Bases of minimal vectors in lattices, II

Jacques Martinet

Abstract. We prove that a Euclidean lattice of dimension n = 5 (resp. 6; resp. 7) having at least 6 (resp. 10; resp. 18) pairs of minimal vectors has a basis of minimal vectors. Résumé. Nous montrons qu’un réseau euclidien de dimension n = 5 (resp. 6; resp. 7) ayant au moins 6 (resp. 10; resp. 18) paires de vecteurs minimaux possède une base de vecteurs minimaux. Titre français : Bases de vecteurs minimaux dans les réseaux, II.

Mathematics Subject Classification (2000). 11H55. Keywords. Euclidean lattices, minimal vectors, bases.

1. Introduction

In this paper, we consider well rounded lattices, i.e. lattices Λ in some Euclidean space E whose set of minimal vectors spans E. We denote by S(Λ) = S the set of minimal vectors of Λ and by 2s its cardinality. Let n = dim E. Our aim is to calculate lower bounds s_{0} for s which ensure that any well rounded lattice with s ≥ s_{0} is generated by its minimal vectors. It is proved in [5] that up to n = 8, every lattice generated by its minimal vectors possesses a basis of minimal vectors. Hence for the dimensions we shall consider here, these two notions are equivalent. We shall prove:

Theorem 1.1. Let Λ be an n-dimensional, well rounded lattice. Suppose that one of the following conditions holds:

1. n ≤ 4;

2. n = 5 and s ≥ 6;

3. n = 6 and s ≥ 10;

4. n = 7 and s ≥ 18. Then Λ possesses a basis of minimal vectors. Moreover, for n = 5, 6 and 7, the lower bounds given above for s are optimal.

Travail effectué avec le soutien de l’Université Bordeaux 1 et du C.N.R.S. (UMR 5251).

Since perfect lattices satisfy the condition s ≥ \frac{n(n+1)}{2}, we recover the result proved by Csóka ([1], 1987), namely that perfect lattices of dimension n ≤ 7 possess bases of minimal vectors, a result which we can check nowadays on the Stacey-Jaquet classification.

Given a well rounded lattice Λ, n independent minimal vectors e_{1}, …, e_{n} of Λ constitute a basis for a sublattice Λ′ of Λ. The set of possible structures for Λ/Λ′, and in particular the list L of indices [Λ : Λ′] are invariants for Λ.

Definition 1.2. The largest possible value ı(Λ) for [Λ : Λ′] is the maximal index of Λ. For a given pair (Λ, Λ′), the smallest integer m such that Λ = ⟨Λ′, e^{(i)}⟩ where Λ′ has a basis (e_{1}, …, e_{n}) with e_{i} ∈ S(Λ) and the e^{(i)} are of the form \frac{a_{1}e_{1}+···+a_{m}e_{m}}{d_{i}} is called the length of (Λ, Λ′).

For instance, if ı(Λ) = 2 (resp. 3), the length of Λ is the smallest integer m such that one can write Λ = ⟨Λ′, e⟩ with e = \frac{e_{1}+···+e_{m}}{d} and d = 2 (resp. d = 3). Note that Λ has a basis of minimal vectors if and only if index 1 occurs in the list L.

Lemma 1.3. (Watson’s index lemma, [6]) Suppose that Λ = ⟨Λ′, e⟩ for a vector

e = \frac{a_{1}e_{1} + ⋯ + a_{m}e_{m}}{d} with a_{i} ≥ 1 and d ≥ 2. Then

∑_{i=1}^{n} a_{i}(N(e − e_{i}) − N(e_{i})) = (( ∑_{i=1}^{n} a_{i}) − 2d) N(e) .

Assume moreover that the vectors e_{i} are minimal. Then we have ∑_{i} a_{i} ≥ 2d and equality holds if and only if all vectors e − e_{i} are minimal.

Proof. The straightforward proof is left to the reader. Some more details can be read in [4], Section 2. □

The following proposition is an example of the results we shall prove in relation with the index:

Proposition 1.4. Let Λ be a well rounded lattice of dimension n, maximal index 2, and length ℓ, having no basis of minimal vectors.

1. If ℓ = n, then s(Λ) = n.

2. If ℓ = n − 1, then s ≤ 2n − 1, and if s ≥ n + 4, we can choose the e_{i} so that either S(Λ) = {±e_{i}, ±(e_{n} + e_{j})}, i = 1, …, n, j = 2, …, s − n, or S(Λ) = {±e_{i}, ±(e_{1} + e_{j} + e_{n})}, i = 1, …, n, j = 1, …, s − n (and then, s ≤ 2n − 2).

Numerical evidence suggests that the bound s ≤ 2n − 1 is optimal from dimension 7 onwards. Theorem 1.1 in dimension 6 will follow from the improvement s ≤ 2n − 3 = 9.

Acknowledgements. I warmly thank Roland Bacher for his close look at a first draft of this paper and his numerous remarks which allowed me to greatly improve the original manuscript.

2. Bounds for the Index

We succinctly recall some results on the index. For the proof, the reader is referred to [4].

