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ERRATUM AND COMPLEMENTS TO MONOGRAPHIE 37 RÉSEAUX EUCLIDIENS, DESIGNS SPHÉRIQUES ET FORMES

MODULAIRES

JACQUES MARTINET

Abstract. We present here errata and complements for some of the papers published in “Monographie 37 de l’Enseignement Mathématique”. The papers in which I am not involved will be considered only if the authors wish to send me their remarks.

N.B. Recent references are those of “Corrected and extended reference list of the book “Perfect Lattices in Euclidean Spaces”.” (This homepage.)

Summary of the book.

(1) B.Venkov (Notes by J. Martinet), Réseaux et designs sphériques.

(2) C. Bachoc, B.Venkov (with an appendix with G. Nebe), Modular forms, lattices and spherical designs.

(3) J. Martinet, B.Venkov (with an appendix by R. Coulangeon), Les réseaux fortement eutactiques.

(4) J. Martinet, Sur certains designs sphériques liés à des réseaux entiers.

(5) R. Coulangeon, Voronoï theory over algebraic number fields.

(6) J. Martinet (with an appendix by C. Batut), Sur l’indice d’un sous-réseau.

(7) R. Bacher, B.Venkov, Réseaux entiers unimodulaires sans racines en dimensions 26 et 27.

(8) P. Engel, L. Michel †, M. Senechal, New geometric invariants for Euclidean lattices.

1. First paper.

Updated bibliography.

The paper [L-S-T] has appeared: [L-S-T] W. Lempken, B. Schröder, P.H. Tiep, Symmetric squares, spherical designs, and lattice minima. With an appendix by Christine Bachoc and Tiep, J. Algebra 240 (2001), 185–208. An English updated version of the book [M] has appeared: [M’] J. Martinet, Perfect Lattices in Euclidean Spaces, Grundlehren 327, Springer-Verlag, Heidelberg (2003).

The paper [R-S] has appeared: [R-S] E. Rains, N.J.A. Sloane, The shadow theory of modular and unimodular lattices, J. Number Theory 73 (1998), 359–389. The paper [N-R-S] has appeared: [N-R-S] G. Nebe, E. Rains, N.J.A. Sloane, The invariants of the Clifford groups, Des. Codes Cryptogr. 24 (2001). See also [N-R-S2] G. Nebe, E. Rains and N. Sloane, Codes and invariant theory, Mathematische Nachrichten 274-275 (2004), 104-116, and also the book by the same authors: Self-Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics nu. 17, Springer-Verlag, Heidelberg, 2006.

Complements. \underline{Section 18}. Using the results of [N-R-S], C. Bachoc (Designs, groups, and Lat-

tices, J. Th. Nombres de Bordeaux 17 (2005), 25–44, issue dedicated to the Journées

Arithmétiques of Graz, 2003) has proved that from dimension 8 onwards, the automorphism groups of the Barnes-Wall lattices have no non-trivial invariants in degrees d ≤ 6. As a consequence, all non-zero orbits of these automorphism groups are spherical 7-designs. In particular, all layers of the Barnes-Wall lattices are spher-

ical 7-designs. [In the language of Section 5, the fundamental degrees for Aut(BW_{n}) (n = 2^{p} ≥ 8) are equal to 2 or ≥ 7.]

\underline{Section 19}. The list of strongly perfect lattices displayed in Tables 19.1 and 19.2 was constructed by Batut and Venkov. No new strongly perfect lattices have been discovered since in dimensions n ≤ 25. In dimension 26, three strongly perfect lattices are known, namely one 3-modular extremal lattice, the integral lattice of minimum 4 and determinant 3 discovered by G. Nebe, and its dual.

G. Nebe and B. Venkov (Low dimensional strongly perfect lattices. I: The 12-dimensional case, L’Enseignement Mathématiques 51 (2005), 129–163) have recently proved that the

Coxeter-Todd lattice K_{12} is the only strongly perfect 12-dimensional lattice.

They have also proved an analogous result in dimension 14, but under an extra condition, and work with E. Nossek has extended this last result to dimensions 13 and 15; see the complements to Perfect Lattices in Euclidean Spaces, Section 16.3 C (ref. [Ne-No-V]). \underline{More on group theory.} C. Bachoc’s written talk of the “Journées Arithmétiques of Graz, 2003” contains the list of all known strongly perfect lattices in dimensions n ≤ 26 whose automorhism groups have fundamental non-trivial degrees ≥ 5.

