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<Section Label="AlgebraicGeometricCodes">
<Heading>Algebraic-geometric codes functions</Heading>

This section of &GAP;'s singular package and the
corresponding code were written by David Joyner,
<Email>wdj@usna.edu</Email>, (with help from Christoph Lossen and Marco
Costantini). It has been tested with <Package>Singular</Package> version 2.0.x.

<P/>

To start off, several new <Package>Singular</Package> commands must be loaded.

The following command loads the necessary <Package>Singular</Package>
and &GAP; commands, the packages <Package>singular</Package> and
<Package>GUAVA</Package> (if not already loaded), and (re)starts
<Package>Singular</Package>.

<Example>
gap> ReadPackage("singular", "contrib/agcode.g");;
</Example>

<ManSection>
<Func Name="AllPointsOnCurve" Arg = "f, F"/>
<Description>

Let <M>F</M> be a finite and prime field. The function
<A>AllPointsOnCurve( f, F )</A> computes a list of
generators of maximal ideals representing rationals
points on a curve <M>X</M> defined by <M>f(x,y)=0</M>.

<Example>
gap> F:=GF(7);;
gap> R2:= PolynomialRing( F, 2 );;
gap> SetTermOrdering( R2, "lp" );; # --- the term ordering must be "lp"
gap> indet:= IndeterminatesOfPolynomialRing(R2);;
gap> x:= indet[1];; y:= indet[2];;
gap> f:=x^7-y^2-x;;
gap> AllPointsOnCurve(f,F);
[ [ x_1 ], [ x_1-Z(7)^0 ], [ x_1+Z(7)^4 ], [ x_1+Z(7)^5 ], [ x_1+Z(7)^0 ],
  [ x_1+Z(7) ], [ x_1+Z(7)^2 ] ]
</Example>

</Description>
</ManSection>

<ManSection>
<Func Name="AGCode" Arg = "f, G, D"/>
<Description>

Let f be a polynomial in x,y over F=GF(p) representing plane curve
<M>X</M> defined by <M>f(x,y)=0</M>, where p is a prime (prime powers
are not yet supported by the underlying <Package>Singular</Package>
function).
Let G, D be disjoint rational divisors on <M>X</M>, where
D is a sum of distinct points, <M>supp(D)={P_1, ..., P_n}</M>.
The AG code associated to
f, G, D is the F defined to be the image of the
evaluation map <M>f \mapsto (f(P_1),...,f(P_n))</M>.
The function <A>AGCode</A> computes a list of
length three, [G, n, k], where G is a generator matrix
of the AG code C, n is its length, and k is its dimension.

<Example>
gap> F:=GF(7);;
gap> R2:= PolynomialRing( F, 2 );;
gap> SetTermOrdering( R2, "lp" );; # --- the term ordering must be "lp"
gap> SingularSetBaseRing(R2);
gap> indet:= IndeterminatesOfPolynomialRing(R2);;
gap> x:= indet[1];; y:= indet[2];;
gap> f:=x^7-y^2-x;;
gap> G:=[2,2,0,0,0,0,0]; D:=[4..8];
[ 2, 2, 0, 0, 0, 0, 0 ]
[ 4 .. 8 ]
gap> agc:=AGCode(f,G,D);
[ [ [ Z(7)^3, Z(7), 0*Z(7), Z(7)^4, Z(7)^5 ],
      [ 0*Z(7), Z(7)^4, Z(7)^0, Z(7)^5, Z(7)^3 ],
      [ 0*Z(7), 0*Z(7), Z(7)^3, Z(7), Z(7)^2 ] ], 5, 3 ]
</Example>

This generator matrix can be fed into the <Package>GUAVA</Package>
command <Ref Oper="GeneratorMatCode" BookName="GUAVA"/> to create a
linear code in &GAP;, which in turn can be fed into the
<Package>GUAVA</Package> command
<Ref Oper="MinimumDistance" BookName="GUAVA"/> to compute the minimum
distance of the code.

<Example>
gap> ag_mat:=agc[1];;
gap> C := GeneratorMatCode( ag_mat, GF(7) );
a linear [5,3,1..3]2 code defined by generator matrix over GF(7)
gap> MinimumDistance(C);
3
</Example>

</Description>
</ManSection>

</Section>

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