EDIT 24 November 2011: It is my educated guess that the sum of eight squares is one of the "odd" borderline lattices. I'm trying to figure out how to check this.
EDIT 24 November 2011: It is my educated guess that the sum of eight squares is one of the "odd" borderline lattices. I'm trying to figure out how to check this.
The computer that hosts my websites is down for the next month, so here is the table of "odd" lattice successes up to five variables. It is probably complete up to dimension 4, in dimension 5 I was getting near the upper bound of Nipp's tables so I am less certain about completeness. There are surely some of these in dimensions 6,7,8, probably not 9,10.
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f11
1
3
5
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f11 f12 f22
1 0 1
1 0 2
1 0 3
1 0 5
2 0 3
2 2 3
3 0 3
-------------------------
1 0 4 borderline
2 2 5 borderline
3 2 3 borderline
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
disc: f11 f22 f33 f23 f13 f12
succeed 4: 1 1 1 0 0 0
succeed 8: 1 1 2 0 0 0
succeed 12: 1 1 3 0 0 0
succeed 12: 1 2 2 2 0 0
succeed 20: 1 1 5 0 0 0
succeed 20: 1 2 3 2 0 0
succeed 24: 1 2 3 0 0 0
succeed 28: 2 2 3 2 2 2
succeed 36: 1 2 5 2 0 0
succeed 36: 1 3 3 0 0 0
succeed 36: 2 2 3 0 0 2
succeed 60: 2 3 3 0 0 2
-----------------------------------------------------------
succeed 16: 1 1 4 0 0 0 borderline
succeed 16: 1 2 2 0 0 0 borderline
succeed 32: 1 3 3 2 0 0 borderline
succeed 32: 2 2 3 2 2 0 borderline
succeed 48: 2 3 3 2 2 2 borderline
succeed 64: 3 3 3 -2 2 2 borderline
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disc: f11 f22 f33 f44 f12 f13 f23 f14 f24 f34
succeed 16: 1 1 1 1 0 0 0 0 0 0
succeed 32: 1 1 1 2 0 0 0 0 0 0
succeed 48: 1 1 1 3 0 0 0 0 0 0
succeed 48: 1 1 2 2 0 0 0 0 0 2
succeed 80: 1 1 2 3 0 0 0 0 0 2
succeed 96: 1 1 2 3 0 0 0 0 0 0
succeed 112: 1 2 2 3 0 0 2 0 2 0
succeed 144: 1 1 2 5 0 0 0 0 0 2
succeed 144: 1 2 2 3 0 0 2 0 0 0
succeed 240: 2 2 3 3 2 2 0 2 0 0
succeed 240: 1 2 3 3 0 0 0 0 2 0
-----------------------------------------------------------
succeed 64: 1 1 1 4 0 0 0 0 0 0 borderline
succeed 64: 1 1 2 2 0 0 0 0 0 0 borderline
succeed 64: 1 2 2 2 0 0 2 0 2 0 borderline
succeed 80: 1 1 1 5 0 0 0 0 0 0 borderline
succeed 128: 1 1 3 3 0 0 0 0 0 2 borderline
succeed 128: 1 2 2 3 0 0 0 0 2 2 borderline
succeed 128: 2 2 2 3 2 2 0 2 0 0 borderline
succeed 144: 1 1 3 3 0 0 0 0 0 0 borderline
succeed 192: 1 2 3 3 0 0 2 0 2 0 borderline
succeed 192: 2 2 2 3 0 0 0 2 2 2 borderline
succeed 256: 1 3 3 3 0 0 2 0 2 -2 borderline
succeed 256: 2 2 3 3 0 2 2 2 2 2 borderline
succeed 256: 2 2 3 3 2 2 0 0 2 0 borderline
succeed 304: 2 2 3 3 0 2 2 0 2 0 borderline
succeed 512: 3 3 3 3 2 2 -2 -2 -2 2 borderline
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disc: f11 f22 f33 f44 f55 f12 f13 f23 f14 f24 f34 f15 f25 f35 f45
succeed 16: 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
succeed 32: 1 1 1 1 2 0 0 0 0 0 0 0 0 0 0
succeed 48: 1 1 1 1 3 0 0 0 0 0 0 0 0 0 0
succeed 48: 1 1 1 2 2 0 0 0 0 0 0 0 0 0 2
succeed 80: 1 1 1 2 3 0 0 0 0 0 0 0 0 0 2
succeed 112: 1 1 2 2 3 0 0 0 0 0 2 0 0 2 0
succeed 144: 1 1 2 2 3 0 0 0 0 0 2 0 0 0 0
-----------------------------------------------------------
succeed 64: 1 1 1 1 4 0 0 0 0 0 0 0 0 0 0 borderline
succeed 64: 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 borderline