2. Third paper.

Erratum. In Table 8.1, a lattice with n = 5, r = s = 12 and minimum 10 has been forgotten; see nu. 5 in Martinet’s catalogue of perfect lattices.

Updated bibliography.

[Bt] C. Batut, Classification of quintic eutactic forms, Math. Comp. 70 (2001), 395–417.

Complements. One more strongly eutactic 6-dimensional lattice has been found with s = r = 15

and minimum 8 (A.-M. Bergé, J. Martinet, Symmetric Groups and Lattices, Monat-

shefte Math. 140 (2003), 179–195, the lattice with Gram matrix C_{6} displayed after Theorem 4.3). More recently, Elbaz-Vincent/Gangl/Soulé have obtained the complete classification of 6-dimensional cells, and consequently, that of strongly eutactic 6-dimensional lattices. The results for dimension 7 could be extracted from their work. See the file devoted to strongly eutactic lattices in Martinet’s home page, and P. Elbaz-Vincent, H. Gangl, C. Soulé, Perfect forms and the cohomology of modular groups, [E-G-S]; preprint at arXiv:math/1001.0789v1.

3. Fourth paper.

Erratum. In the theorem stated in the introduction (page 137 of the book), the lattice O_{23} has been forgotten. The correct statement reads as follows: Theorem. The integral primitive lattices Λ of minimum m ≤ 5 whose set S(Λ) of minimal vectors is a spherical 7-design are ℤ, the root lattice 𝔼_{8}, the shorter Leech lattice O_{23}, the three laminated lattices Λ_{16} (the Barnes-Wall lattice BW_{16}), Λ_{23} and Λ_{24} (the Leech lattice), and the even unimodular lattices of dimension 32 and minimum 4 (which have not been classified). In particular, minimum 5 is not possible.

Complements. The classification of lattices whose sets of minimal vectors constitute a 6-design (or 7 -design) has been solved in dimensions n ≤ 24 (except for an open case if n = 23) by Nebe and Venkov in On lattices whose minimal vectors form a 6-design, preprint (2006); European J. Combin. 30 (2009), 716–724. (Special issue dedicated to Eiichi Bannai’s 60th birthday.)

4. Sixth paper.

Erratum. \underline{p. 173}, Example 3.3. In the three formulae involving e, replace the denominators 2 by d. In Section 9, case \underline{d=6}.

line +2 (p. 192, l. -5 in the book): read i.e., m_{3} ≤ 2.

(6,0,2) (page 193, line 8 in the book), after “donc e_{6} · f′ = 0”, read: ∀ i, e_{i} · f = 1 ⟹ N(f) = 2 and N(f − e_{i} − e_{6}) = 2 e_{i} · e_{j} ≥ 1 ⟹ N(f) ≥ \frac{1+5/2}{3} = \frac{7}{6} < 2, a contradiction; see also Proposition 2.2 a below. In Section 9, case \underline{d=9}, (page 194, line 3 in the book), read: i.e., m_{3} ≤ 2. In Section 9, case \underline{(2,4,2)}, (page 195, line -5 in the book), in “e =”, read −e_{3}. In Section 9, case \underline{(4,3,1)} (page 196, line 12 in the book), read ... + e_{4} + e_{8} instead of ... + e_{4}e + e_{8}. \underline{pp. 198–199}. Several slips in the long proof showing the impossibility of (3, 2, 2, 1) have been found by Achill Schürmann. The proof is otherwise correct, but can now be skipped since all cyclic quotients in dimensions up to 9 have been classified in [K-M-S]. In Section 10, page 200, line -6, read \frac{1}{3} ∑_{i=1}^{8} a_{i}e_{i} instead of \frac{1}{3} ∑_{i=0}^{8} a_{i}e_{i}.

\underline{p.204}, line 18. Actually, all scalar products e_{i} · e_{j}, 3 ≤ i < j ≤ 6 are also equal to y, so that N(e) = \frac{10+12y}{16} . The inequality N(e) ≥ 1 now reads y ≥ \frac{1}{2}, whence y = \frac{1}{2}, which implies N(e − e_{3}) = \frac{3}{4} < 1, a contradiction.

Updated bibliography.

