Reduction Modulo 2 and 3 of Euclidean Lattices, II

Jacques Martinet

Laboratoire A2X, UMR 5465 associée au C.N.R.S. & Université Bordeaux 1

Abstract. This paper is a continuation of Reduction Modulo 2 and 3 of Euclidean Lattices ([7], Journal of Algebra, 2002). Résumé. Reduction modulo 2 et 3 des réseaux euclidiens (II). Cet article fait suite à l’article [7] Reduction Modulo 2 and 3 of Euclidean Lattices, paru en 2002 au Journal of Algebra.

1. Introduction.

Let E be an n-dimensional Euclidean space with scalar product x · y, and let L be the set of lattices (discrete subgroups of rank n) in E. For a lattice Λ ∈ L, we denote by min Λ its minimal norm: min Λ = min_{x∈Λ∖{0}} x · x, and by det Λ the determinant of the Gram matrix (e_{i} · e_{j}) of any ℤ-basis (e_{1}, e_{2}, …, e_{n}) of Λ.

In this paper, we still study short representatives for classes modulo 2 and 3 of a given lattice, indeed only modulo 2 from Section 3 onwards. Section 2 is devoted to root lattices modulo 2 and 3 and Section 3 to laminated lattices modulo 2. In Section 4, we give a few complements which may apply to odd lattices. In Section 5, we show how to attach to classes modulo 2 containing not too large vectors a lattice of codimension 1 having a relatively large minimum. Notably, the Leech lattice produces the 23-dimensional “equiangular” integral lattice of minimum 5 described in [9] whose set of minimal vectors constitutes a spherical (tight) 5-design; see also [1].

We now recall some results which are proved in [7]. We denote by Λ a lattice, and we set n = dim Λ, m = min Λ, and for t > 0, we denote by S_{t} the set of norm t vectors in Λ and we set s_{t} = \frac{1}{2} |S_{t}|. We also denote by m′ the norm of the second layer of Λ: m′ = min_{N(x)>m} N(x).

1991 Mathematics Subject Classification. 11H55, 11H56.

Key words and phrases. Lattices, short vectors, reduction.

I thank the authors of the PARI system (H. Cohen et al.), and especially Christian Batut for his specific lattice programs.

For lattices modulo 2, the basic identity, involving non-zero vectors x and y = x + 2z ≡ x mod 2Λ, is

N(y) + N(x) = 2(N(z) + N(x + z)) .  (1)

Provided that y ≠ ±x, this implies N(y) + N(x) ≥ 4m, with equality if and only if z and x + z = y − z are minimal.

Proposition 1.1. If x ≠ 0 and y ≠ ±x, we have N(y) + N(x) ≥ 4m, and equality holds if and only if z and x + z = y − z are minimal. We then have x · z = −\frac{N(x)}{2}, and x and y are orthogonal.

Proof. The first part is clear. If N(z) = N(x + z) = m, then

2 x · z + N(x) = 0 and y · x = N(x) + 2 x · z = 0 .
□

A complete set T of shortest representatives for non-zero classes modulo 2 yields a weighted formula of the kind

∑_{x∈T} \frac{1}{w(x)} = 2^{n} − 1  (2)

where w(x) = |{y ∈ Λ | N(y) = N(x) and y ≡ x mod 2Λ}| .

In [7], we essentially considered vectors of norm N ≤ 2m. The weight w(x) is equal to 1 if 0 < N(x) < 2m and belongs to the interval [1, n] if N(x) = 2m; some estimations for w beyond norm 2m will be proved in the next sections. This implies the inequality

∑_{0<t<2m} s_{t} + \frac{s_{2m}}{n} ≤ 2^{n} − 1

and various other inequalities of the same kind related to better bounds for w under various hypotheses.

