}=29<32 $$ it is conceivable that a 2 error correcting code with minimum Hamming distance 5 exists.

The optimal $(7,4)$ Hamming code has distance $3.$ You can't, however select a subset of its codewords and obtain a code with minimum distance 5, since its weight distribution (hence distance distribution, being linear) is not supported at 5. It has no weight 5 codewords.