Jacques Martinet Last update: March 21st, 2023

Complements to Perfect Lattices in Euclidean Spaces.

Note. Since the publication of Perfect Lattices in Euclidean Spaces, improvements on several questions considered in this book appeared in the literature, or have been noticed by various researchers who read my book. Besides the erratum, my home page displays an update of the reference list (completion of the original which appeared after the book as well as new references quoted after the original reference list). In what follows, we intend to list some results which improve on those which occur in the book, and either give (an outline of) a proof or refer to the updated reference list. The last page, formerly an appendix to the erratum, is devoted to Craig lattices.

Section 1.10.C. Problems involving tensor products of lattices play an interesting rôle in Arakelov theory. The interested reader could look at the following

two papers by D. Grayson (Comment. Math. Helvetici 59 (1984),600– 634 & 61 (1986), 661–676) and at recent papers by Y. André (Tohoku Math. J. 63 (2011), 629-–649), J.-B. Bost & H.Chen (arXiv:1203.0216v1 [math.NT]), B. Casselman (Asian Journal of Mathematics 8 (2004); issue dedicated to Armand Borel), and É. Gaudron & G. Rémond (arXiv:1109.2812v1 [math.NT]); quoted by Fabien Pazuki in

http://www.math.u-bordeaux1.fr/ fpazuki/GDTGD.html .

Section 2.2.C.

An analogue of the Hermite constant for global function fields is defined in [Hu-Y].

Section 2.3.C.

Generalization of the Mordell inequality to lattices endowed with various algebraic structures have been obtained by Stephanie Vance in [Van1]. In particular, her inequality shows that the maximal Hermite invariant among 16-dimensional lattices over the Hurwitz order is that of the Barnes-Wall lattice Λ_{16}; this makes use of the fact (proved by Sigrist and by Schürmann) that the densest 12-dimensional Hurwitz lattices are Λ_{12}^{max} and Λ_{12}^{min}.

Section 2.6.C.

Achill Schürmann pointed out to me that van der Waerden’s constants δ_{p} quoted in Theorem 2.6.11 are indeed equal to (\frac{5}{4})^{p−4} for all p ≥ 4. Alternative reference:

[vdW’], same title, Acta Math. 96 (1956), 265–309.

He moreover puts ahead a conjecture according to which the exponential bound above could be replaced for n ≤ 8 by the linear bound \frac{n}{4}. This I have proved in an unpublished preprint (Hermite versus Minkowski, November, 2007), now on arXiv as [Mar14]. [Added August 12th, 2021.] I recently noticed the paper [Reg] by S. Regavim (and its joint appendix with L. Hadassi) which throws new light on reduction problems for lattices. What concerns Schürmann’s conjecture (around van der Waerden’s theorem) has been previously solved up to dimension 8 in [Mar14] in a somewhat crude form. However the results of Regavim for dimensions 6 and 7 are more precise.

The methods of [Mar14] can probably be pushed to dimension 9 using [Mar-Schr1], and maybe to dimensions 10 and 11, using unpublished data of M. Dutour-Sikirić, but do not look suitable to attack the problem in large dimensions.

Regavim’s paper reminds me of the existence of the paper [L-L-S], which appeared between the French 1996 and English 2003 editions of my book, and that I should have quoted in my printed book.

Section 2.8.C.

C. Poor and D. Yuen have obtained the exact values of γ′_{n} and the corresponding dual-critical lattice for n = 5, 6, 7; see Section 6.4.C below. [November 14th, 2008.] In a mail dated October 23rd, 2008, T. Watan-

abe wrote: “In the last Monday , we have found that Hermite-Rankin constants of 8-dimensional are immediately determined by the Bergé-Martinet constants of 5 and 7 dimensional, which were determined by Poor and Yuen.”; see [S-Wt-O]; the “Bergé-Martinet” inequalities are those of [B-M1] and are accounted for in this section 2.8.

(Feb. 19, 2009). Recent progress has been made on the constant γ′′_{n} introduced in [B-M1].

(1) On request of the editorial board of “J. Th. Nombres de Bordeaux”, a paper by Marc Gindraux ([Gi’], replacing the inaccessible [Gi] quoted in the book) has been published in the third issue of 2009 of the journal. This is a compact form of a 2002 text intended to be part of a Ph. D., which was never completed (the author no longer works in a university). The constant γn^{′′} is broken into two constants γn,+^{′′} and γn,−^{′′} in order to take care of problems of non-continuity; see “publications by various authors” in my homepage.

(2) The paper [Pe-vZ] by R.A. Pendavingh et S.H.M. van Zam gives a very sharp upper bound for γ′′_{5}, which strongly suggests that the exact value is attained on Blichfeldt’s example.

Section 3.2.C.

Definition 3.22 (3) could be enlarged by introducing the notion of strong semi-eutaxy: Definition 3.2.C1. We say that a system of lines or a lattice is strongly semi-eutactic if it is semi-eutactic with equal nonzero coefficients.

It is easily checked that the set of lines (or vectors) having nonzero corresponding eutaxy coefficients is then a 3-design in the sense of Chapter 16. The first example occurs in dimension 4; see the file strongeut.gp on my home page, which contains the complete classification of strongly (semi-)eutactic lattices up to dimension 6. A more “classical” example is provided by K′_{9}^{∗}; see Section 8..

In the course of the joint research [B-M8] with Anne-Marie Bergé, we noticed the following result on weakly eutactic lattices, which improves Corollary 3.2.6 for such lattices:

Proposition 3.2.C2 (June, 2005). Let Λ be weakly eutactic and let Id = ∑_{s∈S(Λ)/±} ρ_{x} p_{x} be a eutaxy relation for Λ. Then the set of minimal vectors of Λ having strictly positive coefficients ρ_{x} spans E.

Proof. Let T = {x ∈ S | ρ_{x} > 0} and let F ⊂ E be the span of T. Let y ∈ E be orthogonal to all vectors of T. Let us apply to y the

eutaxy relation above. Since p_{x}(y) = \frac{1}{min Λ} (x·y) x, this relation reads

y = \frac{1}{min Λ} ∑_{ρ_{x}<0} ρ_{x} (x · y) x. Taking the scalar product of both sides

with y, we now obtain N(y) = \frac{1}{min Λ} ∑_{ρ_{x}<0} ρ_{x} (x · y)^{2}.

Since the left hand side is ≥ 0 and the right hand side is ≤ 0, both are zero. In particular, we have N(y) = 0, i.e. y = 0. □

Section 3.5.C.

The characterization given in Proposition 3.5.3, (3) of perfection for lattices having a perfect hyperplane section with the same minimum has the following easy generalization: perf Λ = \frac{n(n−1)}{2} + rk S \ S ∩ H (where perf stands for the perfection rank). This formula is itself a special case of the following more general statement:

Proposition 3.5.C3. Let S be a (finite) subset of E, let H be a hyperplane in E, and let S_{0} = S ∩ H. Denote by r_{0} the perfection rank of S_{0}, by r that of S, and by t the rank of S \ S_{0}. Then we have the inequality r ≥ r_{0} + t , and equality holds whenever either S contains exactly t pairs of vectors off S_{0}, or S_{0} is perfect.

[This applies to a lattice Λ and a hyperplane section Λ_{0} of Λ, assuming that min Λ_{0} = min Λ.]

Proof. We normalize all vectors of S to norm 1, and denote by e a unit vector orthogonal to H.