The trivial bound ı ≤ γ_{n}^{n/2} is optimal for all n ≤ 8, but if we disregard the root lattices 𝔻_{4}, 𝔻_{6}, 𝔼_{7}, 𝔼_{8} (which obviously have bases of minimal vectors), we obtain the bounds ı ≤ 2, 3, 4, 8 for n = 5, 6, 7, 8, and we moreover have ı = 1 for n ≤ 4, which proves Theorem 1.1 for dimensions n ≤ 4.

When d = 2 or 3, we may suppose (by reduction modulo d of the a_{i} and negation of some e_{i}) that we have a_{1} = ⋯ = a_{m} = 1 and a_{i} = 0 for m ≤ i ≤ n for some m ≤ n. The index lemma then shows that if Λ is not generated by its minimal vectors, and if ı(Λ) = 2 (resp. 3), then Λ has length ℓ ≥ 5 (resp. ℓ ≥ 7).

We now make and comment two general remarks, namely:

1. if Λ/Λ′ is cyclic of prime order and if Λ is not generated by minimal vectors, then S(Λ) must be reduced to S(Λ′);

2. if [Λ : Λ′] = ı(Λ), then ı(Λ′) = 1. Condition ı(Λ′) = 1 can be expressed in terms of characteristic determinants (in the sense of Korkine and Zolotareff; see [3], Section 5.1). The consideration of determinants of orders 1, 2, 3 show that a lattice of maximal index ı = 1 satisfies the properties below.

1. Minimal vectors have components 0, ±1 on e_{1}, …, e_{n}.

2. Two minimal vectors cannot have components (1, 1) and (1, −1) on a pair (i, j) of indices.

3. Three minimal vectors cannot have components (1, 1, 0), (1, 0, 1) and (0, 1, 1) on a system (i, j, k) of three components. The easy proposition below illustrates the case of these small indices:

Proposition 2.1. Let Λ be a well rounded lattice which is not generated by its minimal vectors, of dimension and length n, and of index 2 (resp. 3). Then s(Λ) = n (resp. s(Λ) ≤ n + ⌊\frac{n}{2}⌋).

Proof. We may write Λ = ⟨Λ′, e⟩ with

e = \frac{a_{1}e_{1} + ⋯ + a_{n}e_{n}}{d} and d = 2 or 3 .

Let x = ∑_{i=1}^{n} x_{i}e_{i} ∈ S(Λ′); by the remarks above, we have x_{i} = 0, ±1.

Suppose first that d = 2. Negating e_{i} and replacing e by e − e_{i} if necessary, we may assume that all x_{i} are equal to 0 or 1. If, say, x = e_{1} + ⋯ + e_{k} were minimal for some k ≥ 2, we could write e = \frac{x+e_{k+1}+···+e_{n}}{2} and Λ would have length ℓ ≤ n + 1 − k, a contradiction.

Suppose next that d = 3. We get rid of sums e_{1} + ⋯ + e_{k} with k ≥ 2 as above, but we must this time also consider sums of the form e_{1} ± e_{2} ± ⋯ ± e_{k}. If, say, x = e_{1} + e_{2} − e_{3} were minimal, we could write e − e_{3} = \frac{x−e_{3}+e_{4}+···+e_{n}}{3} , and Λ would have length ℓ ≤ n − 1. Similarly x = e_{1} − e_{2} and y = e_{1} − e_{3} cannot be both minimal, for we could then write e − e_{1} = \frac{−x−y+e_{4}+···+e_{n}}{3} . Then the minimal vectors of Λ′ others that the ±e_{i} are vectors ±(e_{i} − e_{j}) with disjoint supports {i, j}. This completes the proof of Proposition 2.1. □

This proposition immediately implies Theorem 1.1 for dimension 5, since 5-dimensional lattices have maximal index ı = 1 or 2.

3. Lattices of Index 2

In this section, we prove Proposition 1.4 and complete the proof of Theorem 1.1 for dimension 6. We also prove a preliminary result for dimension 7. Proof of Proposition 1.4. The first assertion of the proposition results from Proposition 2.1. Let us prove (2). We then write as usual

Λ = ⟨Λ′, e⟩ with e = \frac{e_{1} + ⋯ + e_{n−1}}{2} .

As in the proof of Proposition 2.1, we show that S(Λ) = S(Λ′) and that at least one x ∈ S(Λ′) can be written as a sum e_{i_{1}} + ⋯ + e_{i_{k}}. Since systems of components (1, 1), (1, −1) are forbidden, every x ∈ S(Λ′) (up to sign) has then positive components. Moreover, by definition of the length, x must have a non-zero component on e_{n}. If, say, x = e_{1} + e_{2} + e_{3} + e_{n} were minimal, we could write e = \frac{x−e_{n}+e_{4}+···+e_{n−1}}{2} . As a consequence, we may assume that minimal vectors other than the e_{i} are of the form ±(e_{i} + e_{n}) (type I) or ±(e_{i} + e_{j} + e_{n}) (type II). Next we observe that x = e_{1} +e_{2} +e_{n} and y = e_{3} +e_{n} cannot be both minimal, for we could write e + e_{n} = \frac{x+y+e_{4}+···+e_{n−1}}{2} , nor similarly x and y = e_{3} + e_{4} + e_{n}. We now discuss the various possibilities according to the number t of type I vectors.

