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I would like a way to create an $2$x$n$ matrix $M$ with integer entries with a large minimum kernel element $k\neq0$, when $k$ is scaled to contain only integers. If you want a precise statement of the problem it could be set up as follows.

Use the notation $|k|$ and $|M|$ for $L^2$ vector and matrix norms, respectively, of vector $k$ and $2$x$n$ matrix $M$. For a fixed dimension $n$ and a bound $C$ for which $|M|$ < $C$, choose M with integer entries to maximize $$min(|k|/|M|:k\neq0,Mk=0,\:k\: \epsilon\: \mathbb{Z}^n)$$

The type of norm does not matter to me, this is just an example of a precise statement that gets at the idea.

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    what happens for $n=3?$ – 2017-01-28
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    For n=3 it's pretty easy to find good matrices. Even with just one row [2,48,49] the best PSLQ can find is [-22,5,-4]. I found this example by reducing [4*46,5*46,7*46] mod 91. (I picked 4,5,7 somewhat randomly as small numbers). The only vectors that can annihilate [4k%N,5k%N,7k%N] must annihilate [4,5,7] mod N. Since 4, 5 and 7 are small there was no annihilation mod N=91 that was also 0. I was feeling pretty good about the prospects of constructing my "difficult" M until I went to n=4, where I have made little progress. – 2017-01-28
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    Another consideration when n=3 is that a large cross product between the two rows of M is good. In general (including n > 3) perhaps a large wedge product or small angle between the two rows? But also the individual rows should resist integer relation algorithms as I described in my previous comment. It would be nice to tie this all together and/or come up with other heuristics. – 2017-01-29

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