First observe that vectors x_{1}, …, x_{k} ∈ S \ S_{0} are dependent if and only if there exists a relation ∑ λ_{i} p_{x_{i}} = 0 in End^{s}(E)/ End^{s}(H) with coefficients not all zero off H. Indeed, a relation ∑_{x} λ_{x} p_{x} = 0 in End^{s}(E) implies ∑_{x∉H} λ_{x} p_{x} ∘ p_{e} because p_{x} ∘ p_{e} = 0 on H. Applied to e, it reads ∑_{x∉H} λ_{x} (x · e) x = 0, a non-trivial dependence relation since x · e ≠ 0 off H. Conversely, a dependent relation can be written ∑_{x∉H} λ_{x} (x · e) x = 0, and implies ∑_{x} λ_{x} p_{x} ∈ End^{s}(H), since for all y ∈ E, we have

∑_{x} λ_{x} (p_{x} ∘ p_{e})(y) = (e · y) ∑_{x} λ_{x} (x · e) x.

Now choose t independent vectors y_{1}, …, y_{t} ∈ S \ S_{0} and vectors x_{1}, …, x_{r_{0}} ∈ S_{0} such that the p_{x_{i}} constitute a basis for the span of the p_{x}, x ∈ S_{0}. Then the projections p_{x_{i}}, p_{y_{j}} are independent, which shows that r ≥ r_{0} + t.

If S \ S_{0} reduces to the ±x_{i}, then the vector space {p_{x} | x ∈ S} is generated by the p_{x_{i}}, p_{y_{j}}, whence the inequality r ≤ r_{0} + t in this case.

If S_{0} is perfect, then the p_{x}, x ∈ S_{0} span End^{s}(H), so that the p_{x}, x ∈ S \ S_{0} are again linear combinations of the p_{x_{i}} and the p_{y_{j}}. □

The notion of a perfection relation occurs only incidentally in the book (in Remark 6.2.5.). A formal definition should have been given in this section: a perfection relation on a lattice Λ is a relation of the form

∑_{x∈S(Λ)/{±}} a_{x} p_{x}

with real coefficients a_{x}. In terms of matrices, this takes the form ∑_{X∈S(M)/{±}} a′_{X} X^{t}X, with set of coefficients (a′_{X}) proportional to (a_{x}). Taking the trace shows that the a_{x} add to zero.

The set of perfection relations is a real vector space of dimension s − r (s is the kissing number, r is the perfection rank); and the set of eutaxy relations, if non-empty, is an affine space over the space of perfection relations. More details can be found in the paper [B-M10] of the complementary reference list.

Section 3.8.C.

The first part of Corollary 3.8.6 could have been more suitably stated in a form which separates the properties of (dual-) perfection and eutaxy. For a perfect lattice, we can even prove slightly more.

Corollary 3.8.C1. Let Λ be a eutactic (resp. an extreme) lattice. If Λ^{∗} is eutactic (resp. semi-eutactic), then Λ is dual-eutactic (resp. dual-extreme).

Proof. Since Λ^{∗} is semi-eutactic, there exists for Λ^{∗} a eutaxy relation of the form Id = ∑_{y∈S_{1}/±} ρ_{y} p_{y} with strictly positive coefficients ρ′_{y} for some symmetric subset S_{1} of S(Λ^{∗}). Eliminating the identity in this relation and in a eutaxy relation for Λ, we obtain a relation of dual-eutaxy of the form

∑_{x∈S(Λ)/±} ρ_{x} p_{x} = ∑_{y∈S_{1}/±} ρ′_{y} p_{y}

with strictly positive coefficients ρ_{x}, ρ′_{y}.

If S_{0} = S(Λ^{∗}) \ S_{1} is empty, this shows that Λ is dual-eutactic.

Otherwise, we may assume that Λ is extreme. Then the endomorphism ∑_{z∈S_{0}/±} p_{z} is a combination ∑_{x∈S(Λ)} μ_{x} p_{x}. For ε > 0 small enough, the relation

∑_{x∈S(Λ)/±} (ρ_{x} + εμ_{x}) p_{x} = ∑_{y∈S_{1}/±} ρ_{y} p_{y} + ∑_{z∈S_{0}/±} ε p_{z}

is a relation of dual-eutaxy with strictly positive coefficients. □

[The argument indeed shows that if Λ is perfect and possesses a relation of dual-eutaxy ∑_{x∈S(Λ)} ρ_{x} p_{x} = ∑_{y∈S(Λ^{∗})} ρ′_{y} p_{y} with strictly positive ρ_{x} and non-negative ρ′_{y}, then Λ is dual-extreme.]

In general, dual-eutaxy does not imply any eutaxy property. However, the following statement establishes a partial converse to the corollary above:

Proposition 3.8.C2. Let Λ be a eutactic and dual-eutactic lattice. Assume that its automorphism group acts transitively on its set of minimal vectors. Then Λ^{∗} is eutactic.

Proof. Consider a relation of dual-eutaxy

∑_{x∈S(Λ)/±} ρ_{x} p_{x} = ∑_{y∈S(Λ^{∗})/±} ρ′_{y} p_{y}

with strictly positive coefficients ρ_{x}, ρ′_{y}. By averaging on the automorphism group, we obtain a new relation of the form

∑_{x∈S(Λ)/±} p_{x} = ∑_{y∈S(Λ^{∗})/±} ρ′′_{y} p_{y}

with strictly positive coefficients ρ′′_{y}. By Theorem 3.6.6, Λ is strongly eutactic. This shows that the left hand side of the equality above is proportional to the identity, and so is the right hand side. □

The mere condition that Λ should be strongly eutactic does not imply that Λ^{∗} is eutactic. An example is provided by the 9-dimensional lattice Λ found by Baril and referred to in Section 14.5, for which γ^{′2} = \frac{16}{5}, the largest known value in dimension 9 (also attained on Coxeter’s 𝔸_{9}^{2}). Here is a Gram matrix for Λ; see also strongeut.gp, matrix baril9:

( \begin{matrix}6 & −2 & 2 & −1 & −2 & 0 & −2 & −1 & 2 \\ −2 & 6 & −2 & 2 & −1 & −2 & 0 & −2 & −1 \\ 2 & −2 & 6 & −2 & 2 & −1 & −2 & 0 & −2 \\ −1 & 2 & −2 & 6 & −2 & 2 & −1 & −2 & 0 \\ −2 & −1 & 2 & −2 & 6 & −2 & 2 & −1 & −2 \\ 0 & −2 & −1 & 2 & −2 & 6 & −2 & 2 & −1 \\ −2 & 0 & −2 & −1 & 2 & −2 & 6 & −2 & 2 \\ −1 & −2 & 0 & −2 & −1 & 2 & −2 & 6 & −2 \\ 2 & −1 & −2 & 0 & −2 & −1 & 2 & −2 & 6 \end{matrix} )

This lattice is extreme and strongly eutactic, with s = 45; its dual, which has s = 40, is not even weakly eutactic. The automorphism group of Λ acts transitively on S(Λ^{∗}), but has two orbits S_{1} with s_{1} = 5 and S_{2} with s_{2} = 40 on S = S(Λ). Dual eutaxy can be studied using the averaging argument of Proposition 3.8.8: if Λ is dual-eutactic, there exists a relation of dual eutaxy involving sums of projections to vectors of S_{1}, S_{2} and S^{∗}. Using formula 3.8.3’, we check that such a relation must be unique up to proportionality and that

8 ∑_{x∈S_{1}/±} p_{x} + 3 ∑_{x∈S_{2}/±} p_{x} = 4 ∑_{y∈S^{∗}/±} p_{y}

indeed holds.

An attempt for classifying dual-extreme lattices in dimension 5 was done by Anne-Marie Bergé at the time the English edition of this book was prepared. Her aim was to prove that γ′_{5} is attained uniquely on the two perfect lattices P_{5}^{1} and P_{5}^{2} and their duals, at a time when Poor-Yuen’s result was no known. I briefly explain her ideas below.