One can replace [M] by the better reference [M’] J. Martinet, Perfect Lattices in Euclidean Spaces, Grundlehren 327, Springer-Verlag, Heidelberg, 2003, and also [M2] by [M’], Section 6.4.

One should also add to [Ry] the reference [B-R] Baranovskii, S.S. Ryshkov, Derivation of perfect lattices from admissible centerings, Russian Math. Survey 40, 4 (1985), 155–156. (Russian original: 1985.) as well as the recent [K-M-S] W. Keller, J. Martinet, A. Schürmann, On classifying Minkowskian sublattices, Math. Comp. 81 (2012), 1063-1092; preprint at arXiv:0904.3110v3. Maxim Anzin, in an e-mail dated March 23rd, 2004, pointed out to me that the three possible structures which were forgotten in [Zah] were corrected by the author in a preprint written under the name of N. V. Novikova, a preprint (in Russian) that I have never seen. Also, the name Zahareva may occur in print as Zaharova or Zakharova.

Complements. \underline{Section 2}. In a joint work with Anne-Marie Bergé (On Perfection Relations in Lattices, preprint, arXiv: math.NT/0611220 (8 Nov. 2006), 26 pp.); Contemporary Math. 493 (2009), 29–49, we proved the following complement to Watson’s Theorem 2.2: Proposition 2.2 a. If equality holds in Theorem 2.2, we have a_{i} ≤ \frac{d}{2} for all i, and if d ≥ 4, equality holds for at most one index i.

This simplifies some proofs, and in particular immediately shows that the system (6, 0, 2) is impossible.

Whenever Watson’s condition holds, there is a unique perfection relation relating the orthogonal projections to the vectors e_{i} and e′_{i}, namely

∑_{i=1}^{n} a_{i} p_{e_{i}} = ∑_{i=1}^{n} a_{i} p_{e′_{i}} .

This is quoted in Appendix 3 to the joint paper with A.-M. Bergé A generalization of some lattices of Coxeter, Mathematika 51 (2004), 49–61. \underline{Section 7}. For cyclic quotients with order d ≤ 6), all lattices Λ constructed as Λ = ⟨e_{1}, …, e_{n}, e = \frac{a_{1} e_{1}+···+a_{n} e_{n}}{d} ⟩ with e_{1}, …, e_{n} ∈ S(Λ) and a_{i} not divisible by d exist provided Watson’s inequalities hold for all divisors d′ > 1 of d and moreover have for s_{min} and r the value predicted by Watson’s identities, except for a short list of exceptions corresponding to small dimensions n. Here are the results, which were only partially given in the paper.

• d = 2, n ≥ 4 : s = 12 and r = 10 if n = 4, s = r = n if n ≥ 5.

• d = 3, n ≥ 6 : s = 12 and r = 11 if n = 6, s = r = n if n ≥ 7.

• d = 4, n ≥ 7, m_{1} ≥ 4, m_{2} ≠ 0 if n = 7. Except if (m_{1}, m_{2}) = (4, 3), (6, 1) or (8, 0), we have s = n + 8 and r = n + 6 if m_{1} = 4, and s = r = n if m_{1} ≥ 5.

• d = 5, n ≥ 8 (and m_{1} ≥ m_{2}): s = 2n and r = 2n − 1 if (m_{1}, m_{2}) = (4, 4), (6, 2), (8, 1) or (10, 0), and s = r = n otherwise.

• d = 6, n ≥ 9. All systems (m_{1}, m_{2}, m_{3}) satisfying Watson’s conditions

m_{1} + m_{3} ≥ 4 , m_{1} + m_{2} ≥ 6 and m_{1} + 2m_{2} + 3m_{3} ≥ 12

exist and have the predicted values for s_{min} and r except (4, 5, 0) and (5, 1, 3) for which s_{min} = 23 (by repeated application of Watson’s identities). \underline{Section 9}, Proposition 9.1. Zahareva’s identity holds true for any system (m_{1}, m_{2}), with

(m_{1} − 4) ∑_{m_{1}+1≤i≤m_{2}} (N(e′ − e_{i}) − N(e_{i})) + (m_{2} − 4) ∑_{1≤i≤m_{1}} (N(e − e_{i}) − N(e_{i}))

in the right hand side. When m_{1} = m_{2} = 4, there is a unique perfection relation relating the orthogonal projections to the vectors e_{i} and e′_{i} = e − e_{i} or e′ − e_{i}, namely ∑_{i=1}^{n} p_{e_{i}} = ∑_{i=1}^{n} p_{e′_{i}}. There is a similar result for denominator 7, involving m_{1}, m_{2}, m_{3}, and assuming that m_{1} = m_{2} = m_{3} = 3 for the perfection relation. This accounts for the values s = 18, r = 17 in Remark 9.2.