• t ≥ 2, say, e_{1} + e_{n} and e_{2} + e_{n} ∈ S(Λ′). The only possible vector of type II is e_{1} + e_{2} + e_{n}, and then s = n + 3. This proves Proposition 1.4 in this case.

• t = 1, say, e_{1} + e_{n} ∈ S(Λ′). Then all type II vectors must be of the form e_{1} +e_{i} +e_{n}. Replacing e_{n} by e′_{n} = e_{1} +e_{n} and e_{1} by e′_{1} = −e_{1}, we reduce ourselves to the previous case.

• t = 0. We cannot have a system e_{i} + e_{j} + e_{n}, e_{i} + e_{k} + e_{n}, e_{j} + e_{k} + e_{n}, which would imply i(Λ′) ≥ 2 (the vectors above define a non-trivial characteristic determinant). Hence we may assume (up to a permutation of e_{1}, …, e_{n−1}) that all minimal vectors other than the ±e_{i} are of the form e_{1} + e_{i} + e_{n}. □

Proof of Theorem 1.1 for dimension 6. We consider a lattice Λ with maximal index ı. We have ı ≤ 4, and bases of minimal vectors exist if ı = 4 (because Λ is then similar to 𝔻_{6}; see [4], theorem 4.3), if ı = 3 (because equality holds in Watson’s theorem) and of course if ı = 1, and also if ı = 2 and if Λ has length 4 (because it then has a 𝔻_{4}-section) or 6 (by proposition 1.4).

There remains to consider the case where Λ is of the form ⟨Λ′, e⟩ with e = \frac{e_{1}+e_{2}+e_{3}+e_{4}+e_{5}}{2} , that we scale for convenience to minimum 2. Hence, e_{i} · e_{i} = 2. By Proposition 2.1, we may assume that up to sign, the minimal vectors of Λ′ other than the ±e_{i} are the vectors e_{i} + e_{6} or e_{i} + e_{5} + e_{6} for i = 1, …, k, k ≤ 5 or k ≤ 4, and we must prove that k ≤ 3.

Suppose first that e_{1} + e_{6}, …, e_{4} + e_{6} are minimal. Applying the averaging argument of [4], Proposition 8.5 to the group S_{4} acting on {1, 2, 3, 4}, we may reduce ourselves to the case where the scalar products e_{i} · e_{j} only depend on three parameters, namely t = e_{i} ·e_{j} (1 ≤ i < j ≤ 4), u = e_{i} ·e_{5} (1 ≤ i ≤ 4) and v = e_{5} ·e_{6} (we have e_{i} · e_{6} = −1 for 1 ≤ i ≤ 4). An easy calculation shows that

(1) N(e + e_{6}) − 2 = −\frac{3}{2} + 3t + 2u + v;

(2) N(e − e_{5} + e_{6}) − 2 = −\frac{3}{2} + 3t − 2u − v;
(3) N(e − e_{2} − e_{3}) − 2 = \frac{1}{2} − t.

The left hand sides must be strictly positive. Adding (1) and (2), we obtain the lower bound 6t > 3 which contradicts (3).

Suppose next that e_{1} + e_{5} + e_{6}, …, e_{4} + e_{5} + e_{6} are minimal. Using the same averaging argument, we find this time N(e−e_{1})−2 = u+ \frac{1}{2} and N(e−e_{1}+e_{6})−2 = v −u− \frac{3}{2}. Adding these two positive numbers, we obtain the inequality −v −1 > 0, i.e. v < −1. This contradicts N(e_{5} + e_{6}) = 4 + 2v ≥ 2.

That 2s ≤ 18 is the least upper bound for the number of minimal vectors in a well rounded 6-dimensional lattice can be seen by inspecting the Gram matrices displayed in the appendix. □

We now turn to lattices of index 2 and length n − 2 not generated by their minimal vectors. This implies n ≥ 7 by the index lemma. We put special emphasis on dimension n = 7, for which hyperplane sections can be handled using the case n = 6 of Theorem 1.1, the statement of which is more precise than that of Proposition 1.4. This will complete the study of 7-dimensional lattices of index 2. The bounds given in the following proposition are perhaps not optimal.

Proposition 3.1. Let Λ be a well rounded lattice of dimension n, index 2, and length n − 2, having no basis of minimal vectors. Then s ≤ 4n − 5, and if n = 7, we have the sharper bound s ≤ 17.

Proof. Let F be the span of e_{1}, …, e_{n−2} and let F_{1} = F + e_{n−1}, F_{2} = F + e_{n}, F_{3} = F + (e_{n−1} + e_{n}) and F_{4} = F + (e_{n−1} − e_{n}). Since Λ′ has index 1, all minimal vectors of Λ′ are of the form ±e_{i} ± e_{j} ± …, hence (up to sign) belong to one of the affine spaces F_{i}. Moreover, F_{3} and F_{4} cannot both contain minimal vectors of Λ′, because their components on e_{n−1} and e_{n} would define a characteristic determinant equal to ±2; see Section 2. Negating e_{n} if necessary, we may assume that S(Λ′) ∩ F_{4} = ∅.

