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