# A Priori proof that Covering Radius strictly less than $\sqrt 2$ implies class number one

It turns out that each of Pete L. Clark's "euclidean" quadratic forms, as long as it has coefficients in the rational integers $\mathbb Z$ and is positive, is in a genus containing only one equivalence class of forms. In the language of (positive) integral lattices, the condition is that the covering radius be strictly smaller than $\sqrt 2.$ Please see these for background:

Intuition for the last step in Serre's proof of the three-squares theorem

Is the square of the covering radius of an integral lattice/quadratic form always rational?

I have been trying, for some months, to find an a priori proof that Euclidean implies class number one. I suspect, without much ability to check, that any such Euclidean form has a stronger property, if it represents any integral form (of the same dimension or lower) over the rationals $\mathbb Q$ then it also represents it over $\mathbb Z.$ This is the natural extension of Pete's ADC property to full dimension. Note that a form does rationally represent any form in its genus, with Siegel's additional restriction of "no essential denominator." If the ADC property holds in the same dimension, lots of complicated genus theory becomes irrelevant.

EDIT: Pete suggests people look at §4.4 of http://math.uga.edu/~pete/ADCFormsI.pdf .

EDIT 2: It is necessary to require Pete's strict inequality, otherwise the Leech lattice appears.

So that is my question, can anyone prove a priori that a positive Euclidean form over $\mathbb Z$ has class number one?

EDIT 3: I wrote to R. Borcherds who gave me a rough idea, based on taking the sum of a given lattice with a 2-dimensional Lorentzian lattice. From page 378 in SPLAG first edition, two lattices are in the same genus if and only if their sums with the same 2-dimensional Lorentzian lattice are integrally equivalent. I hope someone posts a fuller answer, otherwise I'll be spending the next six months trying to complete the sketch myself. The references: Lattices like the Leech lattice, J.Alg. Vol 130, No. 1, April 1990, p.219-234, then earlier The Leech lattice, Proc. Royal Soc. London A398 (1985) 365-376. Quoted:

But there is a good way to show that some genus of lattices L has class number 1 if its covering radius is small. What you do is look at the sum of L and a 2-dimensional Lorentzian lattice. Then other lattices in the genus correspond to some norm 0 vectors in the Lorentzian lattice. On the other hand, if the covering radius is small enough one can use this to show that a fundamental domain of the reflection group of the Lorentzian lattice has only one cusp, so all primitive norm 0 vectors are conjugate and therefore there is only 1 lattice in the genus of L. This works nicely when L is the E8 lattice for example. There are some variations of this. When the covering radius is exactly sqrt 2 the other lattices in the genus correspond to deep holes, as in the Leech lattice (Conway's theorem). The covering radius sqrt 2 condition corresponds to norm 2 reflections, and more generally one can consider reflections corresponding to vectors of other norms; see http://dx.doi.org/10.1016/0021-8693(90)90110-A for details

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Just a couple of comments: (i) this is a serious research question in the sense that several people who know something about quadratic forms have been trying to answer it for well over a year (and I have too!). Unless we're all missing something pretty obvious, the answer to this question might well be publishable.... –  Pete L. Clark Jul 4 '11 at 5:01
(ii) The reason I refer to my preprint is that I have a more general setup in which it makes sense to ask the same question: namely for quadratic forms over any $S$-integer ring in a global field. The answer is so far known for definite forms over $\mathbb{Z}$ and $\mathbb{F}_q(t)$, by methods which are not, as Will puts it, a priori. It would be great to have an argument which sheds light on the more general case. –  Pete L. Clark Jul 4 '11 at 5:03