By Proposition 1.4 applied in dimension n − 1, there are in F + ℝe_{n−1} and F +ℝe_{n} at most n−2 pairs of minimal vectors besides e_{1}, …, e_{n−1}, e_{n}. Hence each set F_{1}, F_{2} contributes at most n − 2 to s. A result of the same kind holds for F_{3}: if e_{n−1} + e_{n} is minimal, the contributions to s of F_{3} is now at most n − 1 (at most (n−2), plus 1 for e_{n−1}+e_{n}); otherwise, if S(Λ′)∩F_{3} ≠ ∅, let x be one of its elements, and let Λ′′ be the lattice generated by e′_{1} = e_{1}, …, e′_{n−1} = e_{n−1} and e′_{n} = x; applying the previous argument to F′ = F +e′_{n}, we find that there are at most n−2 minimal vectors in F′ besides e_{1}, …, e_{n−1}, x, but this list includes e_{n}. Altogether, this gives for s(Λ) = s(Λ′) the upper bound s ≤ n + 2(n − 2) + (n − 1) = 4n − 5, which proves the first part of the proposition.

The argument for n = 7 is similar, replacing the bound n−2 of Proposition 1.4 by the bound 3 = 9 − 6 of Theorem 1.1 for dimension 6. This time we have s ≤ 7 + 2 × 3 + 4 = 17. □

4. 7-Dimensional Lattices

We begin this section with a lemma which paves the way for the complete study of dimension 7. We keep the previous notation. In particular, e_{1}, …, e_{7} are independent minimal vectors of Λ which constitute a basis for a sublattice Λ′ of Λ.

Lemma 4.1. Let Λ be a 7-dimensional lattice which is not generated by its minimal vectors. Assume that s(Λ) ≥ 14. Then Λ has one of the following forms:

1. Λ = ⟨Λ′, e⟩ with e = \frac{e_{1} + e_{2} + e_{3} + e_{4} + e_{5}}{2} .

2. Λ = ⟨Λ′, e, f⟩ with

e = \frac{e_{1} + e_{2} + e_{3} + e_{4}}{2} and f = \frac{e_{1} + e_{2} + e_{5} + e_{6} + e_{7}}{2} .

3. Λ = ⟨Λ′, e⟩ with e = \frac{e_{1} + e_{2} + e_{3} + e_{4} + e_{5} + 2e_{6} + 2e_{7}}{4} .

Proof. For n = 7, we have ı(Λ) ∈ {2, 3, 4} or ı = 8. but this last case must be discarded since it occurs only for lattices similar to 𝔼_{7}, see the beginning of Section 2. If ı(Λ) = 2 and length(Λ) ≥ 6, we have s ≤ 13 by Proposition 1.4. Hence if ı = 2, we must have length(Λ) = 5, which is case one of the lemma. If ı(Λ) = 3, we have s ≤ 10 by Proposition 2.1. We are thus left with lattices such that [Λ : Λ′] = 4.

If Λ/Λ′ is non-cyclic, we attach to (Λ, Λ′) the binary code (of length 7 and dimension 2), with words (a_{1}, …, a_{7}) mod 2 for \frac{a_{1}e_{1}+···+a_{7}e_{7}}{2} ∈ Λ. Its minimal weight is ≥ 4 by Watson’s inequality, which implies that its weight system is (4^{3}), (4^{2}, 6) or (4, 5^{2}). In the first two cases, Λ has a basis of minimal vectors, whereas the third case corresponds to case (2) of the lemma.

If Λ/Λ′ is cyclic, we can write Λ = ⟨Λ′, e⟩ with e = \frac{e_{1}+...e_{p}+2e_{p+1}+···+2e_{7}}{4} and 4 ≤ p ≤ 6 (see [4], Proposition 5.1). Applying Watson’s identity if p = 6 or writing e = \frac{e′+e_{5}+e_{6}+e_{7}}{2} with e′ = \frac{e_{1}+e_{2}+e_{3}+e_{4}}{2} if p = 4, we see that e − e_{7} is in both cases a minimal vector. Hence we must have p = 5, which is case (3) of the lemma. □

Remark 4.2. The smallest possible value for s is s = 7 in cases (1) and (3), and s = 15 in case (2); see [4], table 11.1.

We now consider lattices of index ı ≥ 4 which are not generated by their minimal vectors. This implies ı = 4, and we are in one of the cases (2), (3) of Lemma 4.1. We keep the notation of the previous sections. We begin with case (2), corresponding to a non-cyclic quotient Λ/Λ′.

Proposition 4.3. Let e_{1}, …, e_{7} be 7 independent minimal vectors of Λ generating a sublattice Λ′ of Λ. Suppose that Λ/Λ′ is non-cyclic of order 4 and that Λ is not generated by its minimal vectors. Then s(Λ) ≤ 15.

(The upper bound s(Λ) ≤ 15 is optimal by Remark 4.2.)

Proof. We have ı(Λ) = 4 and by lemma 4.1, we may write Λ = ⟨Λ, e, f⟩ with

e = \frac{e_{1} + e_{2} + e_{3} + e_{4}}{2} and f = \frac{e_{1} + e_{2} + e_{5} + e_{6} + e_{7}}{2} . We observe that Λ contains

at least 15 pairs of minimal vectors, namely the 7 pairs ±e_{i} and the 8 pairs \frac{±e_{1}±e_{2}±e_{3}±e_{4}}{2} . We are going to show that Λ has no other minimal vectors.