Her basic idea consists in looking at putative dual-extreme lattices in each minimal class. A lattice with perfection rank r = 15 is extreme and dualextreme (there are three such lattices, namely 𝔻_{5}, 𝔸_{5}^{3}, and 𝔻_{5}). We now consider classes and lattices with s ≤ 14. Next observe that if a 5-dimensional lattice has one (resp. at least two) perfect 𝔻_{4} minimal sections, then it has s^{∗} = 1 (resp s ≥ 12 + 12 − 6 = 18, hence is similar to 𝔻_{5}), so that we may restrict ourselves to lattices without a 𝔻_{4} minimal sections, that is, lattices with r = s. Since we have s + s^{∗} ≥ \frac{n(n+1)}{2} + 1 = 16, we may restrict ourselves to classes with r ≥ 8. Using essentially computations by hand, she succeeded in dealing with all classes of perfection rank r ≥ 11. She probably proved more, namely that there does not exist dual-extreme lattices with perfection rank r = 14, 13, 12 or 11.

Up to similarity, lattices in a class with perfection rank r depend on 15 − r parameters, so that the difficulty of the task is a rapidly increasing function of the co-rank 15 − r. To go further, and to check A.-M. Bergé’s results, it would be highly desirable to make algorithmic her procedure.

Sections 4.2.C to 4.5.C (September 202)

The knowledge of the sets of orbits of pairs of minimal vectors having a given scalar products is useful is some applications (e.g., to the theory of graphs). This is done in the book for the lattice E_{8}. Here is the general result:

Proposition. Let Λ be an irreducible root lattice. Then the ordered pairs (x, y) of minimal vectors of Λ having a given scalar product constitute a unique orbit under Aut(Λ) except for 𝔻_{n}, n ≥ 5 where there are two orbits if x · y = 0, with representatives (ε_{1} + ε_{2}, ε_{1} − ε_{2}) and (ε_{1} + ε_{2}, (ε_{3} + ε_{4}).

Sketch of proof. By transitivity we may choose x arbitrarily. We choose x = ε_{0} −ε_{1} for 𝔸_{n} and x = ε_{1} +ε_{2} for 𝔻_{n} and 𝔼_{n}. It suffices to consider the cases x · y = 1 and x · y = 0.

𝔸_{n}. Then x · y = 1 (resp. x · y = 0) amounts to y = ε_{0} − ε_{i} or −ε_{i} + ε_{i}, i ≥ 2 (resp. y = ±(ε_{i} − ε_{j}), 2 ≤ i < j ≤ n), and under the fixator G_{1} of x in {± Id} × S_{n+1}, y is clearly equivalent to ε_{0} − ε_{2} (resp. to ε_{2} − ε_{3}).

𝔻_{n}. We consider the subgroup G = {±1} × S_{n} of Aut(A_{n}) and the fixator G_{1} of x in G. Then x · y = 1 amounts to y = ε_{1} ± ε_{i} or ε_{2} ± ε_{i}, i ≥ 3, and y is clearly equivalent to ε_{1} + ε_{3} under G_{1}.

However, ±(ε_{1} − ε_{2}) and ±ε_{i} ± ε_{j}, 3 ≤ i < j ≤ n belong to different orbits under G_{1}, those of ε_{1} − ε_{2} and ε_{1} + ε_{3}. Since G is the whole automorphism group of 𝔻_{n} for n ≥ 5, we are done if n ≥ 5. If n = 4 we identify 𝔻_{4} with the ring 𝔐 of Hurwitz quaternions equipped with the form Trd(x\overline{y}) and the basis (ε_{ℓ}) to (1, i, j, k). Then right multiplication with ω := \frac{−1+i+j+k}{2} transforms (1+i, 1−i) into (−1+k, i+j) belonging to distinct orbits.

𝔼_{n}, n = 8, 7, 6. Let e = \frac{(ε_{1}−ε_{2})+vp_{3}+ε_{4}+ε_{5}+(ε_{6}−ε_{7}+ε_{8})}{2} and e′ = e + ε_{2}. These are elements of 𝔼_{6} and we have 𝔼_{8} = 𝔻_{8} ∪ (e + 𝔻_{8}). Under the action of the Weyl group W (𝔻_{8}) ∩ 𝔼_{n}, vector y ∈ 𝔼_{n} with x · y = 1 (resp. x · y = 0) are equivalent to ε_{1} + ε_{3} or e′ (resp. ε_{1} − ε_{2}, ε_{3} + ε_{4} or e). Using the formulae displayed in the proof of Theorem 4.4.4, we check that the symmetry along e send ε_{1} + ε_{3}, ε_{1} − ε_{2} and ε_{3} + ε_{4} to elements of e + 𝔻_{8}, which completes the proof of the proposition. □

Section 4.C. Exercises. (July, 2010; January, 2012.)

I should have added a useful third question to Exercise 4.3.1, and written some more exercises on vectors having a given norm in root lattices.

Exercise 4.3.1... 3. Denote by 𝔻_{2} the embedding of 𝔸_{1} ⊥ 𝔸_{1} of the form ⟨ε_{i} + ε_{j}, ε_{i} − ε_{j}⟩ and by 𝔻_{3} that of 𝔸_{3} of the form ⟨ε_{i} − ε_{j}, ε_{i} − ε_{k}, ε_{j} + ε_{k}⟩. Show that every root sublattice of rank n of 𝔻_{n} is isometric to an orthogonal sum of lattices 𝔻_{m_{i}} with m_{i} ≥ 2 and ∑_{i} m_{i} = n, and is well defined up to an automorphism of 𝔻_{n} by the set {m_{i}}. Nota bene. For the complementary exercises for Chapter 4 below, the reader could look at my papers [Mar8] and [Mar10] of the extended reference list on the homepage.

Note in particular that for any n-dimensional lattice of minimum m, if two vectors x, y of norms N ≤ 2m define the same class modulo 2, then either y = ±y, or N(y) = N(x) = 2m and y · x = 0; in the latter case, for a given x, the set of vectors \frac{y±x}{2} with N(y) = 4 and y ≡ x mod 2 constitute a root system of type D D D_{k} rescaled to norm m (k = k(x) may depend on x). Other results for classes modulo 2 or 3 can be found in [Mar8] or [Mar10]. Exercise 4.C1. 1. Show that the vectors of norm 4 in an exceptional lattice Λ constitute a single orbit under Aut(Λ), containing 135 pairs of vectors if Λ ≃ 𝔼_{6}, 378 if Λ ≃ 𝔼_{7}, and 1080 if Λ ≃ 𝔼_{8}.

2. Show that the set of norm 4 vectors congruent to one of them modulo 2 consists of k pairwise orthogonal pairs of vectors with k = 5 if Λ ≃ 𝔼_{6}, k = 6 if Λ ≃ 𝔼_{7}, and k = 8 if Λ ≃ 𝔼_{8}. [One can choose as representatives under Aut(Λ) the system ±2ε_{i} with 1 ≤ i ≤ 5, 1 ≤ i ≤ 6, and 1 ≤ i ≤ 8, respectively.]

3. Show that s_{2} + \frac{s_{4}}{k} is equal to 2^{6} − 1 if Λ ≃ 𝔼_{6}, to 2^{8} − 1 if Λ ≃ 𝔼_{8}, and to 2^{8} − 2 if Λ ≃ 𝔼_{7}.

4. Show that if if Λ ≃ 𝔼_{6} or Λ ≃ 𝔼_{8}, all classes of Λ modulo 2 have representatives of norm N ≤ 4, but that there is one missing class if Λ ≃ 𝔼_{7}.