For lattices modulo 3, the basic identity, involving non-zero vectors x and y = x + 3z ≡ x mod 3Λ, is

N(y) + 2N(x) = 3(2N(z) + N(x + z)) ,  (3 a)

together with its companion identity obtained by exchanging x and y:

N(x) + 2N(y) = 3(2N(z) + N(y − z)) .  (3 b)

We shall this time enumerate the classes modulo 3 up to sign (i.e., we now consider the set T of pairs ±C of classes modulo 3). The weighted formula now takes the form

∑_{x∈T} \frac{1}{w(x)} = \frac{3^{n} − 1}{2} .  (4)

In [7], Theorem 3.13, we proved for the weight the following results:

• w(x) = 1 if 0 < N(x) < m + m′ or m + m′ < N(x) < 2m + m′;

• w(x) = 1 or 3 if N(x) = m + m′;

• 1 ≤ w(x) ≤ n + 1 if N(x) = 2m + m′.

[w = 3 (resp. w = n + 1) corresponds to a configuration 𝔸_{2} (resp. 𝔸_{n+1}^{∗}.)]

In the sequel, weighted formulae will be displayed in the following form: for norms N where the weight may take values larger than 1, an expression such as

(\frac{(a_{1} + a_{2} + ⋯)}{w} + \frac{b_{1} + b_{2} + ⋯}{w′} + ⋯)

means that a_{1}, a_{2}, … are the number of pairs of vectors in the various norm N orbits with weight w, etc.

As noticed in [7] (Proposition 2.9 and Table I), lattices having mod 2 representatives of norm N < 2m constitute an open set in L. The corresponding mod 3 result applies to lattices having representatives of norm N < 2m + m′; this results from [7], Propositions 3.7 and 3.8; it notably applies to 𝔸_{n} (n ≤ 3) and 𝔻_{4}; see next section. Finally, we shall essentially consider only irreducible lattices; indeed, the bound 2m for classes modulo 2 does not hold for reducible lattices to within the three exceptions 𝔸_{1} ⊥ 𝔸_{1}, 𝔸_{2} ⊥ 𝔸_{1} and 𝔸_{2} ⊥ 𝔸_{2} (up to scale). It is easy to verify that the similar list for mod 3 lattices reduces to 𝔸_{1} ⊥ 𝔸_{1} and 𝔸_{2} ⊥ 𝔸_{1}.

2. Root Lattices Modulo 2 and 3.

In this section, we consider irreducible root lattices, indeed lattices isometric to 𝔸_{n} (n ≥ 1), 𝔻_{n} (n ≥ 4) or 𝔼_{n} (n = 6, 7, 8). We have m = 2, and disregarding the trivial cases of 𝔸_{1} and 𝔸_{2}, m′ = 4, hence 2m + m′ = 4m = 8. We denote by (ε_{0}, ε_{1}, …, ε_{n}) (resp. (ε_{1}, …, ε_{n})) the canonical basis for ℤ^{n+1} (resp. ℤ^{n}) and set

𝔸_{n} = {x ∈ ℤ^{n+1} | ∑ x_{i} = 0} and 𝔻_{n} = {x ∈ ℤ^{n} | ∑ x_{i} ≡ 0 mod 2} .

Proposition 2.1. Up to signs, shortest representatives for classes modulo 2 or 3 of 𝔸_{n} or 𝔻_{n} are the vectors which are of one of the following forms:

(1) ε_{i_{1}} ± ε_{i_{2}} ± ⋯ ± ε_{i_{k}}, of norm 2k, for 𝔸_{n} and 𝔻_{n}, modulo 2 and 3.

(2) 2ε_{i}, of norm 4, for 𝔻_{n} modulo 2.

(3) 2ε_{i_{1}} + ⋯ + 2ε_{i_{ℓ}} + ε_{i_{ℓ+1}} + ⋯ + ε_{i_{ℓ+k}} − ε_{i_{ℓ+k+1}} − ⋯ − ε_{i_{3ℓ+2k}} (ℓ > 0), of

norm 6ℓ + 2k, for 𝔸_{n} modulo 3.
(4) 2ε_{i_{1}} ± ε_{i_{2}} ± ⋯ ± ε_{i_{k}} (k > 0), of norm k + 4, for 𝔻_{n} modulo 3.

