Modular Lattices (Footnotes to Table)

 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 ( and Neil J. A. Sloane, (

Footnotes to Table of Modular Lattices


(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.