succeed 64: 1 1 2 2 2 0 0 0 0 0 2 0 0 0 2 borderline
succeed 64: 1 2 2 2 2 0 0 0 0 0 0 0 2 2 2 borderline
succeed 96: 1 1 1 2 3 0 0 0 0 0 0 0 0 0 0 borderline
succeed 112: 1 1 1 2 4 0 0 0 0 0 0 0 0 0 2 borderline
succeed 128: 1 1 1 3 3 0 0 0 0 0 0 0 0 0 2 borderline
succeed 128: 1 1 2 2 3 0 0 0 0 0 0 0 0 2 2 borderline
succeed 128: 1 2 2 2 3 0 0 2 0 2 0 0 2 0 0 borderline
succeed 144: 1 2 2 2 3 0 0 2 0 2 0 0 0 0 2 borderline
succeed 144: 1 1 1 2 5 0 0 0 0 0 0 0 0 0 2 borderline
succeed 160: 1 1 2 2 3 0 0 0 0 0 0 0 0 0 2 borderline
succeed 192: 1 1 2 3 3 0 0 0 0 0 2 0 0 2 0 borderline
succeed 192: 1 2 2 2 3 0 0 0 0 0 0 0 2 2 2 borderline
succeed 208: 1 1 2 3 3 0 0 0 0 0 0 0 0 2 2 borderline
succeed 240: 1 1 2 3 3 0 0 0 0 0 2 0 0 0 0 borderline
succeed 256: 1 1 3 3 3 0 0 0 0 0 2 0 0 2 -2 borderline
succeed 256: 1 2 2 3 3 0 0 0 0 2 2 0 2 2 2 borderline
succeed 256: 1 2 2 3 3 0 0 2 0 2 0 0 0 2 0 borderline
succeed 256: 2 2 2 2 3 0 0 0 0 0 0 2 2 2 2 borderline
succeed 256: 2 2 2 3 3 2 2 0 2 0 0 2 0 0 2 borderline
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In the "borderline" cases, one may insert extra points at precisely located points to create an overlattice, still integral but possibly "even," that does better than the original as far as covering radius smallness. That is the real end of the project, finding all the borderline guys and describing these overlattices.
The computer that hosts my websites is down for the next month, so here is the table of "odd" lattice successes up to five variables. It is probably complete up to dimension 4, in dimension 5 I was getting near the upper bound of Nipp's tables so I am less certain about completeness. There are surely some of these in dimensions 6,7,8, probably not 9,10.
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
f11
1
3
5
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
f11 f12 f22
1 0 1
1 0 2
1 0 3
1 0 5
2 0 3
2 2 3
3 0 3
-------------------------
1 0 4 borderline
2 2 5 borderline
3 2 3 borderline
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
disc: f11 f22 f33 f23 f13 f12
succeed 4: 1 1 1 0 0 0
succeed 8: 1 1 2 0 0 0
succeed 12: 1 1 3 0 0 0
succeed 12: 1 2 2 2 0 0
succeed 20: 1 1 5 0 0 0
succeed 20: 1 2 3 2 0 0
succeed 24: 1 2 3 0 0 0
succeed 28: 2 2 3 2 2 2
succeed 36: 1 2 5 2 0 0
succeed 36: 1 3 3 0 0 0
succeed 36: 2 2 3 0 0 2
succeed 60: 2 3 3 0 0 2
-----------------------------------------------------------
succeed 16: 1 1 4 0 0 0 borderline
succeed 16: 1 2 2 0 0 0 borderline
succeed 32: 1 3 3 2 0 0 borderline
succeed 32: 2 2 3 2 2 0 borderline
succeed 48: 2 3 3 2 2 2 borderline
succeed 64: 3 3 3 -2 2 2 borderline
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
disc: f11 f22 f33 f44 f12 f13 f23 f14 f24 f34
succeed 16: 1 1 1 1 0 0 0 0 0 0
succeed 32: 1 1 1 2 0 0 0 0 0 0
succeed 48: 1 1 1 3 0 0 0 0 0 0
succeed 48: 1 1 2 2 0 0 0 0 0 2
succeed 80: 1 1 2 3 0 0 0 0 0 2
succeed 96: 1 1 2 3 0 0 0 0 0 0
succeed 112: 1 2 2 3 0 0 2 0 2 0
succeed 144: 1 1 2 5 0 0 0 0 0 2
succeed 144: 1 2 2 3 0 0 2 0 0 0
succeed 240: 2 2 3 3 2 2 0 2 0 0
succeed 240: 1 2 3 3 0 0 0 0 2 0
-----------------------------------------------------------
succeed 64: 1 1 1 4 0 0 0 0 0 0 borderline
succeed 64: 1 1 2 2 0 0 0 0 0 0 borderline
succeed 64: 1 2 2 2 0 0 2 0 2 0 borderline
succeed 80: 1 1 1 5 0 0 0 0 0 0 borderline