Set f′ = \frac{e_{3} + e_{4} + e_{5} + e_{6} + e_{7}}{2} . Since Λ = ⟨Λ′, e, f⟩ = ⟨Λ′, e, f′⟩ is not gen-

erated by its minimal vectors, none of the cosets f + Λ′, f′ + Λ′ contains minimal vectors. We first prove that the only minimal vectors in the coset Λ′ are the ±e_{i}.

Indeed, the arguments used to prove Proposition 1.4 show that we may reduce ourselves to the case where there exists a minimal vector of the form x = e_{i_{1}} + ⋯ + e_{i_{r}} with r ≥ 2, and that the support of x must not be contained in any of the sets {1, 2, 3, 4}, {1, 2, 5, 6, 7} or {3, 4, 5, 6, 7}. This implies that the support T of x intersects all three complementary sets {1, 2}, {3, 4} and {5, 6, 7} of the supports of f′, f and e. In particular , we have r ≥ 3. We cannot have T ⊂ {1, 2, 3, 4} because \frac{e_{1}+e_{2}+e_{3}+e_{4}}{2} ∉ 2Λ. Hence we may assume that either x = e_{1} +e_{3} +e_{5} +… with r ≥ 3 components or x = e_{1} + e_{2} + e_{3} + e_{5} + … with r ≥ 4 components. The case r = 4 (and a fortiori r > 4) is impossible: otherwise f = \frac{x+e_{2}−e_{3}+e_{7}}{2} or f = \frac{x−e_{3}+e_{6}+e_{7}}{2} would be minimal. Suppose now r = 3. The existence of the 8 pairs of minimal vectors \frac{±e_{1}±e_{2}±e_{3}±e_{4}}{2} implies e_{i} · e_{j} = 0 and a straightforward computation yields N(e_{1} + e_{5}) + N(e_{3} + e_{5}) = N(e_{5}) + N(e_{1} + e_{3} + e_{5}) which shows the minimality of e_{1} + e_{5} and e_{3} + e_{5} and contradicts the inequality r ≥ 3 established above.

We now prove that \frac{±e_{1}±e_{2}±e_{3}±e_{4}}{2} are the only minimal vectors in the coset e + Λ. Let x ∈ S(Λ) ∩ (e + Λ), say, x = \{\frac{a_{1}e_{1}+···+a_{7}e_{7}}{2} \}; we have a_{i} ≡ 1 mod 2 if i ≤ 4 and a_{i} ≡ 0 mod 2 if i > 4. For an i with a_{i} ≠ 0, consider the lattice Λ′′

generated by x and the e_{j} with j ≠ i. We have ±e_{i} = \frac{−2x + ∑_{j≠i} a_{j}e_{j}}{|a_{i}|}, whence

|a_{i}| = [Λ : Λ′′] ≤ 4, and equality must be excluded, since it corresponds to the case “p = 4” in the proof of Lemma 4.1, (3). Hence a_{i} = 0, ±2 for i = 5, 6, 7, say, a_{i} = 2a′_{i}, and at least one of them is non-zero, so that we may assume that a′_{7} = 1. Replacing e_{7} in f by its expression as a combination of x and the a_{j} with j ≠ 5, we obtain for f an expression of the form

f = \frac{∑_{1≤i≤4} b_{i}e_{i} + ∑_{5≤i≤7} c_{i}e_{i}}{4}

with odd b_{i} (b_{i} = 2 − a_{i} or −a_{i}) and even c_{i} (2(1 − a′_{5}), 2(1 − a′_{6}), 2x). The impossible case “p = 4” again shows up. □

Proposition 4.4. Let e_{1}, …, e_{7} be 7 independent minimal vectors of Λ generating a sublattice Λ′ of Λ. Suppose that Λ/Λ′ is cyclic of order 4 and that Λ is not generated by its minimal vectors. Then s(Λ) ≤ 17.

Proof. By Lemma 4.1, we may write Λ in the form Λ = ⟨Λ′, e⟩ where e = \frac{e_{1}+···+e_{5}+2(e_{6}+e_{7})}{4}. Set f = \frac{e_{1}+e_{2}+e_{3}+e_{4}+e_{5}}{2} . Then e = \frac{f+e_{6}+e_{7}}{2} , and the cosets of Λ modulo Λ′ are those of ±e, f and 0. We again have ı(Λ) = 4. Assuming that Λ is not generated by its minimal vectors, we are going to show that Λ has at most 17 pairs of minimal vectors. We observe that the cosets of ±e do not contain any minimal vector, and we consider successively Λ′ and f + Λ′, and denote by s′ and s_{f} the contributions to s of Λ′ and f + Λ′ respectively.