\underline{Section 9, Remarque 9.2}, page 200 in the book (November 24th, 2008). The complete classification for dimension 9 has been obtained in the joint work [K-M-S] with Wolfgang Keller and Achill Schürmann (On classifying Minkowskian sublattices; preprint at arXiv: 0904.3110v1). The new structures for quotients Λ/Λ′ which occur are (7), (8), (9), (10), (12), (6, 2) and (4, 2, 2), which all exist in the lattice Λ_{9}, and (4, 4), which exists only in a perfect, strongly eutactic lattice with s = 81. The proofs make intensive use of linear programming. \underline{Sections 9 and 10} (July 3rd, 2006).

The bound [Λ : Λ′] ≤ 9 for 8-dimensional lattices (except for 𝔼_{8}, for which an elementary quotient of order 16 exists) was stated without a proof by Watson. For the sake of completeness, we present a detailed, handy-computational proof as an appendix.

5. Eighth paper.

Updated bibliography.

[EMS01] Marjorie Senechal’s home page http://www.math.smith.edu/ senechal/books.html refers to a 284 pages, 2003 preprint, available as an I.H.E.S. 2004 preprint at http://www.ihes.fr/ .

Appendix: complements for dimension 8

In dealing with dimension 8 (Sections 9 and 10), we have left aside the possible large indices: all the details have been written for ı ≤ 9, and we just trusted Watson’s statement without proofs for ı > 9. Here we shall provide proofs. Note that if one makes use of the results on Hermite’s constant in dimension 8, there just remains to consider the range 10 ≤ ı ≤ 15, since γ_{8}^{4} = 16 and is attained only on lattices similar to 𝔼_{8}. We must thus consider for Λ/Λ′ the structures (10), (11), (12), (6, 2), (13), and (15), (note that (14) is impossible because index 7 does not exist in dimension 8).

[Since γ_{8} and the corresponding critical lattices can be easily classified using the corresponding results in dimensions 6 and 7 (a short proof due to A.-M. Bergé is written in [M], Theorem 6.6.1), it is reasonable not to bother with larger indices. However, using only the value of g_{5} and the bound for g_{8} obtained by repeated use of Mordell’s inequality (according to Watson’s philosophy), one would have to consider the larger range 10 ≤ ı ≤ 18. Again, ı = 18 is impossible (because ı = 9 exists only for 𝔼_{8}) and quotients Λ/Λ′ of type (16) and (8, 2) are excluded (because denominator 8 does not exist in dimension 8). We are then left with the extra structures (4, 4), (4, 2, 2), and (17).

We first consider the case when Λ/Λ′ is cyclic, i.e.

Λ = ⟨Λ′, e = \frac{a_{1}e_{1} + ⋯ + a_{n}e_{n}}{d} ⟩

for some d in the range 10 ≤ d ≤ 15 (or d = 17) 1 ≤ a_{i} ≤ d′ = ⌊\frac{d}{2}⌋. We denote by m_{i} the number of j ∈ {1, …, 8} such that a_{j} = i ; we have m_{1} + ⋯ + m_{d′} = 8, and set σ_{1} = m_{1} + 2m_{2} + ⋯ + d′m_{d′} and for a prime to d, we denote by σ_{a} the transform of σ_{1} resulting from the transformation e ↦ a e. Watson’s inequalities then read σ_{a} ≥ 2d. If equality holds for, say, σ_{1}, all vectors e − e_{i} are minimal, so that if m_{i} ≠ 0, then the denominator d − i is possible in dimension 8. If d = 10, we have m_{5} = 0 (because index 5 does not show up in dimension 7), m_{1} and m_{3} cannot be both zero, and a denominator 10 − 1 = 9 or 10 − 3 = 7 shows up, a contradiction. When it is proved that index 10 is impossible, we may again apply the same argument with d = 11, which will show inductively that the inequality must be strict in Watson’s inequality for all d ≥ 10.

