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The linear code C over the finite field GF(q) associated
with the lattice points of the polygon P.
To achieve this, after a translation so that the lattice
points of P lie in the first quadrant, as close to the
origin as possible, these points must lie in the
box [0, q - 2] x [0, q - 2].
Then the code is the monomial evaluation code where
each point (a, b) corresponds to the monomial xayb,
and these monomials are evaluated at the points of
the torus (GF(q) * )2.
ToricCode(S, q) : SetEnum[TorLatElt], RngIntElt -> Code
The linear code C over the finite field GF(q) associated
with the lattice points in S. (Note that the points will
be translated to lie within a box at the origin of the first
quadrant, as is usual.)
We construct the toric code based on the lattice points in the polygon
with vertices (3, 0), (5, 0), (3, 3), (1, 5), (0, 3), (0, 1).
> P := Polytope( [[3,0], [5,0], [3,3], [1,5], [0,3], [0,1]] );
> C := ToricCode(P, 7);
> [ Length(C), Dimension(C), MinimumDistance(C) ];
[ 36, 19, 12 ]
We can compare this with the current database of best known linear
codes.
> BKLCLowerBound(Field(C), Length(C), Dimension(C));
12
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