Moreover, the weights of the vectors above are 1 in case (1), n in case (2), \binom{3ℓ+k}{ℓ} in case (3), and k in case (4).

Proof. (Sketch.) If x = ∑_{j} a_{j}ε_{j} ∈ Λ (Λ = 𝔸_{n} or 𝔻_{n}) has some large component a_{i}, consider a transformation of the form x ↦ x ± 2(ε_{i} − ε_{j}) or x ↦ x ± 3(ε_{i} − ε_{j}) if Λ = 𝔸_{n}, x ↦ x ± 2(ε_{i} ± ε_{j}) or x ↦ x ± 3(ε_{i} ± ε_{j}) if Λ = 𝔻_{n} (and then also x ↦ x±4ε_{i} or x ↦ x±6ε_{i} if x has a single component). That the vectors listed above are among the shortest representatives in their class is easy to verify; we leave to the reader the calculation of the weights. □

For exceptional lattices, we have:

Proposition 2.2. For 𝔼_{6} mod 2, 𝔼_{7} mod 3 and 𝔼_{8} mod 2 and 3, all classes possess representatives of norm ≤ 2m = 4 or 2m + m′ = 8. For 𝔼_{7} mod 2 (resp. 𝔼_{6} mod 3), there is one missing class, whose smallest representatives have norm 6 (resp. 12); this is the set of minimal vectors in 2 𝔼_{7}^{∗} (resp. 3 𝔼_{6}^{∗}), of weight 28 (resp. 27).

Proof. Modulo 2, we use the general bound w ≤ n and its refinement w ≤ n − 1 (proved in [7] before Theorem 2.4) which applies to lattices such that \frac{(2m)^{n}}{det(Λ)} is not a square, together with the fact that the sum must not exceed 2^{n} − 1. This immediately gives us the following three formulae for vectors of norm 2 and 4:

E_{6} mod 2: 36 + \frac{135}{5} = 63 = 2^{6} − 1 ;

E_{7} mod 2: 63 + \frac{378}{6} = 126 = (2^{7} − 1) - 1 ;

E_{8} mod 2: 120 + \frac{1080}{8} = 255 = 2^{8} − 1.

The same device applies for classes modulo 3. For integral lattices, the bound w ≤ n + 1 can be refined to w ≤ n whenever the scaled copy of 𝔸_{m}^{∗} to norm 2m + m′ (here, 8) is not integral; this applies to 𝔼_{6} and 𝔼_{7}. We also need the fact that norm 6 vectors in 𝔼_{7} share out among two orbits (in all other cases, primitive vectors of norm N ≤ 8 constitute a single orbit). These two orbits have s′_{6} = 28 and s′′_{6} = 1008 pairs of vectors; the first one is 2S(𝔼_{7}^{∗}). These remarks show that the weighted formulae for vectors of norm N ≤ 8 are:

E_{6} mod 3 : 36 + 135 + \frac{360}{3} + \frac{432}{6} = 363 = \frac{3^{6} − 1}{2} - 1 ;

E_{7} mod 3 : 63 + 378 + (28 + \frac{1008}{3}) + \frac{2016}{7} = 1093 = \frac{3^{7} − 1}{2} ;

E_{8} mod 3 : 120 + 1080 + \frac{3360}{3} + \frac{8640}{9} = 3280 = \frac{3^{8} − 1}{2}.

There remains to characterize the two missing classes. For 𝔼_{7} mod 2, consider x ∈ 2S(𝔼_{7}^{∗}). We have N(x) = 2^{2} · \frac{3}{2} = 6. Let y ≡ x mod 2 in 𝔼_{7}. We have N(y) ≡ N(x) ≡ 2 mod 4. Hence if y were shorter than x, it would have norm 2. But pairs of norm 2 vectors in 𝔼_{7} constitute an orbit of 63 > 28 different classes mod 2, a contradiction.

