Ok, thanks to @Seva giving me the right name for these, I found out that:

• The sets I'm describing are Perfect Difference sets.
• The one-dimensional version I'm working with are equivalent to projective planes.
• https://en.wikipedia.org/wiki/Projective_plane says that there is no plane for $j=6$ $(j^2+j+1)$, which is equivalent to my case where $k=7$ $(k^2+k+1)$.

So, not an error in my code, proven impossible prior to 1938 (as a different problem) and related to projective planes in 1938.

Ok, thanks to @Seva giving me the right name for these, I found out that:

• The sets I'm describing are Perfect Difference sets.
• The one-dimensional version I'm working with are equivalent to projective planes.
• https://en.wikipedia.org/wiki/Projective_plane says that there is no plane for $j=6$ $(j^2+j+1)$, which is equivalent to my case where $k=7$ $(k^2-k+1)$.

So, not an error in my code, proven impossible prior to 1938 (as a different problem) and related to projective planes in 1938.