In arXiv:1204.1181
a classification scheme for reflexive Gorenstein cones was presented.
An essential role for that scheme is played by basic weight systems,
i.e. weight systems containing no weights of 1/2 or 1 and no two weights whose
sum is 1.
We use the following notation:

d ... the dimension of the cone,

r ... the index of the cone,

d_{CY} = d  2r ... the "CalabiYaudimension" of the cone,

q_{i} = n_{i}/k ... the weights.
The files presented below contain lines of the form
k n_{1} ... n_{d} M: # # N: # # H: ...
if the cone determined by the weight system is reflexive, and of
the form
k n_{1} ... n_{d} M: # # F: #
otherwise, where

k n_{1} ... n_{d} encodes the weights as
q_{i} = n_{i}/k,

M: # # gives the numbers of lattice points and vertices of the
support of the cone,

N: # # gives the numbers of lattice points and vertices of the
support of the dual cone,

H: ... gives the "stringy Hodge numbers" in the format
h_{11} ... h_{1,dCY1} [Euler number]
(nonstandard h_{0i}),
where "nonstandard h_{0i}" indicates any h_{0i}'s that
do not follow the standard pattern
h_{00} = h_{dCYdCY} = 1,
h_{0i} = 0 for 0 < i < d_{CY},

F: # gives the number of facets of the support of the cone.
Weight systems for (d,r) with halfinteger r are listed as the weight systems
of type (d+1,r+1/2) that are obtained by adding one weight of 1/2;
e.g. (1/6, 1/6, 1/6) of type (3, 1/2) becomes (1/6, 1/6, 1/6, 1/2) of type
(4, 1) etc.
The basic weight systems for d_{CY} = 2 are listed in the following files:
The first 95 = 48 + 47 correspond to K3 hypersurfaces.
Each of the weight systems gives rise to a reflexive Gorenstein cone.
The basic weight systems for d_{CY} = 3 are listed in the following files:

d=4, r=1/2
(97,036)

d=5, r=1
(86,990)

d=6, r=3/2
(168,107)

d=7, r=2
(34,256)

d=8, r=5/2
(6,066)

d=9, r=3
(1)
The first 184,026 = 97,036 + 86,990 correspond to CalabiYau hypersurfaces, and
also to reflexive Gorenstein cones of index 1.
The cases for d > 5 do not necessarily give rise to reflexive Gorenstein cones.
The sublists of reflexive cones are