Similarly, the 27 pairs of vectors in 3S(𝔼_{6}^{∗}), of norm 3^{2} \frac{4}{3} = 12, cannot be congruent mod 3 to a shorter vector, for such a vector would have norm 6 (because y ≡ x mod 3𝔼_{6} ⇒ N(y) ≡ N(x) mod 3), and norm 6 vectors constitute an orbit of \frac{360}{2} = 120 > 27 vectors. (Incidentally, this shows that all norm 10 vectors in 𝔼_{6} are congruent mod 3 to a norm 4 vector.) □

3. Laminated Lattices Modulo 2.

These lattices, that we shall consider only in the range 1 ≤ n ≤ 24, were defined inductively by Conway and Sloane; see [3], Chapter 6 for a precise definition. They have minimum 4. There is one lattice in each dimension, denoted by Λ_{n}, except for n = 11, 12, 13 where there are two, three, and three lattices respectively, characterized by their kissing number, and denoted by an extra superscript min, mid or max. The aim of this section is to show that the list of laminated lattices for which it was proved in [7], Section 2, that all classes modulo 2 contain representatives of norm N ≤ 8 is actually complete up to dimension 24.

Theorem 3.1. Laminated lattices of dimension n ≤ 24 possessing representatives of norm N ≤ 8 for all classes modulo 2 are those of dimension n ≤ 6, 8 ≤ n ≤ 10 and n = 24.

Before proceeding to the proof, we state and prove a lemma:

Lemma 3.2. Let L be an integral lattice of minimum 3 and let Λ = L_{even} be its even part. Assume that there exist in L two non-orthogonal pairs of minimal vectors. Then Λ has minimum 4 and contains a class modulo 2 of minimum 12.

proof of 3.2. We have min L_{even} ≥ 4, and if x, y are non-orthogonal, nonproportional minimal vectors in L, we have x · y = ±1, hence N(x ∓ y) = 4, whence N(L_{even}) = 4.

Let e ∈ S(L) and let f = 2e. Since [L : Λ] = 2, we have Λ = ⟨L, e⟩ = ⟨L, \frac{f}{2} ⟩, hence

L ∖ Λ = \{\frac{x}{2} | x ≡ f mod 2Λ\} .

Since min L = min L∖Λ = 3, we have N(x) ≥ 12 on the whole class of f modulo 2. Since N(f) = 12, this completes the proof of the lemma. □

proof of Theorem 3.1. In dimensions n ≤ 8, the laminated lattices are scaled copies of root lattices, namely 𝔸_{n} (n = 1, 2, 3), 𝔻_{n} (n = 4, 5) and 𝔼_{n} (n = 6, 7, 8), and Theorem 3.1 follows from the results of Section 2. For n = 24, Λ_{24} is the Leech lattice, and the result is a theorem of Conway (see [3], Chapter 12). The case of dimensions 9 and 10 is dealt with in [7], Section 2, Table III. We are thus left with the 18 laminated lattices of dimension n ∈ [11, 23]. We now show how Lemma 3.2 can be used to deal with 16 of them.

Recall that O_{23} stands for the unimodular 23-dimensional lattice of minimum 3. The lattice ℤ ⊥ O_{23} can be defined as a Kneser–neighbour of Λ_{24} through a norm 4 vector, which shows that (O_{23})_{even} is isometric to Λ_{23}. Now, it is shown in [2] that the antilaminations of O_{23} (the descending chain of the densest cross–sections) produce a unique lattice (denoted by O_{n}) in dimensions 23 to 14. Since the antilaminations of Λ_{23} also produce the Λ_{n} series in these dimensions, we have (O_{n})_{even} ≃ Λ_{n} for 14 ≤ n ≤ 23. Then we find two 13-dimensional lattices, which allows again to deal with Λ_{13}^{max} and Λ_{13}^{min} (Λ_{13}^{mid} is a dead–end for laminated lattices). We can even consider dimensions 12 and 11. Explicitly, in the notation of [2], we have Λ_{13}^{max} ≃ (O_{13b})_{even}, Λ_{13}^{min} ≃ (O_{13a})_{even}, Λ_{12}^{max} ≃ (O_{12b})_{even}, Λ_{12}^{mid} ≃ (O_{12a})_{even}, and Λ_{11}^{max} ≃ (O_{11})_{even}. As for Λ_{13}^{mid}, it is also the even part of an integral norm 3 lattice, discovered by Plesken and Pohst ([8]), indeed the lattice with s = 84 of their list. Lemma 3.2 shows that all the sixteen lattices listed above contain a class of minimum 12.

