Toric Codes

ToricCode(P, q): TorPol, RngIntElt -> Code

The linear code \(C\) over the finite field \({\bf F}_{q}\) 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 \([0,q-2] \times [0,q-2]\). Then the code is the monomial evaluation code where each point \((a,b)\) corresponds to the monomial \(x^ay^b\), and these monomials are evaluated at the points of the torus \(({\bf F}_{q}^*)^2\).

ToricCode(S, q): SeqEnum[TorLatElt], RngIntElt -> Code

ToricCode(S, q): SetEnum[TorLatElt], RngIntElt -> Code

The linear code \(C\) over the finite field \({\bf F}_{q}\) 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: Toric Code (ex-deddf3)

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 ]

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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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