Before looking at the possible values of d, we write down two general inequalities. First, Watson’s bound implies that m_{1} +(8−m_{1})d′ ≥ 2d+1, i.e. m_{1} ≤ \frac{4d−6}{d−3} < 5

for d ≥ 11 odd (and similarly m_{1} ≤ 4 holds for d ≥ 10 even).

Next, if d is odd, adding Watson’s inequalities for e and 2e mod Λ′, we can sharpen m_{1} ≤ 4 to m_{1} ≤ 3 for d ≥ 11 odd: for k even (resp. odd) the coefficient of m_{k} is bounded from above by k + \frac{k}{2} ≤ \frac{3(d−1)}{4} (resp. by k + \frac{d−k}{2} = \frac{d+k}{2} ≤ \frac{3d−1}{4}), so that m_{1} + \frac{3d−1}{4} (8 − m_{1}) ≥ 2(2d + 1), i.e. m_{1} ≤ \frac{8(d−2)}{3d−13} < 4 for d ≥ 11. [If Watson’s

equality could hold, the bound m_{1} ≤ 3 would be correct only for d ≥ 13.]

Using the action of (ℤ/dℤ)^{×}/{±1}, we see that the inequalities above hold for all m_{k} with k prime to d. In particular, when d is a prime, m_{k} ≤ 3 holds for all k.

We now consider the cyclic quotients Λ/Λ′ of order d = 10 to 15.

\underline{d = 10}. Recall that m_{5} = 0, and consider the two Watson inequalities

σ_{1} = m_{1} + 2m_{2} + 3m_{3} + 4m_{4} ≥ 21 and σ_{3} = 3m_{1} + 4m_{2} + m_{3} + 2m_{4} ≥ 21 and their averaging 2(m_{1} + m_{3}) + 3(m_{2} + m_{4}) ≥ 21. We have m_{5} = 0 because index 5 is not possible in dimension 7, hence 2(m_{1} + m_{3}) + 3(8 − (m_{1} + m_{3})) ≥ 21, i.e. m_{1} + m_{3} ≤ 3, which contradicts the lower bound m_{1} + m_{3} ≥ 4 given by 5e. \underline{d = 11}. We have five inequalities, obtained by performing permutations on σ_{1} = m_{1} + 2m_{2} + 3m_{3} + 4m_{4} + 5m_{5} ≥ 23, σ_{2} = 2m_{1} + 4m_{2} + 5m_{3} + 3m_{4} + m_{5} ≥ 23, σ_{3} = 3m_{1} + ⋯ ≥ 23, … Adding the first two inequalities above, we obtain (∗) 3m_{1} + 6m_{2} + 8m_{3} + 7m_{4} + 5m_{5} ≥ 46. We have m_{i} ≤ 3. If max m_{i} = 3, we may assume that m_{1} = 3, but then the maximal value of σ_{1} + σ_{2} is 41 < 46 (attained on (3, 0, 2, 2, 1)). Hence max m_{i} = 2, so that the system (m_{i}) is a permutation of 2^{4}, 0) or (2^{3}, 1^{2}). In the first case, we may assume that m_{3} = 0, which implies σ_{1} + σ_{2} ≤ 44 < 46. In the second case, we may assume that the value 1 occurs for m_{3} but not for m_{1}, which implies σ_{1} + σ_{2} ≤ 46, with equality only on (2, 1, 1, 2, 2) (and then, σ_{2} = 21 < 23) or (2, 2, 1, 2, 1) (and then, σ_{1} = 22 < 23). \underline{d = 12}. Considering denominators 6 and 4, we see that m_{6} = 0 and m_{4} ≤ 1. Averaging the two inequalities σ_{1} = m_{1} + 2m_{2} + 3m_{3} + 4m_{4} + 5m_{5} ≥ 25 and σ_{5} = 5m_{1} + 2m_{2} + 3m_{3} + 4m_{4} + m_{5} ≥ 25, we obtain σ = \frac{σ_{1}+σ_{5}}{2} = 3(m_{1} + m_{3} + m_{5}) + 2m_{2} + 4m_{4} ≥ 25. Setting t = m_{1} + m_{3} + m_{5}, we have σ = 3t + 2m_{2} + 4m_{4} ≤ 3t + 2(7 − t) + 4 = t + 18, and σ ≥ 25 holds only if t = 7 and m_{4} = 1, whence m_{2} = 0. But this contradicts σ_{4} = m_{1} + 2m_{2} + m_{3} + m_{5} ≥ 8. \underline{d = 13}. The proof roughly follows the one we gave for d = 11. We have σ = σ_{1}+s_{2}+s_{3} = 6m_{1}+12m_{2}+13m_{3}+10m_{4}+10m_{5}+12m_{6} ≥ 27×3 = 81. If max m_{i} = 3, we may assume that m_{1} = 3, so that σ ≤ 6×3+13×3+12×2 = 81, value attained uniquely on the systems (3, 0, 3, 0, 0, 2), (3, 1, 3, 0, 0, 1) and (3, 2, 3, 0, 0, 0), for which σ_{1} ≤ 24 < 27. There remains to consider systems which are a permutation of (2^{4}, 0^{2}), (2^{3}, 1^{2}, 0) or (2^{2}, 1^{4}).