To complete the proof of Theorem 3.1, it suffices to consider the two lattices Λ_{11}^{min} and Λ_{12}^{min}. We have shown using PARI-GP that vectors of norm N ≤ 8 do not represent all classes. For the sake of completeness, we display below the weighted formulae for vectors of norm N ≤ 8 for the four lattices Λ_{9}, Λ_{10}, Λ_{11}^{min} and Λ_{12}^{min}. The notation is that of section 2. The first two numbers are s_{4} and s_{6}; we then give for each weight the numbers of vectors in a given orbit with this weight. (Note that two congruent vectors may belong to different orbits.)

Λ_{9} : 136 + 128 + \frac{1 + 8}{9} + \frac{560 + 512}{8} + \frac{448}{4} = 511 .

Λ_{10} : 168 + 384 + \frac{3 + 24}{9} + \frac{768 + 288}{8} + \frac{48 + 192}{5} + \frac{1152}{4} = 1023 .

Λ_{11}^{min} : 216 + 816 + \frac{54}{9} + \frac{1032}{8} + \frac{960}{5} + \frac{1920}{4} + \frac{384}{3} = 1967 .

Λ_{12}^{mid} : 312 + 1728 + \frac{12 + 96}{9} + \frac{768 + 192 + 24}{8} + \frac{768 + 3072}{5} +

\frac{1536 + 2304}{4} = 3903 .

We observe that there are (2^{11} − 1) − 1967 = 80 missing classes in the case of Λ_{11}^{min} and (2^{12} − 1) − 3903 = 192 in the case of Λ_{12}^{min}. □

4. Odd Lattices Modulo 2.

In this section, we consider as previously a lattice Λ of dimension n and minimum m. Our aim is to study the contribution of norm 2m+1 vectors. Such vectors of course do not exist if Λ is even. In the proposition below, the rôle of the dual of an 𝔸_{k} lattice resembles the one it plays for norm 2m + m′ vectors with respect to Λ mod 3.

Theorem 4.1. Let Λ be integral.

(1) Vectors of norm 2m + 1 (if any) are minimal in their class modulo 2.

(2) If min Λ is odd, each class contains at most 2m + 2 pairs of such vectors, and when this bound is attained, their configuration is that of S(𝔸_{2m+1}^{∗}).

Proof. Let x ∈ S_{2m+1}(Λ), and let y ≡ x mod 2Λ, say, y = x + 2z. By Proposition 1.1, we have N(y) ≥ 2m − 1. Since y ≡ x mod 2 ⟹ N(y) ≡ N(x) mod 4, we must have N(y) ≥ N(x), which proves the first part of Theorem 4.1.

Suppose now that N(y) = N(x) = 2m + 1. Writing −y = x − 2(x + z), we see that changing y into −y amounts to exchanging z and −(x + z). In the sequel, we shall assume that N(z) = m + 1 and N(x + z) = m. With this choice we have x · z = −(m + 1), hence x · y = 2m + 1 − 2(m + 1) = −1.

Lemma 4.2. Let ±x_{1}, …, ±x_{r}, r ≥ 2 be a system of norm 2m + 1 vectors in Λ belonging to the same class modulo 2 Λ. Then for a convenient choice of the x_{i} among x_{i}, −x_{i}, the scalar products x_{i} · x_{j}, j ≠ i all have the same value, namely +1 if m is even, and −1 if m is odd.