Exercise 4.C2. (The missing class in 𝔼_{7} modulo 2.) 1. Show that the

norm 6 vector y = ε_{1} + ε_{2} + ε_{3} + ε_{4} + ε_{5} + ε_{6} ∈ 𝔼_{7} is not congruent to a vector of smaller norm.

2. Show that the class C_{x} of x modulo 2 consists of

(a) the 16 pairs ±ε_{1} ± ε_{2} ± ε_{3} ± ε_{4} ± ε_{5} ± ε_{6} having an even number of minus signs, and

(b) the 12 pairs ε_{7} − ε_{8} + 2ε_{i}, i = 1, 2, 3, 4, 5, 6.

3. Show that C_{x} = 2 S(𝔼_{7}^{∗}).

4. Prove that there are 1008 pairs of norm-6 vectors off 2S(𝔼_{7}^{∗}), that they constitute one orbit under Aut(𝔼_{7}), and that every minimal vector of 𝔼_{7} is congruent to 16 such pairs of vectors. (See Exercise 4.4.9.) Exercise 4.C3. 1. Show that the automorphism group of 𝔼_{6} acts transitively on its norm 6 vectors.

2. Show that the norm 6 vectors of the class C of ε_{1} + ε_{2} consists of

(a) the 6 pairs ε_{1} − ε_{2} ± 2ε_{i}, i = 3, 4, 5, and

(b) the 4 pairs ±ε_{3} + ±ε_{4} + ±ε_{5} + ±(ε_{6} − ε_{7} + ε_{8}) having an even number of minus signs.

3. Show that the configuration of these 10 pairs of vectors is that of S(𝔸_{5}^{2}). (The Coxeter lattice 𝔸_{n}^{2} is defined in Section 5.2.)

Exercise 4.C.4. E_{7}^{∗} modulo 2 (scaled to minimum 3).

1. Show that the norm of x ∈ 𝔼_{7}^{∗} is of the form 4k + 3 if x ∈ 𝔼_{7}^{∗} \ 𝔼_{7} and 4k if x ∈ 𝔼_{7}, k ≥ 0.

2. Show that there is one orbit of vectors for each of the norms 3, 4, 7, having 28, 63 and 288 pairs of vectors, respectively.

3. Prove that vectors of norm 7 appear in sets of 8 pairs ±x consisting of one class modulo 2 𝔼_{7}^{∗}, with configuration 𝔸_{7}^{∗}.

4. Prove that these vectors represent all non-zero classes modulo 2, according to the formula 28 + 63 + \frac{288}{8} = 2^{7} − 1.

Section 5.1.C.

Not to assume that r is integral is useless!

Section 5.2.C.

A way of using Coxeter lattices to increase γ(𝔸_{n}) is to consider lattices ⟨𝔸_{m}^{r}, 𝔸_{n}⟩ for m ≤ n and r | m+1. In particular lattices with r | n of Proposition 5.2.5 are isometric to ⟨𝔸_{n−1}^{r}, 𝔸_{n}⟩. This applies notably to the dense lattices 𝔸_{8}^{2} and 𝔸_{9}^{3} and shows immediately the existence of cross-sections 𝔼_{7} and 𝔼_{8}, respectively. One can even insert in 𝔸_{n} two or more copies of Coxeter lattices of lower dimensions provided n is large enough.

Section 5.3.C. (May 2nd, 2015)

The way we construct the lattices P_{n} = 𝔸_{n}^{(2)} in this section is a special case of the following general procedure. Consider an n-dimensional lattice Λ with successive layers of norm m_{0} = 0, m_{1} = min Λ, m_{2}, m_{3}, ..., a finite additive Abelian group A of order m, and a surjective homomorphism φ : Λ → A. Then L := ker φ is a lattice of index m^{2} in Λ, hence det(L) = m^{2} det(Λ). If φ is non-zero on S(Λ) we have min L ≥ m_{2}. In case of equality (which is in practice the usual situation) we have γ(L) = γ(Λ) × \frac{m_{2}}{m_{1}} m^{−2/n}. Otherwise we shall have a

multiplicator \frac{m_{3}}{m_{1}} m^{−2/n}, etc. When A is cyclic, L can be viewed as the lattice orthogonal modulo m to a convenient vector of Λ^{∗}.

Taking Λ = 𝔸_{n} equipped with its Korkine-Zolotareff basis (e_{i}), we must get rid of the vectors e_{i} and e_{i} − e_{j}, i < j, which needs φ(e_{i}) ≠ 0 and φ(e_{j}) ≠ φ(e_{i}) for i < j, hence |A \ {0}| ≥ n, i.e., m ≥ n + 1 ; and if m = n + 1, all choices of φ are equivalent under the automorphisms of 𝔸_{n} induced by permutations of the e_{i}; for instance, we may take φ(e_{i}) = i mod m, a map induced by φ(ε_{i}) = i on the canonical basis (ε_{0}, ε_{1}, …, ε_{n}) of ℤ^{n+1}.

Taking A = ℤ/mℤ we obtain Barnes’s lattices P_{n}. A generalization of this construction yields the Craig lattices 𝔸_{n}^{(r)} of minimum ≥ m_{r}, a lower bound which is sometimes strict, as one can see in the appendix.

I could have made more precise Proposition 5.3.6. Proposition 5.3.6.C. For n ≥ 11, the group G_{n}, of order 2(n+1)φ(n+1), is the full automorphism group of P_{n}.

A similar construction can be done with 𝔻_{n}. The smallest possible modulus is m = 2n − 2, which can be achieved taking for φ the restriction of the function defined on ℤ^{n} by φ(ε_{i}) = i−1 mod m. Conjecturally, these lattices are perfect for n ≥ 9, satisfy S(L^{∗}) = S(𝔻_{n}^{∗}) [= S(ℤ^{n})] for n ≥ 14, and are never strongly eutactic. I have not tested them for eutaxy.

Constructing the even part of an odd lattice is also a construction of type “ker φ”; other such constructions occur in further chapters (e.g., modulo 3 in Section 8.4).

Section 5.4.C.

Variants of the Craig lattices and analogues defined using ideals 𝔭_{p}^{i}𝔭_{q}^{j} in cyclotomic fields ℚ(ζ_{pq}) (p, q primes) have been considered by Flores et al. in [F-I-P] and [FIPLO]. A different kind of generalization has been considered by Hao Chen in [Chen].

Section 6.4.C.

In lemma 6.4.13, the ten minimal vectors (up to sign) outside Λ′ are e, e − e_{i} (i = 1, 2, 3), and e − e_{j} − e_{k} (j = 1, 2, 3; k = 4, 5). The matrix B_{2} corresponds to the more symmetric choice e−e_{i} −e_{j}, 1 ≤ i < j ≤ 5.

[January 23rd, 2007.] Conjecture 6.4.16 (and the natural ones for dimensions 6 and 7) has been solved by Cris Poor and David Yuen ([P-Y4], 2006): γ′_{5} is attained exactly on 𝔻_{5}, 𝔻_{5}^{∗}, 𝔸_{5}^{3}, 𝔸_{5}^{3}^{∗} = 𝔸_{5}^{2}, γ′_{6} on 𝔼_{6} and 𝔼_{6}^{∗}, and γ′_{7} on 𝔼_{7} and 𝔼_{7}^{∗}. We have

(γ′_{5})^{2} = 2 , (γ′_{6})^{2} = \frac{8}{3} and (γ′_{7})^{2} = 3 .

Their proofs make use of a new function satisfying a convenient convexity property (rather “concavity”), a type one function. Curiously, proofs are shorter in dimensions 6 and 7, in relation with the fact that the two densest perfect lattices in these dimensions are dual to each other. In dimension 5, the proof is by inspection of the minimal classes (classified by Batut), with however an extra trick thanks to which they avoid the consideration of all cases.