succeed 128: 1 1 3 3 0 0 0 0 0 2 borderline
succeed 128: 1 2 2 3 0 0 0 0 2 2 borderline
succeed 128: 2 2 2 3 2 2 0 2 0 0 borderline
succeed 144: 1 1 3 3 0 0 0 0 0 0 borderline
succeed 192: 1 2 3 3 0 0 2 0 2 0 borderline
succeed 192: 2 2 2 3 0 0 0 2 2 2 borderline
succeed 256: 1 3 3 3 0 0 2 0 2 -2 borderline
succeed 256: 2 2 3 3 0 2 2 2 2 2 borderline
succeed 256: 2 2 3 3 2 2 0 0 2 0 borderline
succeed 304: 2 2 3 3 0 2 2 0 2 0 borderline
succeed 512: 3 3 3 3 2 2 -2 -2 -2 2 borderline
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disc: f11 f22 f33 f44 f55 f12 f13 f23 f14 f24 f34 f15 f25 f35 f45
succeed 16: 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
succeed 32: 1 1 1 1 2 0 0 0 0 0 0 0 0 0 0
succeed 48: 1 1 1 1 3 0 0 0 0 0 0 0 0 0 0
succeed 48: 1 1 1 2 2 0 0 0 0 0 0 0 0 0 2
succeed 80: 1 1 1 2 3 0 0 0 0 0 0 0 0 0 2
succeed 112: 1 1 2 2 3 0 0 0 0 0 2 0 0 2 0
succeed 144: 1 1 2 2 3 0 0 0 0 0 2 0 0 0 0
-----------------------------------------------------------
succeed 64: 1 1 1 1 4 0 0 0 0 0 0 0 0 0 0 borderline
succeed 64: 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 borderline
succeed 64: 1 1 2 2 2 0 0 0 0 0 2 0 0 0 2 borderline
succeed 64: 1 2 2 2 2 0 0 0 0 0 0 0 2 2 2 borderline
succeed 96: 1 1 1 2 3 0 0 0 0 0 0 0 0 0 0 borderline
succeed 112: 1 1 1 2 4 0 0 0 0 0 0 0 0 0 2 borderline
succeed 128: 1 1 1 3 3 0 0 0 0 0 0 0 0 0 2 borderline
succeed 128: 1 1 2 2 3 0 0 0 0 0 0 0 0 2 2 borderline
succeed 128: 1 2 2 2 3 0 0 2 0 2 0 0 2 0 0 borderline
succeed 144: 1 2 2 2 3 0 0 2 0 2 0 0 0 0 2 borderline
succeed 144: 1 1 1 2 5 0 0 0 0 0 0 0 0 0 2 borderline
succeed 160: 1 1 2 2 3 0 0 0 0 0 0 0 0 0 2 borderline
succeed 192: 1 1 2 3 3 0 0 0 0 0 2 0 0 2 0 borderline
succeed 192: 1 2 2 2 3 0 0 0 0 0 0 0 2 2 2 borderline
succeed 208: 1 1 2 3 3 0 0 0 0 0 0 0 0 2 2 borderline
succeed 240: 1 1 2 3 3 0 0 0 0 0 2 0 0 0 0 borderline
succeed 256: 1 1 3 3 3 0 0 0 0 0 2 0 0 2 -2 borderline
succeed 256: 1 2 2 3 3 0 0 0 0 2 2 0 2 2 2 borderline
succeed 256: 1 2 2 3 3 0 0 2 0 2 0 0 0 2 0 borderline
succeed 256: 2 2 2 2 3 0 0 0 0 0 0 2 2 2 2 borderline
succeed 256: 2 2 2 3 3 2 2 0 2 0 0 2 0 0 2 borderline
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
In the "borderline" cases, one may insert extra points at precisely located points to create an overlattice, still integral but possibly "even," that does better than the original as far as covering radius smallness. That is the real end of the project, finding all the borderline guys and describing these overlattices.
Alright, make this a separate answer... there are tables of positive forms available up to dimension 5. So I revised some C++ programs and found the "odd" lattices that satisfy Allcock's rule, the ambient vector space is covered when one places an open ball of radius 1 around each lattice point of odd norm, then an open ball of radius $\sqrt 2$ around each lattice point of even norm (such as the origin). A lattice is called "borderline" if those open balls do not suffice but closed balls of the same radii do the job.
So I am posting the table, so far (dimension 1,2,3,4,5), at this LINK. One can see that these are integral lattices as the matrix entries listed (one matrix per line) gives inner products always integral, meaning all f_ij are even when i,j are distinct, but the lattices are "odd" in that at least one diagonal entry f_ii is odd.