proof of 4.2. For i = 2, …, r, define z_{i} by x_{i} = x_{1} + 2z_{i}. Taking x = x_{1} and y = x_{i} in the calculation we made in the course of the proof of Theorem 4.1, we see that we may choose the signs of the x_{i} so that x_{1} · x_{i} = −1 for all i ≥ 2. We then have

x_{i} · x_{j} = (x_{1} + 2z_{i}) · (x_{1} + 2z_{j}) = 2m + 1 − 4(m + 1) + 4 z_{i} · z_{j}

hence

z_{i} · z_{j} = \frac{2m + 3 + x_{i} · x_{j}}{4}

for 2 ≤ i < j ≤ r. We must have x_{i} · x_{j} + 2m + 3 ≡ 0 mod 4, whence the result for 2 ≤ i < j ≤ n. Negating x_{1} if m is even yields the desired result in all cases. □

End of proof of 4.1. Since m is odd, we may assume by Lemma 4.2 that x_{i} · x_{j} = −1 for all pairs (i, j) with j ≠ i. Since

N(x_{1} + ⋯ + x_{r}) = r(2m + 1) − 2 \binom{r}{2} = r (2m + 2 − r) ≥ 0 ,

we have r ≤ 2m + 2, and x_{1}, …, x_{r} generate a canonical section of 𝔸_{2m+1}^{∗} scaled to norm 2m + 1. Since the vectors \frac{x_{i}+x_{j}}{2} belong to Λ, we are done. □

Example 4.3. Let Λ be 𝔼_{7}^{∗} scaled to minimum 3. This is an integral lattice, whose norms are the positive integers congruent to 0 or −1 modulo 4. We have s_{3} = 28, s_{4} = 63 and s_{7} = 288, hence

s_{3} + s_{4} + \frac{s_{7}}{8} = 28 + 63 + 36 = 127 = 2^{7} − 1 .

Theorem 4.1 hence shows that the shortest vectors in classes modulo 2 of Λ are those of norm 3, 4, and 7.

5. Lattices of Codimension 1.

In this section, we assume that Λ is integral. We explain how to construct lattices of dimension n−1 from a vector e ∈ Λ of norm μ in the range m ≤ μ < 2m. (Everything also works if μ = 2m, but the configuration of minimal vectors of the lattices we are going to construct are then uninteresting orthogonal configurations.) We denote by C the class of e modulo 2.

Lemma 5.1. Let e be as above and let x ≡ e mod 2Λ. Then one of the following conditions holds:

(1) x is proportional to e.

(2) N(x) > 4m − μ.

(3) N(x) = 4m − μ and x is orthogonal to e.

Proof. Write x = e + 2z. Proposition 1.1 shows that if x is not proportional to e, then N(x) ≥ 4m − N(e), and that if equality holds, then e · z = −\frac{N(e)}{2}, which implies e · x = e · (e + 2z) = 0. □

Lemma 5.2. L = C ∪ 2Λ is a lattice of determinant 2^{2n−2} det(Λ).

Proof. We have C ∪ 2Λ = 2Λ ∪ (e + 2Λ). Hence L is a lattice containing 2Λ to index 2, which shows that det(L) = 2^{−2} det(2Λ). □

By Lemma 5.1, min L = μ and S(L) = {±e}. To obtain a lattice with a larger minimum, we consider L_{e} = (ℝ e)^{⊥} ∩ L.

Proposition 5.3. L_{e} = (ℝ e)^{⊥}∩L is an (n−1)-dimensional lattice of minimum M ≥ 4m − μ and determinant 2^{2n−2}μ det(Λ) if μ is even, and 2^{2n}μ det(Λ) if μ is odd.

Proof. Only the last assertion needs a proof. Given a primitive vector e′ ∈ L^{∗}, the determinant of L′ = L ∩ (ℝ e′)^{⊥} is det(L′) = det(L) N(e′) (see [6], Proposition 1.3.4; N(e′) is the determinant of the 1-dimensional lattice (ℝ e′)∩L^{∗}). Here we must determine a generator e′ of ℝ e ∩ L^{∗}. We have L = ⟨2Λ, e⟩ and (2Λ)^{∗} = \frac{1}{2} Λ^{∗}, hence

L^{∗} = \{\frac{y}{2} | y ∈ Λ^{∗}, y · e ≡ 0 mod 2\} .