Section 6.6.C.

In this overview of our knowledge on 8-dimensional perfect lattices, I mentioned the problem of eutaxy only in the case of Laïhem lattices, which had been dealt with by Jaquet ([Ja7]) a few months after Laïhem’s thesis had been completed. Cordian Riener ([Rie]) has recently established the status of the 10916 perfect lattices. His data confirm Jaquet’s result. We summarize them in the table below:

Eutaxy for Perfect 8-Dimensional Lattices

Type  Laïhem Baril Napias Batut total
Eutactic  383  16 1929 66 2394
Semi-eut.  21  0  5  2  28
Weakly eut. 771  37 7608 78 8494
Total  1175 53 9542 146 10916

In the table above, “semi-eutactic” means “semi-eutactic, non-eutactic”; similarly, “weakly eutactic” means “weakly eutactic, non-semi-eutactic”.

Among these 2394 eutactic lattices, only 7 are strongly eutactic, namely the Laïhem lattices lh(2), lh(3), lh(8), lh(271), lh(1172) ≃ 𝔼_{8}, lh(1174) ≃ 𝔻_{8}, and lh(1175) ≃ 𝔸_{8}. Their duals are also strongly eutactic except lh(3), for which s^{∗} = 3, and lh(8).

We now consider the property of being dual-extreme (or dual-eutac-tic, this amounts to the same since these lattices are perfect). A necessary condition is that S(Λ^{∗}) should have rank 8. This condition is satisfied by only 8 lattices: the 7 Laïhem lattices lh(2), lh(6), lh(8), lh(271), lh(1172), lh(1174), and lh(1175), and the Napias lattice nap(6920). Since the duals of 5 of them are strongly eutactic (see above), we are left with lh(6), the dual of which is only (strongly) semi-eutactic, and lh(8) and nap(6920), the duals of which are not weakly eutactic. We have calculated the orbit structures of (Λ, Λ^{∗}) for these three lattices: (3, 2), (3, 1) and (5, 1) respectively, and applied the method used in Section 3.8.C above to deal with the 9-dimensional Baril lattice. The conclusion is that lh(6) and lh(8) are dual-extreme and that nap(6920) is not. (For lh(6), one can use Corollary 3.8.C1.)

Work by Mathieu Dutour, Achill Schürmann and Frank Vallentin (e-mail from A. Schürmann, June 30th, 2005) has thrown new light on dimension n = 8 (and also n = 9, for which they found more than 500 000 lattices). In particular, they showed that:

1. All neighbours of the known perfect lattices except perhaps 𝔼_{8} belong to the list of previously known lattices.

2. Every lattice of the list is a neighbour of some lattice of the list with kissing number s ≤ 39.

3. Every lattice of the list except 𝔸_{8} and nap(7014) is a neighbour of 𝔼_{8}.

In an e-mail dated October 5th, 2005, Schürmann wrote “... we finally managed, with the help of Mathieu Dutour’s code for the so called adjacency decomposition method, to finish the classification of 8-dimensional perfect forms.” The corresponding manuscript now exists on arXiv, Article math.NT/0609388, Sept. 18, 2006 ([D-S-V1]). This work proves that the list of 10916 perfect, 8-dimensional lattices referred to in my book (and for which Gram matrices can be downloaded from my home page) is complete. In particular, Conjecture 6.6.7 is true (all perfect 8-dimensional lattices have a basis of minimal vectors). The classification also shows that all perfect 8-dimensional lattices have a hexagonal section with the same norm; in particular, all have an even minimum in any scale which make them integral.

[Integral perfect lattices with an odd minimum exist in dimension n = 1, n = 7 and also n ≥ 10 by [B-M7], and do not exist for n = 2, 3, 4, 5, 6 and 8. The case of dimension 9 remains open. However, Riener checked that the (more than) 500,000 perfect 9-dimensional lattices quoted above have an even minimum in any scale which make them integral. Anne-Marie Bergé and myself have conjectured that this is general, and even that all perfect 9-dimensional lattices do have hexagonal sections with the same minimum.]

[updated on March 21st, 2023.] Over two billions of perfect lattices in dimension 9 have been found by van Woerden ([vW1]). Also lower and upper bounds for the number of perfect lattices in a given dimension have been obtained by Bacher ([Bc5], 2017) and by van Woerden ([vW2], 2018).

Section 7.4.C.

For relations between the Voronoi graph and minimal classes, see Section 9.1.C below.

Section 8.4.C.

In Theorem 8.4.2, for n = 2h + 1 odd, we have s(L_{n}) = \frac{9h(h−1)}{2} + \frac{3h}{2}. This accounts for the existence of two orbits of minimal vectors, that of the section L_{n−1}, and its complementary set in S(L_{n}).

Section 8.7.C. (a) See the erratum.

(b) January 12th, 2023. The lattices K′_{n}, n ≤ 24 are defined in dimensions n ≤ 12 inside K′_{12} = K_{12}, then by symmetry in dimensions 24 − n. They differ from laminated lattices only if 3 ≤ n ≤ 21. One can define K′_{21} as the orthogonal of a minimal square in Λ_{23}^{∗} (using instead a minimal hexagonal lattice yields Λ_{21}). An alternative construction is as follows: taking the orthogonal of a norm 6 vector in Leech’s Λ_{24} we obtain a lattice Λ′_{23} well-defined up to isometry; we then obtain by “antilaminations” first a lattice Λ′_{22}, next a lattice Λ′_{21} ≃ L′_{21}.

The lattices Λ′_{23} and Λ′_{22} are perfect, strongly eutactic, and have strongly eutactic duals, hence are extreme and dual-extreme, Section 9.1.C.

After the proof of Theorem 9.1.5, I remark that perfect classes (those which contain a perfect lattice, i.e., those of maximal perfection rank \frac{n(n+1)}{2}) reduce to the similarity class of one perfect lattice. Here I should have also remarked that classes of rank \frac{n(n+1)}{2} − 1 are represented by Voronoi paths. Voronoi paths are listed in Section 6.5. There is one path per orbit of their perfect endpoints. However it may happen that two distinct orbits define equivalent minimal classes. (March 20th, 2023.) In dimensions n ≤ 7, this occurs exactly with the two paths (i, j) = (1, 9) and (1, 10), two out of the eleven paths P_{7}^{1}—P_{7}^{1}. What happens for n = 8is not known. A more convenient description of the Voronoi graph would be to classify the edges by their minimal class rather than by isometries of their endpoints.

Section 9.4.C.

In the joint research with A.-M. Bergé referred to above in connection with Corollary 3.2.6, we were lead to use a stronger form of Corollary 9.4.2. Here is a more precise statement:

Proposition 9.4.C1 (June, 2005). On a minimal class C consisting of lattices whose minimal vectors do not span E (a “non-well– rounded class”), the Hermite invariant has no maximum nor minimum. Moreover, inf_{Λ∈C} γ(Λ) = 0 .

Proof. Only the “moreover” part needs a proof. Let Λ ∈ C, let F ⊂ E be the span of S(Λ), and let r = dim F. For λ ≥ 1, let u_{λ} be the linear map which is the identity on F and multiplication by λ on F^{⊥}, and set Λ_{λ} = u_{λ}(Λ). For x ∈ E with components y on F and z on F^{⊥}, we have u_{λ}(x) = x+λy, hence N(u_{λ}(x)) = N(x)+λ^{2}N(y) ≥ N(x). Hence S(Λ_{λ}) = S(Λ), so that Λ is still in C. But det(u_{λ}) = λ^{n−r}, hence

γ(Λ_{λ}) = γ(Λ) · λ^{−2(n−r)/n} ⟶_{λ→∞} 0 . □

Section 9.5.C. (December, 2013)

By Theorem 9.5.2, the fields of definition of all weakly eutactic lattices up to dimension 5 are totally real. Batut (private communication) has made an exploration of all minimal classes lying below that of 𝔸_{6}. He found eutactic lattices which are defined over non-totally real fields, notably on a cubic field with mixed signature.