This is the more important case of "even" lattices, where all inner products are integral and all vector norms are even. We follow pages 131-134 in Wolfgang Ebeling, Lattices and Codes, available for sale at LINK \$45.00 for paperback. Given an even positive lattice$\Lambda$with covering radius below$\sqrt 2,$form the integral Lorentzian lattice$L = \Lambda \oplus U.$Elements are of the form$ (\lambda, m,n) $where$\lambda \in \Lambda, \; m,n \in \mathbb Z.$The norm on$L$is given by $$(\lambda, m,n)^2 = \lambda^2 + 2 mn.$$ We infer the inner product $$(\lambda_1, m_1,n_1) \cdot (\lambda_2, m_2,n_2) = \lambda_1 \cdot \lambda_2 + m_1 n_2 + m_2 n_1.$$ We choose a particular set of roots (elements of norm 2) beginning with any$\lambda \in \Lambda$by $$\tilde{\lambda} = \left( \lambda, 1, 1 - \frac{\lambda^2}{2} \right).$$ Let the group$ R \subseteq \mbox{Aut}(L)$be generated by reflections in all the$\tilde{\lambda}$and by$\pm 1.$Given any$ l \in L,$a primitive null vector, we have$l \cdot l = 0,$and so$l \in l^\perp.$Furthermore, if$k \in l^\perp$as well, then$ (k + l)^2 = k^2.$As a result, we may form the lattice spanned$\langle l \rangle$by$l$itself, then form another lattice with norm, $$E(l) = l^\perp / \langle l \rangle.$$ It took me a bit of doing to confirm (a calculus exercise along with Cauchy-Schwarz) that$E(l)$is positive definite, all norms are positive except for the$0$class. Given any root$r \in L,$meaning$r^2 = 2,$we get the reflection $$s_r (z) = z - ( r \cdot z) r.$$ Also$ s_r^2(z) = z.$It is an exercise to show that, when$ s_r(z) = y,$then$E(z)$and$E(y)$are isomorphic. There is rather more than first appears to the question: In particular, every even lattice in the same genus as$\Lambda$occurs as some$E(u),$where$u \in L$is a primitive null vector. Furthermore, taking$w = (0, 0,1),$we find that$E(w) = \Lambda.$So, all we really need to do to prove class number one is show that every primitive null vector can be taken to$w = (\vec{0}, 0,1),$by a sequence of operations in$R.$Given some primitive null vector $$z = (\xi, a,b),$$ so that$ 2 a b = - \xi^2.$Note that, in order to have a primitive null vector, if one of$a,b$is 0, then$\xi = 0$and the other one of$a,b$is$\pm 1.$In the first case, suppose$|b| < |a|.$Then$ | 2 a b| = \xi^2 < 2 a^2,$so in fact $$\left( \frac{\xi}{a} \right)^2 < 2.$$ We choose the root$ \tilde{\lambda} = \left( 0, 1, 1 \right).$Then$z \cdot \tilde{\lambda} = b + a,$and $$s_{\tilde{\lambda}} (z) = z - ( \tilde{\lambda} \cdot z) \tilde{\lambda} = (\xi, -b,-a).$$ Therefore, we may always force the second case, which is$|a| \leq |b|.$We assume that$a \neq 0,$so that$ b = \frac{- \xi^2}{2a}.$Now we have$ 2 a^2 \leq | 2 a b| = \xi^2,$so$ \left( \frac{\xi}{a} \right)^2 \geq 2.$From the covering radius condition, there is then some nonzero vector$\lambda \in \Lambda$such that the rational number $$\left( \frac{\xi}{a} - \lambda \right)^2 < 2.$$ We form the root from this$\lambda,$as in $$\tilde{\lambda} = \left( \lambda, 1, 1 - \frac{\lambda^2}{2} \right).$$ For convenience we write $$a' = \frac{a}{2} \left( \frac{\xi}{a} - \lambda \right)^2$$ From the equation$z \cdot \tilde{\lambda} = a - a',$we see that$a' \in \mathbb Z.$Well, we have $$s_{\tilde{\lambda}} (z) = (\xi - ( a - a') \lambda, a',b'),$$ where $$b' = b - ( a - a') \left( 1 - \frac{\lambda^2}{2} \right) = \frac{- \xi^2}{2a} - ( a - a') \left( 1 - \frac{\lambda^2}{2} \right).$$ Now, if$a'=0,$then$ s_{\tilde{\lambda}} (z) = (0,0,\pm 1),$and an application of$\pm 1 \in R$takes us to$w = (0, 0,1),$with$E(w) = \Lambda.$If, instead,$a' \neq 0,$note that our use of the covering radius condition shows that$ | a'| < | a|,$while$a,a'$share the same$\pm$sign, that is their product is positive. So$a - a'$shares the same sign, and$| a - a'| < |a|.$From$2 a b = - \xi^2$we know that$b$has the opposite sign. But$2 a' b' = - (\xi - ( a - a') \lambda)^2,$so$b'$has the opposite sign to$a'$and$a-a'$and the same sign as$b.$Now,$\lambda^2 \geq 2,$so $$1 - \frac{\lambda^2}{2} \leq 0,$$ and$ ( a - a') \left( 1 - \frac{\lambda^2}{2} \right)$has the same sign as$b$and$b'.