If μ is even, e ∈ L^{∗}; if μ is odd, 2e ∈ L^{∗}. Since e is primitive in Λ (because N(e) < 4m), a congruence e · y ≡ 0 mod a may not hold on Λ^{∗} for an integer a > 1. This shows that if μ is even (resp. odd), e (resp. 2e) is primitive in L^{∗}. This completes the proof of the proposition. □

Proposition 5.4. Let Λ_{e} = L_{e} if μ ≡ 1 mod 2, Λ_{e} = \frac{1}{\sqrt{2}} L_{e} if μ ≡ 2 mod 4, and Λ_{e} = \frac{1}{2} L_{e} if μ ≡ 0 mod 4. Then Λ_{e} is an integral lattice, and we have det(Λ_{e}) = 2^{2n}μ det(Λ) if μ ≡ 1 mod 2, det(Λ_{e}) = 2^{n−2}μ det(Λ) if μ ≡ 2 mod 4, and det(Λ_{e}) = \frac{μ}{4} det(Λ) if μ ≡ 0 mod 4.

Proof. The assertions concerning the determinant of Λ_{e} follow immediately from Proposition 5.3. It thus suffices to prove that x, y ∈ L_{e} ⟹ x · y ≡ 0 mod (4, μ) . Write x = 2z (resp. x = e + 2z) if x ∈ 2Λ (resp. x ∉ 2Λ), and similarly y = 2t or y = e + 2t. If both x and y belong to 2Λ, then x · y ≡ 0 mod 4. If, say, x ∈ 2Λ and y ∉ 2Λ, we again have x · y = 2(z · e) + 4 z · t = 4 z · t ≡ 0 mod 4. Finally, in the remaining case, we have

x · y = μ + 2e · z + 2e · t + 4 z · t and 2(e · z) ≡ 2(e · t) ≡ −μ mod 4 ,

hence x · y ≡ −μ mod 4. □

We now give some examples. In all cases we shall consider, the minimum of Λ_{e} is equal to the lower bound given in Proposition 5.3.

If m = 2 and if Λ is even, the only possible choice is μ = 2. Take for Λ an irreducible root lattice of dimension n ≥ 2. Then norm 6 vectors in Λ belong to one or two classes modulo 2, and exactly one such class C contains a norm 2 vector e. When Λ is isometric to 𝔸_{n} (n ≥ 2), 𝔻_{n} (n ≥ 4), 𝔼_{6}, 𝔼_{7}, and 𝔼_{8}, Λ_{e} has minimum 3, and s(Λ_{e}) is equal to n − 1, 2(n − 2), 10, 16, and 28 respectively. The lattice corresponding to 𝔼_{8} is a scaled copy of 𝔼_{7}^{∗}, and 𝔼_{7} and 𝔼_{6} yield lattices similar to Coxeter’s 𝔻_{6}^{+} and 𝔸_{5}^{2}.

If m = 4, we may choose μ = 4 or μ = 6, obtaining in general a lattice Λ_{e} of minimum M_{e} = 3 if μ = 4 and M_{e} = 5 if μ = 6 (and no other value if Λ is even).

Theorem 5.5. Let Λ be an integral lattice of minimum 4.

(1) If μ = 4, then min Λ_{e} ≥ 3 and det(Λ_{e}) = det(Λ).

(2) If μ = 6, Λ_{e} is an integral lattice whose norm 5 vectors have mutual scalar products ±1. In particular, directions of norm 5 vectors constitute an equiangular family of lines.

Proof. If μ = 4, the result is an immediate consequence of Proposition 5.4. Let now μ = 6, and let x = e + 2z and y = e + 2t (y ≠ ±x) be two norm 10 vectors in L_{e}. We have x · y = −6 + 4 z · t (see the proof of Proposition 5.4). Since z and t are minimal in Λ, we have z · t ∈ {4, 2, 1, 0, −1, −2, −4}. Since y ≠ ±x, we have |x · y| ≤ 5, which implies z · t = 2 or 1, hence x · y = ±2. This proves that non-proportional norm 5 vectors in Λ_{e} have scalar product ±1, hence that they generate an equiangular family of lines. □

We now consider the important special case of Λ = Λ_{24} (the Leech lattice).