(July, 2017). Weakly eutactic lattices belonging to Voronoi paths have totally real fields of definition. Actually such a minimal class has a parametrization by matrices M(t) = A + t(B − A), t ∈ [0, 1]. These matrices may be written in the form M(t) = −t A(\frac{t−1}{t}I_{n} − A^{−1}B), which shows that if θ is a root of p := det(M(t)) then \frac{θ−1}{θ} is an eigenvalue of A^{−1}B (and indeed θ ↦ \frac{θ−1}{θ} puts these roots in one-to-one correspondence with the set of eigenvalues ≠ 1 of A^{−1}B). Hence all roots of p, and consequently of its derivative p′ are real, the field of definition of the weakly eutactic lattice (if any) is totally real. (Moreover, since p has no roots in [0, 1], we see that p′ has at most one root in (0, 1), which gives a direct proof of the uniqueness of weakly weakly eutactic in minimal classes of perfection co-rank 1.)

The question of the field of definition also arises for dual-eutactic lattices. In dimension 4, the class a_{8} contains both a eutactic lattice, defined on ℚ(\sqrt{3}), and a dual-eutactic lattice, defined on the cubic field K of discriminant −244, a quadratic discriminant, so that K is (up to conjugacy) the cubic subfield of the Hibert class field of the quadratic field ℚ(\sqrt{−61}) (which has class number h = 6); see Exercise 9.5.1.

Section 9.7.C.

The minimal classes in dimensions 6 and 7 have been classified in a work by Ph. Elbaz-Vincent, H. Gangl and C. Soulé on the K-theory of ℤ. The data can be read in [E-G-S2]. There are 5 634 classes in dimension 6 and 10 722 899 in dimension 7, the classification of which has been completed some months after the 2005 Oberwolfach meeting.

[In the notation of the tables of [E-G-S2], the perfection rank is n + 1.]

Minimal classes are identified by the n × n positive, definite matrix S^{t}S (called the Bacher matrix by Anne-Marie Bergé and the barycenter matrix in [E-G-S2]): two systems of minimal vectors define the same class if and only if their Bacher matrices are equivalent over Gl_{n}(ℤ), that is, define isometric lattices.

Whereas Batut classified all cells containing a weakly eutactic lattice, using a specific gradient algorithm and using the Bacher matrix only for calculating the automorphism group of the class, the authors of [E-G-S2] restricted themselves to minimal classes, using all the strength of the Bacher matrix. (The classification of (weakly) eutactic lattices, though possible using their data, would need heavy calculations.)

From the Bacher matrix (or barycenter matrix) Bc of a class C, one can easily see whether C contains a strongly (semi-) eutactic lattice (definition 3.2.C1). The recipe is given by the proposition below, the proof of which is straightforward from the definitions.

Proposition 9.7.C1. (A.-M. Bergé, J. Martinet).

Let C be a minimal class, let S be the set of minimal vectors of a lattice Λ ∈ C, let Bc = S^{t}S be the corresponding Bacher matrix, and let S_{1} be the set of minimal vectors of the form associated with Bc^{−1}.

(1) C contains a strongly eutactic lattice if and only if S_{1} = S. Then Λ is strongly eutactic, and Bc^{−1} is a Gram matrix for Λ.

(2) C contains a strongly semi-eutactic lattice if and only if S_{1} is contained in S. Then Λ is strongly semi-eutactic, and S_{1} is the set of of minimal vectors of Λ having non-zero eutaxy coefficients.

Using this device, Elbaz-Vincent and Gangl were able during the 2005 Oberwolfach meeting to obtain the complete list of the 6-dimen-sional strongly (semi-) eutactic lattices: 21 strongly and 6 semi-strongly eutactic lattices. (Only 20 + 3 were previously known.) The file strongeut.gp of my homepage contains the complete classification of strongly eutactic or semi-eutactic lattices up to dimension 6 and examples in dimensions 7, 8, 9, and 10.

One of the main problems in this theory is:

What kind of information can be extracted from Bc? On November 27., 2012, while I was delivering a talk in the algorith-

mic seminar in Bordeaux, Gabriel Nebe pointed out to me the existence of a paper by Plesken ([Pl3] in the complementary bibliography) in which the author describes an algorithm which, given Bc and s, outputs an n × s matrix S solving the equation S^{t}S = Bc. On can thus go from Bc back to a minimal class. However one cannot obtain directly from Bc the perfection rank, nor forecast the nature of Λ with respect to (weak) eutaxy.

Also, is there a simple interpretation of the standard invariants of Bc (minimum, kissing number, Smith invariant) in terms of the corresponding minimal class?

The question of the connection which could exist between the Ash and the Bavard formulae has been solved in a 2007 joint work with Anne-Marie Bergé ([B-M8]). It is shown that if a minimal class C does not contain any weakly eutactic lattice, then inf_{C} γ is attained at a unique (up to similarity) lattice L ∈ ∂C, that L is weakly eutactic, but not eutactic, and that Bavard’s weighted sum with signs restricted to the set cells for which inf_{C} γ is attained on L is zero.

[In the statement above, semi-eutactic lattices play no special rôle inside the set of all weakly eutactic lattices, in contrast to e.g. Corollary 3.8.C1.]

Section 11.6.C. (March, 2020)

Consider the matrices M(t), P and the Moebius function φ below:

M(t) = ( \begin{matrix}1 & −1+2t & −t \\ −1+2t & 1 & −t \\ −t & −t & 1 \end{matrix} ) , P = ( \begin{matrix}1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 1 \end{matrix} ) , and φ(t) = \frac{1−t}{1+t} ,

where M(t), \frac{1}{3} < t < \frac{1}{2}, represents the dual-minimal class C with s = s^{∗} = 4 of Section 9.2. Then the equalities

tP M(t)P = \frac{4t(1−t)}{1+t} M(φ(t)) and ^{t}P = P

show that C is globally isodual of symmetric type. The fixed point of φ (t = −1 + \sqrt{2} = 0.414 …) defines the ccc lattice; see Exer. 6.3.1.

The same kind of result holds with the matrix displayed after Theorem 9.2.2 for the dual-minimal class with s = s^{∗} = 3, taking

φ(t) = −\frac{t}{t+1} and P = ( \begin{matrix}0 & 0 & −1 \\ 0 & −1 & 0 \\ −1 & 0 & 0 \end{matrix} ); see Exer. 9.2.2.

Section 13.1–3.C. (December, 2013)

In the usual Voronoi algorithm, because minimal classes of perfection co-rank 1 are necessarily well rounded, dead ends never occur. To get rid of them (at least in the equivariant algorithm), it suffices to restrict oneself to well rounded (= bounded) minimal classes: bounded minimal classes are the natural setting of the theory; see also Proposition 9.4.C1.

Section 13.3.C. (September, 2016)

In a previous edition of the file erratang.tex I wrote that one should suppress “∈ 𝔸^{−1}T 𝔸^{−1}”. This is indeed correct.

In the case of the action of a group G the sum in this displayed formula is invariant by the transpose of G, not by G itself, except when G = ^{t}G, e.g. in the case of the regular representation.

