From the answer of @WłodzimierzHolsztyński, I see a misunderstanding in the terms "cover" and "tile". In particular, I think the Apollonian packing "fills" the plane, in the sense that the closure of the packing equals the plane. But it does not "cover" the plane, in the sense that there are points that do not belong to any ball. You need to define the terms "cover" and "tile" more properly.

I think his "such" does not insist on convexity.

FYI, Tumarkin pointed me to the 1985 paper of Vinberg "Hyperbolic reflection groups". In Part 4 of Sec. 5 (Ch. II), he described a construction due to Makarov, which seems to be a sequence of incommensurable Coxeter polytopes in dimension 4 and 5.

And ... it is not yet the upperbound. I think the upperbound is 15, 13 is the new lowerbound. I'll write to the author of phftables.com, so that he can update it.

Actually, it is asked on a chinese forum. The problem is just replace "four" by "six". The posts are in chinese, but the second page contains the program.

Here is the link bbs.tianya.cn/post-71-144903-1.shtml

don't say that ... in that case mine also sucks ...

In fact, I just saw a construction of (5;13,3,3), found by exhaustion (it seems), and I just checked it by program. Here it is: ['000000', '011111', '000112', '001120', '022210', '101200', '112022', '120111', '202101', '210200', '220221', '221011', '221202'] It is possible that I misunderstand something.

By the way: This PHF(6;12,3,3), is it "optimal"? Is there a PHF(6;13,3,3)?

This answer is perfect for me. Thank you.