Let x ∈ S(Λ′), x ≠ ±e_{i}. Because ı(Λ) = 4, x is up to sign a sum e_{i} ± e_{j} ± … with i < j < . . .. As in the proof of Proposition 4.3, the signs of e_{6}, e_{7} do not matter, but those of e_{1}, …, e_{5} do. If x = e_{i} + e_{j} ± … with i, j ≤ 5, or if x = ⋯+e_{6}+e_{7}, then Λ has length ℓ ≤ 6. If x = e_{i}+e_{j} with i ≤ 5 < j or x = e_{i}−e_{j} with i, j < 5, then we are in the case p = 6 of Lemma 4.1. Finally, up to permutations of {1, …, 5} and {6, 7}, we may assume that x = e_{1} − e_{2} + e_{6}. Now another vector y = e_{i} − e_{j} + e_{6} cannot be minimal: indeed, if y = e_{1} − e_{3} + e_{6} (resp. e_{3} − e_{4} + e_{6}) were minimal, we would have e + e_{6} − e_{1} = \frac{−x−y−e_{1}+e_{4}+e_{5}+2e_{7}}{4} (resp. e − e_{6} − e_{1} = \frac{x+y+e_{5}+2(e_{2}+e_{4}+e_{7})}{4}), and Λ would have length 6. Hence we have s′ ≤ 9 and if equality holds, the pairs of minimal vectors in Λ′ other than the ±e_{i} are one pair ±(e_{i} − e_{j} ± e_{6}) and one pair ±(e_{i′} − e_{j′} ± e_{7}).

We now turn to the coset f + Λ, whose elements are of the form x = \frac{a_{1}e_{1}+···+a_{5}e_{5}+a_{6}e_{6}+a_{7}e_{7}}{2} with a_{i} ≡ 1 mod 2 for 1 ≤ i ≤ 5 and a_{6} ≡ a_{7} ≡ 0 mod 2.

If a_{i} ≠ 0, we may write ±e_{i} = ^{\frac{2x−∑_{j≠i} a_{j}e_{j}}{|a_{i}|}}. Assume that x is minimal, and let Λ′′ be the lattice generated by x and the e_{j} with j ≠ i. We have [⟨Λ′, f⟩ : Λ′′] = |a_{i}| hence [Λ : Λ′′] = 2|a_{i}|, which implies 2|a_{i}| ≤ ı(Λ) = 4, i.e. a_{i} ∈ {0, ±1, ±2}. Negating some vectors among {e, f, e_{6}, e_{7}} if necessary, we may assume that there are 0, 1 or 2 coefficients a_{i} equal to −1 with i ≤ 5 and that a_{6}, a_{7} ∈ {0, 2}. Up to sign, we have x = f′ + b_{6}e_{6} + b_{7}e_{7}, where f′ = f, f − e_{i} or f − e_{i} − e_{j} and b_{i} = 0 or 1. If f′ = f + … then e = \frac{f′+e_{6}+e_{7}}{2} has length ℓ ≤ 3 with respect to ⟨Λ′, f⟩, which is plainly impossible. Similarly, if f′ = f − e_{i} + …, then e = \frac{f′+e_{i}+...}{2} has length ℓ ≤ 4 with respect to ⟨Λ′, f⟩, which is again impossible. Finally, the only possibility is f′ = f − e_{i} − e_{j} (with b_{6} = b_{7} = 0), which implies the upper bound s_{f} ≤ 10.

Taking into account s′ and s_{f}, we obtain the bound s ≤ 19, that we shall now slightly sharpen.

We observe that two vectors f′ = f − e_{i} − e_{j} and x = e_{i} − e_{j} + e_{k}, i < j ≤ 5 and k = 6, 7 cannot be both minimal. Indeed, with i = 1, j = 2 and k = 6, we may write e − e_{2} = \frac{f′+x+e_{7}}{2} . Hence if s_{f} = 10 or s_{f} ≤ 8, we have s ≤ 17. There remains to consider the case where all vectors f′ = f − e_{i} − e_{j} except, say, f − e_{4} − e_{5} and also x = e_{4} −e_{5} +e_{6} and y = e_{4} −e_{5} +e_{7} are minimal. We choose the scale in which N(e_{i}) = 4. Writing that the nine vectors f − e_{i} − e_{j} (1 ≤ i < j ≤ 5, (i, j) ≠ (4, 5)) are minimal, we express the Gram matrix for ⟨e_{1}, …, e_{5}, f⟩ as a function of one parameter t such that e_{i} · e_{j} = t for 1 ≤ i < j ≤ 3 and e_{i} · e_{j} = 2 − t for 1 ≤ i ≤ 3 and j = 4, 5 and for i = 4 and j = 5. We then apply the averaging argument of Proposition 8.5 in [4] referred to above with respect to the group generated by the permutations of {1, 2, 3}, the transposition (6, 7), and the transformation which exchanges e_{4} and −e_{5} and negates e_{6} and e_{7}. We can express a Gram matrix as a function of three parameters: t, u such that e_{4} · e_{6} = e_{4} · e_{7} = u and e_{5} · e_{6} = e_{5} · e_{7} = −u, and v = e_{6} · e_{7}, and we have e_{i} · e_{j} = 0 for i = 1, 2, 3 and j = 6, 7. Then N(e_{4} − e_{5} + e_{6}) = 8 + 2t + 4u is minimal if and only if u = −1 − \frac{t}{2}, but we then have N(e − e_{4} − e_{6} − e_{7}) = 1 + \frac{t+v}{2} ≤ 3 since |t| and |v| are bounded from above by 2. Hence s_{f} = 9 implies s′ ≤ 8. This completes the proof of the proposition. □