$From $$b' = b - ( a - a') \left( 1 - \frac{\lambda^2}{2} \right)$$ we conclude that$| b'| \leq |b|.$So, these steps have$|a| + |b|$strictly decreasing, until such time that one of them becomes 0, and we have arrived at$w.$So, actually, the covering radius hypothesis implies that all primitive null vectors are mapped to$w$by a finite sequence of reflections, so that all the$E(z)$are in fact isomorphic to$E(w) = \Lambda.$That is, all lattices in the genus are in fact isomorphic, and the class number is one. - Bravo! Don't forget to put this on meta at MO success stories. – David Roberts Jul 22 '11 at 3:57 Thank you, David – Will Jagy Jul 22 '11 at 4:05 EDIT, Tuesday, July 26. I have convinced myself, with my own C++ programs, that the sum of five squares does satisfy the "easier" Borcherds-Allcock condition, although it fails Pete L. Clark's criterion. Four squares works, I believe that six and seven squares will work, while eight squares will be "borderline," with the worst behavior happening at the point with all coordinates being 1/2. Now, as Prof. Nebe has designed a new algorithm for covering radius, and Magma uses this now, I can at least hope that she gets interested enough to design an algorithm for this hybrid condition. Otherwise it is going to take me forever to find all of the "odd" lattices that succeed. Not at all by the way, the original impetus for Prof. Nebe's involvement was a request by Richard Parker for the positive integral lattices P for which it would be possible to calculate Aut(P + U) where U is the unimodular two-dimensional "even" Lorentzian lattice... I finally figured out that Nebe gets every lattice to be "even" by doubling all coefficients, this particularly happening to the "odd" lattices such as the sum of$k$squares. If we were to leave the odd lattices as they are, her condition (and Pete's) would be that the covering radius be strictly less than 1! The reality check is that both Nebe and Pete include the sum of three squares, but both exclude the sum of five squares. For comparison, see LINK Before continuing, let me point out that this strong condition happens only a few times, $$x^2, 3 x^2, \; x^2 + y^2, x^2 + 2 y^2, 2 x^2 + 2 x y + 3 y^2, \; x^2 + y^2 + z^2, x^2 + 2 y^2 + 2 y z + 2 z^2,$$ in dimension no larger than 3. Note that the earlier proof for "even" lattices does, in fact, apply to the "odd" lattices. Both Richard Borcherds and his student, Daniel Allcock of U. T. Austin, pointed out that I did not have quite the "correct" condition. In the Duke Math. J., vol. 103 (2000) pp. 303-333, see LINK section 6, Allcock discusses what he calls a "well-covering," which is a covering by balls of varied radius, where the radius of the ball around a given lattice point is decided by the norm of the lattice point. In particular, he says (but not in the paper) that the correct condition for odd'' lattices is that$\mathbb R^n$is covered by {\bf open} balls of radius 1 around all lattice points of odd norm, then radius$\sqrt 2$around lattice points of even norm, where the norm of some$x \in \Lambda$is$x \cdot x.$This is called a "strict well-covering." This condition is milder than Pete's. Given such a positive odd lattice$\Lambda$with a strict well-covering as described, form the integral Lorentzian lattice$L = \Lambda \oplus U.$Elements are of the form$ (\lambda, m,n) $where$\lambda \in \Lambda, \; m,n \in \mathbb Z.$The norm on$L$is given by $$(\lambda, m,n)^2 = \lambda^2 + 2 mn.$$ We infer the inner product $$(\lambda_1, m_1,n_1) \cdot (\lambda_2, m_2,n_2) = \lambda_1 \cdot \lambda_2 + m_1 n_2 + m_2 n_1.$$ Note that the sublattice of all points with even norm has index 2, and that norm 0 vectors in$L = \Lambda \oplus U$can only be created from such points with even norm. We choose a particular set of roots (elements of norm 2) beginning with any$\lambda \in \Lambda$of even norm by $$\tilde{\tilde{\lambda}} = \left( \lambda, 1, 1 - \frac{\lambda^2}{2} \right).$$ However, starting with points$\beta \in \Lambda$of odd norm, we create root-ones by $$\tilde{\beta} = \left( \beta, 1, \frac{ 1 - \beta^2}{2} \right),$$ which are elements of norm 1 in$L.