Smallest possible length of a linear code

Question

I want to find smallest n such there exists a binary [n,50,3]-linear code. I managed to narrow the interval down, but not to narrow it down to one value. Based on Singleton bound I can conclude that n is greater or equal to 52 and since only MDS binary codes trivial ones from which neither one has distance equal to 3 I can increase my lower limit to 53. On the other side I know there is Hamming code Ham(6,2), which is [63,57,3] code, from which I can construct a [63-7,57-7,3] code, so I set my upper limit to 63-7=56. I haven't manage to go any further so far. Do you know any methods to solve these kinds of tasks, maybe even more generic methods (e.g. not only for binary codes)?

Answer 1

Extended hints:

1. Your code has a parity check matrix $H$ with $n$ columns and $r=n-50$ rows.
2. For the code to have minimum distance three the columns of $H$ must be non-zero and distinct.
3. Each column has $r$ bits, and in view of the previous item $2^r-1$ possible values.
4. Also, in view of item 2, we need $2^r-1\ge n>50$, so $r\ge6$.