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Toric Codes

ToricCode(P, q) : TorPol, RngIntElt -> Code
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) : SeqEnum[TorLatElt], RngIntElt -> Code
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.)

Example CodeAlG_ToricCode (H153E4)

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));
11
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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012