Existence of code with parameters $[22, 15, 3]_2$ I need to verify the existence of a code with parameters as in the title ($n=22,k=15,d=3,q=2$). I have seen this post but neither of the bounds works to conclude anything for this case.
I have also tried to actually construct a concrete code with these parameters, but with no success. Attempted as well some Hamming-like codes with these parameters but I can't get the distance to be 3, always get it of 2.
Could you give some clever way on how to look at this problem? Thanks in advance.

Actually, according to Markus Grassl's codetables see here a code with those parameters and distance $d=4$ exists.
Click on query form under linear block codes and enter the $n,k$ you desire. The answer comes back as below:

Bounds on linear codes [22,15] over GF(2); lower bound:4; upper bound: 4

Luckily there is even a construction given. It is a series of steps:

Regarding modification of codes such as shortening, subcodes etc, see Jon Hall's Chapter 6 of coding theory notes available online here.
Note: The Plotkin sum is also referred to as $(u|u+v)$ construction in the literature.