Proof of Theorem 1.1 for dimension 7. Lemma 4.1 shows that it suffices to consider lattices which are either of index 2 and length 5, or of index 4, cyclic or not. The bound s ≤ 17 for lattices which are not generated by their minimal vectors then results from the three propositions 3.1, 4.3 and 4.4. The fact that there exists such lattices with s = 17 is seen on the Gram matrix displayed in the appendix, which defines a lattice having a 5-dimensional section similar to Coxeter’s perfect lattice 𝔸_{5}^{3} (the perfect lattice P_{5}^{2} in Conway and Sloane’s notation; see [3], Table 6.5.5).

□

5. Appendix

In this appendix, we display Gram matrices for lattices Λ of dimension n ≤ 7 with given maximal index ı and length ℓ, having the largest possible kissing number among those which are not generated by their minimal vectors. We discard the case where n = 7, ı = 2 and ℓ = 5, for which the optimal value for s (14, 15, 16 or 17) is not known, see the remark below.

We disregard the trivial case where ı = 2 and ℓ = n, the minimal class of which (in the sense of [3], Section 9.1) is that of the centred cubic lattice. This is the only possibility if n = 5. If n = 6, the only other possibility is ı = 2 and ℓ = n − 1 = 5. If n = 7 and ı ≥ 3, then ℓ = 7, and we are left with 5 cases, namely: ı = 2, ℓ = 6 or 5; ı = 3; ı = 4, cyclic or non-cyclic.

Matrices are constructed starting with the Gram matrix A′ of a basis of minimal vectors for a lattice Λ′ such that Λ is of the form ⟨Λ′, e⟩ (resp. ⟨Λ′, e, f⟩) and then replacing the first basis vector by e (resp. the first and the last ones by e and f respectively).

We denote by T a set of representatives up to sign for S(Λ)∖{±e_{i}}.

Dimension 6. ı = 2, ℓ = 5 ; s_{max} = 9. We display two matrices A6a and A6b for the two minimal classes defined by T = {e_{1} + e_{6}, e_{2} + e_{6}, e_{3} + e_{6}} and T = {e_{1} + e_{6}, e_{2} + e_{6}, e_{1} + e_{2} + e_{6}} respectively:

A6a = ( \begin{matrix}75 & 36 & 36 & 21 & 21 & −35 \\ 36 & 48 & 12 & 0 & 0 & −24 \\ 36 & 12 & 48 & 0 & 0 & −24 \\ 21 & 0 & 0 & 48 & −6 & 1 \\ 21 & 0 & 0 & −6 & 48 & 1 \\ −35 & −24 & −24 & 1 & 1 & 48 \end{matrix} ) , A6b = ( \begin{matrix}5 & 2 & 2 & 2 & 2 & −2 \\ 2 & 4 & 0 & 0 & 0 & −2 \\ 2 & 0 & 4 & 0 & 0 & 0 \\ 2 & 0 & 0 & 4 & 0 & 0 \\ 2 & 0 & 0 & 0 & 4 & 0 \\ −2 & −2 & 0 & 0 & 0 & 4 \end{matrix} ) .

Dimension 7. ı = 2, ℓ = 6, T = {e_{1} + e_{7}, …, e_{6} + e_{7}}, s_{max} = 13; ı = 3, T = {e_{1} − e_{2}, e_{3} − e_{4}, e_{5} − e_{6}}, s_{max} = 10.

A7a2 = ( \begin{matrix}27 & 9 & 9 & 9 & 9 & 9 & −12 \\ 9 & 8 & 2 & 2 & 2 & 2 & −4 \\ 9 & 2 & 8 & 2 & 2 & 2 & −4 \\ 9 & 2 & 2 & 8 & 2 & 2 & −4 \\ 9 & 2 & 2 & 2 & 8 & 2 & −4 \\ 9 & 2 & 2 & 2 & 2 & 8 & −4 \\ −12 & −4 & −4 & −4 & −4 & −4 & 8 \end{matrix} ) , A7a3 = ( \begin{matrix}20 & 9 & 9 & 9 & 9 & 9 & 6 \\ 9 & 18 & 0 & 0 & 0 & 0 & 0 \\ 9 & 0 & 18 & 9 & 0 & 0 & 0 \\ 9 & 0 & 9 & 18 & 0 & 0 & 0 \\ 9 & 0 & 0 & 0 & 18 & 9 & 0 \\ 9 & 0 & 0 & 0 & 9 & 18 & 0 \\ 6 & 0 & 0 & 0 & 0 & 0 & 18 \end{matrix} ) .

ı = 4. We display two matrices A7a4 and A7b4 for the two minimal classes defined by T = \{\frac{e_{1}±e_{2}±e_{3}±e_{4}}{2} \} and non-cyclic quotient, with s_{max} = 15, and S = S(P_{5}^{2}) ∪ {±e_{6}, ±e_{7}} and cyclic quotient, with s_{max} = 17, respectively.