Corollary 5.6. Let Λ be the Leech lattice Λ_{24}.

(1) If μ = 4, Λ_{e} is the unimodular lattice O_{23} (min = 3, s = 2300).

(2) If μ = 6, Λ_{e} is the integral lattice of minimum 5 with s = 276 (= \frac{23·24}{2}) which is dual (up to scale) to the lattice M_{23}[2] of [9], Table 19.2.

Proof. If μ = 4, Λ_{e} is a unimodular lattice of minimum M ≥ 3, hence isometric to O_{23} ([3], Table 16.7).

If μ = 6, we use the fact that Λ_{e} has minimum 5 and kissing number \frac{|a_{10}|}{|a_{6}|} = 276,

where we denote as in [7] by a_{6} (resp. a_{10}) the unique orbit of vectors of norm 6 (resp. 10) in Λ_{24}. Theorem 9.1 of [9] (and the results of [5] on equiangular families of lines) now shows that Λ_{e}, as an integral lattice of minimum 5 with equiangular directions of minimal vectors and maximal possible value of s, is similar to M_{23}[2]^{∗}.

[ For μ = 4, since \frac{|b_{12}|}{|a_{4}|} = 2300 (notation of [7]), we recover the equality s(O_{23}) = 2300.]

□

Remark 5.7. The integral scaled copies L of M_{23}[2]^{∗} (of minimum 5) and L′ of M_{23}^{∗} (of minimum 15) which occur in Table 19.2 of [9] have the same configurations of minimal vectors. Indeed, L′ contains to index 2 a lattice isometric to \sqrt{3} L. The successive layers of L (resp. L′) have norms 5, 8, 9, 12, … (resp. 15, 20, 24, …). This shows that L contains a class modulo 3 of minimum 60, which produces vectors of norm 3 · \frac{60}{9} = 20 in L′.

Similarly, using the parity class of O_{23} (of minimum 15; see [4], where Elkies proves a much more general result), we obtain after rescaling an integral lattice of minimum 12. This lattice is indeed proportional to Λ_{23}^{∗}.

References

[1] E. Bannai, A. Munemasa, B. Venkov, The Nonexistence of Certain Tight Spherical Designs (with an appendix by Y.-F. S. Peterman), preprint.

[2] C. Batut, J. Martinet, A Catalogue of Perfect Lattices, http://www.math.u-bordeaux.fr/ martinet .

[3] J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, Grundlehren 290, Springer-Verlag, Heidelberg (1988). Third edition: 1993.

[4] N. Elkies, Lattices and codes with long shadows, Math. Res. Lett. 2 (1995), 643–651.

[5] P. W. H. Lemmens, J. J. Seidel, Equiangular lines, J. Algebra 24 (1973), 494–512.

[6] J. Martinet, Perfect Lattices in Euclidean Spaces, Grundlehren 327, Springer-Verlag, Heidelberg (2003). (English corrected and updated edition of a French version, Masson (now Dunod), Paris, 1996.)

[7] J. Martinet, Reduction Modulo 2 and 3 of Euclidean Lattices, J. Algebra 251 (2002), 864–887.

[8] W. Plesken, M. Pohst, Constructing integral lattices with prescribed minimum. I , Math. Comp. 45 (1985), 209–221 and S5–S16.

[9] B. Venkov, Réseaux et “designs” sphériques (Notes by J. Martinet), in Réseaux euclidiens, designs sphériques et groupes, L’Ens. Math., Monographie 37, J. Martinet, ed., Genève (2001), 10–86.

J. Martinet A2X, Institut de Mathématiques Université Bordeaux 1 351, cours de la Libération F–33405 TALENCE cedex

E-mail address: martinet@math.u-bordeaux.fr