Section 14.4.C. (July, 2016)

We refer to Rogers’s packing bounds to obtain reasonable upper bounds for the Hermite invariant beyond dimension n = 8. Three kinds of improvements have been obtained since the book was written:

(1) In [Cn-El], Cohn and Elkies have improved Rogers’s bounds in the range 4 ≤ n ≤ 36. The inequalities they prove rely on the choice of a function on ℝ^{n} having convenient properties together with its Fourier transform. In dimensions 8 and 24 they prove bounds very close to those which occur for 𝔼_{8} and Λ_{24}, respectively, and conjecture that these bounds should be attained for a well chosen function.

(2) In [Cn-Km1], Cohn and Kumar have proved that the Leech lattice is the unique critical lattice (up to scale) in dimension n = 24.

(3) Using modular forms Maryna Viazovska ([Viaz]) constructed a function which shows that the density of sphere packings in dimension 8 is bounded above by that of E_{8}; and the case of dimension 24 was then soon dealt with similarly ([C-K-V-al]; see also [Oe4]).

Independently of any application to lattice theory, it should be noticed that Hales (see [Hl1]) has found the exact bound for dimension 3 (“Kepler’s conjecture”). As a consequence the density of any sphere packing in one of the dimensions 2, 3, 8 and 24 is bounded above by the density of the densest known lattice packing.

Section 14.5.C. (Updated on June 14 th, 2021, then February 14 th, 2022.)

Thanks to Cohn–Kumar’s [Cn-Km1]; see above, Section 14.4.C (resp. Poor–Yuen [P-Y4]; see above, Section 6.4.C) the results displayed in Table 14.5.1 for n = 24 (resp. for n = 5, 6, 7) are now known to be optimal, and attained exactly on the self-dual Leech lattice (resp. on the four extreme lattices 𝔻_{5}, 𝔸_{5}^{3}, 𝔼_{6}, 𝔼_{7} and their four duals).

In the other dimensions the known upper bounds for γ′_{n} come from upper bounds for γ_{n}, via the inequality γ′_{n} ≤ γ_{n}. This upper bound is strict if the dual of a critical lattice is not critical, a fact we a priori do not know. In this respect the case of dimension 9 is particular: using Theorem 2.8.7 (2) of [M] and the now known exact value of γ′_{5}, we can show the inequality γ′_{9} < 2 (≤ γ_{9}).

Here are three remarks concerning the search of lattices Λ on which γ′ takes high values. (Since γ′^{2} is rational on rational lattices, we shall list values of γ′^{2} rather than of γ′, though we cannot exclude that γ′_{n} be an algebraic irrationality for some n.)

(1) The set S(Λ) carries the structure of a spherical design of level some well-defined odd integer ℓ (if n ≥ 2; if n = 1, ℓ = ∞); conjecturally, n = 1, 3, 5, 7 or 11. Lattices with ℓ ≥ 5 were called strongly perfect by Venkov. In particular he proved that if ℓ ≥ 5 (resp. ℓ ≥ 7, then we have

γ′(Λ)^{2} ≥ \frac{n+2}{3} (resp. γ′(Λ)^{2} > \frac{n+2}{3}) .

This “Venkov bound” \frac{n+2}{3} is an interesting value to give an idea of what can be expected for γ′^{2}.

(2) By contrast with the Hermite invariant γ, when searching for high values of γ′ on sections of a lattice, the densest sections are often not the best choices.

(3) It often happens that for a lattice Λ having an odd minimum m, Λ and its even sublattice Λ_{even} have the same dual. Then passing from Λ to Λ_{even} improves γ′ by a factor at least \frac{m+1}{m}, generally, \frac{4}{3} if m = 3; \frac{8}{5} for a lattice generated by vectors of norm 5 with paiwise scalar poducts ±5 or ±1, as in the example with n = 13 and m = 5, where Λ_{even} is Conway-Sloane’s example in [S-S9]. After rescaling such lattices produce integral lattices of minimum 4.

On a lattice Λ which is not self-dual, Λ and Λ^{∗} are non-similar lattices on which γ′ has the same value. We shall generally quote only one of them. Self-dual lattices will be t-modular for some t (because we shall meet only rational lattices). Denoting by m the minimum of Λ, we then have γ′(Λ)^{2} = \frac{m^{2}}{t}.

Next we consider the results of Table 14.5.1 in the range 9—23. Our largest known value of γ′ occurs on several lattices when n = 9,

10, 11, 14, 17, 19, 20. If some of them is not dual-extreme, the given lower bound is strict. This is the case exactly for n = 17 and n = 19: for n = 17 (γ′^{2} = 6), on even sublattices of lattices of minimum 3 or 5; for n = 19 (γ′^{2} = \frac{48}{7} = 6, 857...), on even sublattices of lattices of minimum 3, or on cross-sections of the “Kleinian”, 7-modular lattice of dimension 20, with γ′^{2} = \frac{64}{7} = 9.142....

Modular lattices occur with (m, t) = (4, 4) (n = 10, two lattices), (4, 3) (n = 12, 14), (4, 2) (n = 16), and (8, 7) (n = 20, three lattices).

Strongly perfect lattices give examples for n = 10, 12, 14, 16, 18, 20, 21 and 23 (and 2, 4, 8, 24).

We now illustrates the remarks (2) and (3) above. The densest 2-dimensional sections of O_{23}^{∗} ≃ O_{23} are represented by matrices ( \begin{matrix}3 & 1 \\ 1 & 3 \end{matrix} ) and ( \begin{matrix}3 & 0 \\ 0 & 3 \end{matrix} ), defining by orthogonality the lattice O_{21} and a new lattice O′_{21}, respectively. Taking the successive densest hyperplane sections we obtain a well-defined descending series O′_{21}, ..., O′_{16} distinct from O_{21}, ..., O_{16}, the next element of which is isometric to O_{15}.

By orthogonlity to any vector of the fourth level of O′_{20}^{∗} (resp. to a conveniently chosen vector of the third level of O′_{18}^{∗}), we obtain a lattice L′_{19odd} (resp. L′_{17odd}). Denote by L′_{19} and L′_{17} their respective even parts (L′_{17} is isometric to one of the Plesken-Pohst lattices for minimum 4). We have

min L′_{19} = 4 , min L′_{19}^{∗} = \frac{12}{7} , min L′_{17} = 4 , min L′_{17}^{∗} = \frac{3}{2} . It turns out that both the sets of minimal vectors of their duals are of rank 12, strictly smaller than their dimensions, so that these lattices (indeed, extreme) are not dual-extreme. We thus have the strict inequalities

γ′_{19} > \frac{48}{7} = 6.857... and γ′_{17} > 6 .

We now list lattices having the largest known value of γ′_{n}, skipping dimensions 13, 17, and 19 yet considered above.

In some dimensions we know a unique lattice to within duality, having a rather large γ′ invariant, that can be conjectured to be the unique dual-critical lattice in its dimension. These are the modular lattices K_{12} and Λ_{16}, and pairs (L, L^{∗}) with L = K′_{18}, K′_{21}, Λ_{22} and Λ_{23}.

We are now left with dimensions 9, 10, 11 and 14. The lower bound given for n = 9 (γ′^{2} ≥ \frac{16}{5}) is attained on the Coxeter lattice 𝔸_{9}^{(2)} and on the Baril lattice described in Section 3.8.C above of this complement (and their duals).

In dimension 10 we have γ′^{2} ≥ 4, value attained on four lattices mentioned above: the 4-modular lattices \sqrt{2}𝔻_{10}^{+} and Souvignier’s Q_{10}

(isometric to the even sublattice of ⋀^{2}𝔻_{5}); a section of Λ_{11}^{mid}), and the pair of strongly perfect lattices (K′_{10}, K′_{10}^{∗}).

In dimension 11, we also have γ′^{2} ≥ 4, value attained on three pairs (L, L^{∗}) of non-isodual lattices, with representatives K_{11}, Coxeter’s 𝔸_{11}^{3} and Λ_{11}^{mid}, the orthogonal to Λ_{13}^{mid} in the Leech lattice (not a laminated lattice).