• In the first case, we may assume that m_{3} = 0 and m_{1} = 2, whence σ ≤ (6 + 12 + 12 + 10) × 2 = 80 < 81.

• In the second case, we may assume that m_{3} = 0 or 1 and m_{1} = 2. If m_{3} = 0, then σ ≤ (6 + 12 + 12) × 2 + 10 = 80. Let now m_{3} = 1 (and m_{1} = 3). If m_{2} = 0, then σ ≤ (6 + 10 + 12) × 2 + (13 + 10) = 79 ; If m_{2} = 1, then σ ≤ 81 with equality only on the systems (2, 1, 1, 0, 2, 2) and (2, 1, 1, 2, 0, 2), for which σ_{2} ≤ 22 ; finally, if m_{2} = 2, then σ_{1} ≤ (1 + 2 + 6) × 2 + (3 + 5) = 26.

• In the third case, we may assume that m_{1} = 2 and m_{3} = 1, which implies σ ≤ (6 + 12) × 2 + (13 + 12 + 10 + 10) = 81, with equality only if m_{2} = 2, and then σ_{1} = 26, or if m_{6} = 2, and then σ_{2} = 26. \underline{d = 15}. We have m_{5} = 0 and m_{2} + m_{6} ≤ 2. Adding σ_{1}, σ_{2}, s_{4}, s_{7}, we obtain the inequality 14(m_{1} + m_{2} + m_{4} + m_{7}) + 18(m_{3} + m_{6}) ≥ 124, whose L H S is bounded from above by 14 × (8 − 6) + 18 × 2 = 120.

[\underline{d = 17}. This superfluous case can be dealt using the methods used for d = 13.]

We now turn to non-cyclic quotients Λ/Λ′, indeed of type (6, 2), and write

Λ = ⟨Λ′, e = \frac{a_{1}e_{1} + ⋯ + a_{n}e_{n}}{6} , f = \frac{b_{1}e_{1} + ⋯ + b_{n}e_{n}}{2} ⟩

with a_{i} ∈ {1, 2, 3} and b_{1} ∈ {0, 1}. We associate a system (m_{1}, m_{2}, m_{3}) with e, a system (m′_{1}, m′_{2}, m′_{3}) with e′ = e + f (whose numerator has coefficients a_{i} + 3b_{i}), and a system (m′′_{1}, m′_{2}, m′′_{3}) with 2e + f, which generates the third subgroup of order 6 in Λ/Λ′. Note that m_{3}, m′_{3}, m′′_{3} are non-zero. If a_{i} = 3 and b_{i} = 1 for some i, then we have m′_{1} + m′_{2} + m′_{3} ≤ 7, and since denominator 6 is impossible in dimension 7, all a_{i} + 3b_{i} must be even (they cannot all be divisible by 3 !), which implies f ≡ 3e mod Λ_{0}, hence that Λ/Λ′ has order only 6. Hence b_{i} = 0 whenever a_{1} = 1 or 2. But 2e is congruent modulo Λ′ to a vector of the form \frac{±e_{1}±···±e_{m_{1}+m_{2}}}{3} , so that e_{1}, …, e_{m_{1}+m_{2}}, 2e + f define a lattice of index 6 in dimension 8 − m_{3} < 8, a contradiction.

[The superfluous cases where Λ/Λ′ would be of type (4, 4) or (4, 2, 2) can be dealt with using the remark they define several structures of type (4, 2) and that in each case, a vector with denominator 4 has exactly 7 components not divisible by 4.]