$Note that when an English soccer goalkeeper boots the ball down the center of the pitch and attack remains central, this is referred to as root-one football. I'm sure I have that right. Let the group$ R \subseteq \mbox{Aut}(L)$be generated by reflections in all the$\tilde{\tilde{\lambda}}$and all the$ \tilde{\beta}$and by$\pm 1.$Given any root$r \in L,$meaning$r^2 = 2,$we get the reflection $$s_r (z) = z - ( r \cdot z) r.$$ Also$ s_r^2(z) = z.$Given any root-one$ \tilde{\beta} \in L,$meaning$ \tilde{\beta}^2 = 1,$we get the reflection $$s_{\tilde{\beta}} (z) = z - 2( \tilde{\beta} \cdot z) \tilde{\beta}.$$ Also$ s_{\tilde{\beta}}^2(z) = z.$Given some primitive null vector $$z = (\xi, a,b),$$ so that$ 2 a b = - \xi^2,$and$b = \frac{- \xi^2}{2 a} \in \mathbb Z.$Note that, in order to have a primitive null vector, if one of$a,b$is 0, then$\xi = 0$and the other one of$a,b$is$\pm 1.$In the first case, suppose$|b| < |a|.$Then$ | 2 a b| = \xi^2 < 2 a^2,$so in fact $$\left( \frac{\xi}{a} \right)^2 < 2.$$ We choose the root$ \tilde{\lambda} = \left( 0, 1, 1 \right).$Then$z \cdot \tilde{\lambda} = b + a,$and $$s_{\tilde{\lambda}} (z) = z - ( \tilde{\lambda} \cdot z) \tilde{\lambda} = (\xi, -b,-a).$$ Therefore, we may always force the second case, which is$|a| \leq |b|.$We assume that$a \neq 0,$so that$ b = \frac{- \xi^2}{2a}.$Now we have$ 2 a^2 \leq | 2 a b| = \xi^2,$so$ \left( \frac{\xi}{a} \right)^2 \geq 2.$From the covering radius condition, there is then some nonzero vector$\lambda \in \Lambda$such that the rational number $$\left( \frac{\xi}{a} - \lambda \right)^2 < 2.$$ If$\lambda$has even norm, everything proceeds as before. However, it is possible the only point close enough is$\beta \in \Lambda$of odd norm. We create the root-one$ \tilde{\beta} = \left( \beta, 1, \frac{ 1 - \beta^2}{2} \right).$By Allcock's strict well-covering condition, $$\left( \frac{\xi}{a} - \beta \right)^2 < 1.$$ For convenience we write $$a' = a \left( \frac{\xi}{a} - \beta \right)^2$$ From the equation$ 2 z \cdot \tilde{\beta} = a - a',$we see that$a' \in \mathbf Z.$Well, we have $$s_{\tilde{\beta}} (z) = (\xi - ( a - a') \beta, a',b'),$$ where $$b' = b - ( a - a') \left( \frac{1 - \beta^2}{2} \right) = \frac{- \xi^2}{2a} - ( a - a') \left( \frac{1 - \beta^2}{2} \right) .$$ Now, if$a'=0,$then$ s_{\tilde{\beta}} (z) = (0,0,\pm 1),$and an application of$\pm 1 \in R$takes us to$w = (0, 0,1),$with$E(w) = \Lambda.$If, instead,$a' \neq 0,$note that our use of the strict well-covering condition shows that$ | a'| < | a|,$while$a,a'$share the same$\pm$sign, that is their product is positive. So$a - a'$shares the same sign, and$| a - a'| < |a|.$From$2 a b = - \xi^2$we know that$b$has the opposite sign. But$2 a' b' = - (\xi - ( a - a') \beta)^2,$so$b'$has the opposite sign to$a'$and$a-a'$and the same sign as$b.$Now,$\beta^2 \geq 1,$so $$\frac{1 - \beta^2}{2} \leq 0,$$ and$ ( a - a') \left( \frac{1 - \beta^2}{2} \right)$has the same sign as$b$and$b'.$From $$b' = b - ( a - a') \left( \frac{ 1 -\beta^2}{2} \right)$$ we conclude that$| b'| \leq |b|.$So, these steps also have$|a| + |b|$strictly decreasing, until such time that one of them becomes 0, and we have arrived at$w = (0, 0,1).$- The football (soccer) phrase is actually "route one football" not "root one". – Allan MacLeod Nov 24 '11 at 16:11 Thank you, Allan. I put a fair amount of effort into setting that up, then no-one noticed for months. You have revived my faith in the human spirit. On current news, I think it would be nice if the entire Premier League lined up and slapped Mario Balotelli with fish, youtube.com/watch?v=IhJQp-q1Y1s – Will Jagy Nov 24 '11 at 20:47 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.

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.

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.

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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
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

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.

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