**Keywords**: modular lattices, tables, minimal norm, quadratic forms

Part of the **Catalogue of Lattices**
which is a joint project of
**Gabriele Nebe**,
RWTH Aachen university
(nebe@math.rwth-aachen.de)
and
**Neil J. A. Sloane**,
**(njasloane@gmail.com)**.

### Constructions

**(b) One of the following 2-dimensional lattices:**

- n = 2, N = 2: the lattice [1, 0; 0, 2]
- n = 2, N = 3, genus "+": the lattice [1, 0; 0, 3]
- n = 2, N = 5, genus = "-": the lattice [2, 1; 1, 3]
- n = 2, N = 5, genus = "-": the lattice [1, 0; 0, 5]
- n = 2, N = 7: the lattice [2, 1; 1, 4]
- n = 2, N = 11, genus = "-": the lattice [2, 1; 1, 6]
- n = 2, N = 11, genus = "+": the lattice [3, 1; 1, 4]
- n = 2, N = 14, genus = "-": the lattice [3, 1; 1, 5]

**(d) Direct sum:** If L and M are N-modular
so is the direct sum L + M.
If L has genus l (= +-1) and M has genus m (= +-1)
then L + M has genus lm.

**(B) Constructed by C. Bachoc:**
see Reference [Bach97] below.

**(C) Construction A applied to an additive trace self-dual
code** of length n and min distance d over GF(4) produces
a 3-modular lattice in dim 2n and minimal norm = min{d, 4}
- see References [Bach97], [RaSl98] below.

**(Q) Constructed (or mentioned) by H.-G.
Quebbemann:**
see References [Queb95], [Queb97] below.

### Upper Bounds

**(g) By a complete investigation of the genus**,
R. Scharlau and B. Hemkemeier [SchHe94] show that no even lattice exists.
From [RaSl98], an extremal lattice would necessarily be even.
So minimal norm 6 is impossible.
**(S) Shadow plus integrality:**
The minimal norm cannot be greater than shown,
or else one of the following conditions fails:
theta functions of lattice and shadow have nonnegative integer
coefficients, theta function of lattice is congruent to 1 mod 2,
theta function of shadow is congruent to 0 or 1 mod 2;
shadow may contain at most two vectors of norm
less than half the minimal norm of the lattice.

**References**

- [Bach97] C. Bachoc,
Applications of coding theory to the construction of modular lattices,
*J. Combin. Theory*, **A 78** (1997), 92-119.
- [Queb95] H.-G. Quebbemann, Modular lattices in Euclidean spaces,
*J. Number Theory* **54** (1995), 190-202.

- [Queb97] H.-G. Quebbemann, Atkin-Lehner eigenforms
and strongly modular lattices,
*L'Enseign. Math.* **43** (1997), 55-65.

- [RaSl98] E. M. Rains and N. J. A. Sloane,
The shadow theory of modular and unimodular lattices,
J. Number Theory, 73 (1998), 359-389.
- [SchHe94] R. Scharlau and B. Hemkemeier,
*Classification of integral lattices with large class number*,
Preprint No. 94-102, Univ. Bielefeld, 1994.
- [SchVe95] R Scharlau and B. B. Venkov,
The genus of the Barnes-Wall lattice,
*Comm. Math. Helv.*, **69** (1994), 322-333.

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