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.
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