In dimension 14, the value γ′^{2} = \frac{16}{3} is attained on the 3-modular, strongly perfect lattice Q_{14}, but also on Λ_{14}, This extreme lattice, which has a non-eutactic dual, is nevertheless dual-extreme, as checked by direct calculation. Questions.

(1) Are the values of γ′^{2} strictly smaller than \frac{n+2}{3} (the Venkov bound for 5-sesigns) in dimensions 9, 11, 13, 17 and 19 ?

(2) Is the lower bound γ′_{n}^{2} ≥ 4 exact in dimensions 10 and 11 ? I expect the answers to be positive, with a modicum of doubt for

n = 17, where the known lower bound is close to Venkov’s.

Beyond dimension 24, good examples are sparse, principally derived from modular lattices. This applies in particular for n = 26, (m, ℓ) = (6, 3), γ′^{2} = 12 ; n = 32, (m, ℓ) = (6, 2), γ′^{2} = 18 ; n = 48, (m, ℓ) = (6, 1), γ′^{2} = 36 ; n = 72, (m, ℓ) = (8, 1), γ′^{2} = 64 . Other reasonably good examples are obtained using cross-sections of the previous lattices, for instance, we have the lower bound γ′_{25}^{2} ≥ 9. Also even sublattices of some unimodular lattices of minimum 3, produce examples with γ′^{2} = 11 for n = 27 and γ′^{2} = 12 for n = 28, 29, 31. However, Nebe’s extremal 5-modular lattice of dimension 28 has γ′^{2} = \frac{64}{5} = 12.8, and Bachoc’s 2-modular, 32-dimensional lattice over the Hurwitz quaternions has sections of dimension 31 with γ′^{2} = \frac{27}{2} = 13.5 .

Section 16.2.C.

G. Nebe and B. Venkov have proved in [Ne-V3] that the Coxeter-Todd lattice K_{12} is the unique strongly perfect lattice in dimension 12. They proved in [Ne-V5] that the isodual lattice Q_{14} is the only 14dimensional lattice L such that both L and L^{∗} are strongly perfect; this result has been extended to dimensions 13 and 15 in [Ne-No-V]; see also [Nos1] for some complements, and to dimension 16 by Hu and Nebe ([Hu-Ne1],[Hu-Ne2]).

[One conjectures that the hypothesis on the dual lattice is not necessary.]

In [Hu-Ne1], the authors construct two strongly perfect “companion lattices” to the Barnes-Wall lattices in dimensions n = 2^{2m} ≥ 16. In particular, the 16-dimensional examples constitute the first enlargement of the list in dimensions ≤ 23 established circa 2000 by Batut and Venkov. Question Are these lattices 7-designs, or only 5-designs ?

G. Nebe and B. Venkov also classified (in [Ne-V4]) lattices whose set of minimal vectors is a higher design up to dimension n = 24, except possibly for n = 23 — but the appearance of other 6-designs in dimension 23 is very unlikely. This completes [Mar7], devoted to integral lattices of minimum m ≤ 5. [Warning: in the statement of the classification theorem of [Mar7], the lattice O_{23} has been forgotten.]

Section 16.3.C.

C. Bachoc ([Bac4]) has proved using group theoretical arguments that all layers of the Barnes-Wall lattices are 7-designs in all dimensions n ≥ 8.

On May 3rd, 2018, in the arXiv paper [Hu-Ne], Hu and Nebe have announced the construction of new infinite series of 7-designs sandwiched into the Barnes-Wall lattices and their duals in dimensions 16, 64, 256,… In particular this gives in dimension 16 new strongly perfect lattices, the first discovery in dimensions 1–23 since the list constructed in 1999 by Batut and Venkov for [Ven3].

Section 16.4.C.

For even unimodular lattices Λ the corrected formula in Theorem 16.4.1 which bounds the minimum (see the erratum), namely min Λ ≤ 2+2 ⌊\frac{(ℓ+1) n}{48}⌋, reads min Λ ≤ 2+2 ⌊\frac{n}{24}⌋. It has been proved by Rains and Sloane ([R-S]) that this bound also holds for odd unimodular lattices except for the lattice O_{23}, of minimum 3; and Gaulter ([Ga]) has shown that for dimensions n ≡ 0 mod 24, this bound is strict on odd unimodular lattices.

The last sentence (... n = 80 with ℓ = 1) refers to [Bac-Ne2] where the authors construct two unimodular lattices of minimum 8. At this date (1998), extremal even unimodular extremal lattices were known to exist in dimensions n = 8k up to n = 64 and in dimension 80. On August 11th, 2010, Gabriele Nebe announced that she had constructed an even unimodular lattice of dimension 72 and minimum 8, now published as [Ne6].

At the date of December 13th, 2013, four even unimodular lattices are known in dimension 48. Two of them are quoted in [C-S], and the other two have been found by Nebe; see [Ne5], [Ne8], [Ne9]. Also four lattices are now known in dimension 80; see [Ste-Watk] and [Watk].

[(Unpublished.) Using the four known even, extremal, 2-modular lattices of dimension 32 and minimum 6, I have constructed four probably distinct extremal even unimodular lattices in dimension 64. The construction, an analogue of a construction of Barnes-Wall lattices, consists in doubling the dimension, obtaining alternatively 1- and 2-modular lattices, but not extremal beyond dimension 64.]

.../...

More on Craig’s lattices (appendix to Section 5.4)

Table 5.4.12 was calculated using the PARI system. We reproduce it below, then list the s invariants of some more Craig lattices of minimum 2r with r ≤ ⌊\frac{p+1}{4}⌋, or their norm N when it exceeds 2r.

Table 5.4.12. Lattices 𝔸_{p−1}^{(r)} with 29 ≤ p ≤ 47 and 3 ≤ r ≤ (p + 1)/4.

3  4  5  6  7  8  9  10  11  12
29 6496 7917 5684 1421 580
31 8835 11625 7905 5735 930 465
37 19092 31968 29304 14430 13320 1665 888
41 29520 57605 61008 36490 31980 4920 N = 20 1066
43 36421 74046 86688 63812 25929 28896 2709 N = 22 903
47 52969 122153 154583 141611 63779 69184 4324 N = 22 N = 24 1081

Complement 1. Beyond (p+1)/4. \underline{p = 29}. r = 8 : N = 20, s = 3248. \underline{p = 37}. r = 10 : N = 24, s = 777. \underline{p = 41}. r = 11 : N = 28, s = 2460. \underline{p = 43}. r = 12 : N = 28, s = 129. \underline{p = 47}. r = 13 : N = 34, s = 4324. \underline{p = 53}. r = 14 : N = 36, s = 4134.

Complement 2. Exceptional pairs (p,r). For p ≤ 53 and r ≤ (p + 1)/4, the only exceptions to the rule N(𝔸_{p−1}^{(r)}) = 2r are the following ones, for which N(𝔸_{p−1}^{(r)}) = 2r + 2 : p = 41, r = 9, N = 20, s = 10086 ; p = 43, r = 10, N = 22, s = 3612 ; p = 47, r = 10, N = 22, s = 12972 ; p = 47, r = 11, N = 24, s = 3243 ; p = 53, r = 11, N = 24, s = 12402 ; p = 53, r = 12, N = 26, s = 1696 .

Complement 3. Larger values of p. \underline{p = 59} : r = 3, N = 6, s = 138591 ;

r = 4, N = 8, s = 424328 ;

r = 5, N =10, s = 759684 .

\underline{p = 61} : r = 3, N = 6, s = 159820 ; r = 4, N = 8, s = 504165 ;

r = 5, N =10, s = 924516 .