A7a4 = ( \begin{matrix}4 & 2 & 2 & 2 & 0 & 0 & 2 \\ 2 & 4 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 4 & 0 & 0 & 0 & 2 \\ 2 & 0 & 0 & 4 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 4 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 4 & 2 \\ 2 & 0 & 2 & 2 & 2 & 2 & 5 \end{matrix} ) , A7b4 = ( \begin{matrix}9 & 4 & 4 & 4 & 4 & 4 & 4 \\ 4 & 8 & 2 & 2 & 2 & 0 & 0 \\ 4 & 2 & 8 & 2 & 2 & 0 & 0 \\ 4 & 2 & 2 & 8 & 2 & 0 & 0 \\ 4 & 2 & 2 & 2 & 8 & 0 & 0 \\ 4 & 0 & 0 & 0 & 0 & 8 & 0 \\ 4 & 0 & 0 & 0 & 0 & 0 & 8 \end{matrix} ) .

[In the cyclic case, there might exist other minimal classes with s = 17.]

Remark 5.1. Extending A6a to dimension 7 by e_{i} · e_{7} = −e_{i} · e_{6} (i = 1, 2, 3, 4, 5), e_{6} · e_{7} = −24 and e_{7} · e_{7} = 48, we obtain a lattice of index 2 and length 5 with s = 14 and extra minimal vectors e_{i} + e_{6}, e_{i} − e_{7} (i = 1, 2, 3) and e_{6} + e_{7}.

Remark 5.2. Let Λ be an 8-dimensional, well-rounded lattice with s ≥ \frac{n(n+1)}{2} = 36 and let Λ_{0} be the sublattice generated by its minimal vectors. I could show that one of the following conditions holds for Λ:

1. Λ has a basis of minimal vectors.

2. ı(Λ) = 4, ı(Λ_{0}) = 2 and [Λ : Λ_{0}] = 2.

3. ı(Λ) = 6, ı(Λ_{0}) = 3 and [Λ : Λ_{0}] = 2.

I conjecture that assertion (1) is indeed the correct one, and even that the hypothesis s ≥ 36 could be slightly weakened. This conjecture implies that 8-dimensional perfect lattices have bases of minimal vectors, a result which is actually a consequence of the recent classification theorem of Dutour–Schürmann– Vallentin ([2]).

Lattices of maximal index 2 play a crucial rôle in the remark above. They are of two different types according to whether they are generated or not by their minimal vectors. Recall that for n ≥ 3 and r | n + 1, the Coxeter lattice 𝔸_{n}^{r} is the unique lattice L such that 𝔸_{n} ⊂ L ⊂ 𝔸_{n}^{∗} and [L : 𝔸_{n}] = r.

If [Λ : Λ_{0}] = ı(Λ) = 2 and s(Λ) ≥ \frac{n(n+1)}{2}, then (by a theorem of Korkine and Zolotareff; see [3], Section 6.1), Λ_{0} is similar to the root lattice 𝔸_{n}. A look at the structure of 𝔸_{n} modulo 2 then shows that Λ is similar to a lattice of the form ⟨𝔸_{m}^{2}, 𝔸_{n}⟩ for some odd m ≤ n. Now (see [3], Sections 5.1 and 5.2), we

Bases of minimal vectors in lattices, II 11

have S(𝔸_{m}^{2}) = S(𝔸_{m}) if and only if m ≥ 9. This gives us a complete description of lattices with ı(Λ) = 2 which are not generated by their minimal vectors; in particular, they exist if and only if n ≥ 9.

For perfect lattices with ı(Λ) = 2 which are generated by their minimal vectors, I only have the following conjecture:

Conjecture 5.3. A perfect lattice of maximal index 2 which is generated by its set of minimal vectors has dimension at most 7. (More optimistic form: replace “perfect” by “s ≥ \frac{n(n+1)}{2}”.)

[In Conway and Sloane’s notation (see [3], Section 6.5), the perfect lattices with ı = 2 which are generated by their minimal vectors are P_{4}^{1}, P_{5}^{1}, P_{5}^{2}, P_{6}^{5}, P_{6}^{6} and P_{7}^{32}.]

References

[1] G. Csóka, There exists a basis of minimal vectors in every n ≤ 7 dimensional perfect lattice, Ann. Univ. Sc. Budapest Eőtvős, Sect. Math. 30 (1987), 245–258.

[2] M. Dutour Sikirić, A. Schürmann, F. Vallentin, Classification of eight dimensional perfect forms, Math. arXiv, NT/0609388.

[3] J. Martinet, Perfect Lattices in Euclidean Spaces, Grundlehren 327, Heidelberg (2003).

[4] J. Martinet. Sur l’indice d’un sous-réseau in “Réseaux euclidiens, designs sphériques et formes modulaires”, J. Martinet éditeur, Monographie Ens. Math. 37, Genève (2001), 165–213.

[5] J. Martinet. Bases of minimal vectors in lattices, I , this journal.

[6] G.L. Watson, On the minimum points of a positive quadratic form, Mathematika 18 (1971), 60–70.

Jacques Martinet Institut de Mathématiques 351, cours de la Libération 33405 Talence cedex France

e-mail: Jacques.Martinet@math.u-bordeaux1.fr