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##          examples for curves.gap
##12-5-2004
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Read("/home/wdj/gapfiles/curves/curves.gap");


F:=GF(11);
R1:=PolynomialRing(F,["a"]);;
var1:=IndeterminatesOfPolynomialRing(R1);; a:=var1[1];;
b:=X(F,"b",var1); 
var2:=Concatenation(var1,[b]);
R2:=PolynomialRing(F,var2);
c:=X(F,"c",var2); 
var3:=Concatenation(var2,[c]);
R3:=PolynomialRing(F,var3);
f:=a^2*b+b^5;
h:=HomogenizeMultivariatePolynomial(f,R2,R3);
DehomogenizeMultivariatePolynomial(h,R3,R2);

R4:=PolynomialRing(F,[a,c]);
HomogenizeCoordinates([Z(11),Z(11)^2],R4,R3);
##  [ Z(11), Z(11)^0, Z(11)^2 ]
HomogenizeCoordinates([Z(11),Z(11)^2],R2,R3);
##  [ Z(11), Z(11)^2, Z(11)^0 ]

cdiv:=[ 1, -2, 1, -1 ];
sdiv:=[ Z(11)^2, Z(11)^3, Z(11)^7, Z(11) ];
crvP1:=AffineCurve(b,R2);
div:=DivisorOnAffineCurve(cdiv,sdiv,crvP1);
div.coeffs;
div.curve;
div.support;

div1:=DivisorOnAffineCurve([1,2,3,4],[Z(11)^2,Z(11)^3,Z(11)^7,Z(11)],crvP1);
DivisorDegree(div1);
div2:=DivisorOnAffineCurve([1,2,3,4],[Z(11),Z(11)^2,Z(11)^3,Z(11)^4],crvP1);
DivisorDegree(div2);
div3:=DivisorAddition(div1,div2);
DivisorDegree(div3);

DivisorGCD(div1,div2);

DivisorIsEffective(div1);
DivisorIsEffective(div2);

ndiv1:=DivisorNegate(div1);
zdiv:=DivisorAddition(div1,ndiv1);
DivisorIsZero(zdiv);
div1:=DivisorOnAffineCurve([1,2,3,4,5],
   [Z(11)^2,Z(11)^3,Z(11)^6,Z(11),Z(11)^9],crvP1);
div2:=DivisorOnAffineCurve([2,1,3,4,5],
   [Z(11)^3,Z(11)^2,Z(11)^6,Z(11),Z(11)^9],crvP1);
DivisorEqual(div1,div2);

Pts:=[[Z(11),-Z(11)]];
crv:=AffineCurve(a+b,R2);
OnCurve(Pts,crv);

f:=(a^4+Z(11)^6*a^3-a^2+Z(11)^7*a+Z(11)^0)/(a^4+Z(11)^4*a^3+Z(11)*a^2+Z(11)^7*a+Z(11));;
divf:=DivisorOfRationalFunction(f,R2);
denf:=DenominatorOfRationalFunction(f); RootsOfUPol(denf);
numf:=NumeratorOfRationalFunction(f); RootsOfUPol(numf);
B:=RiemannRochSpaceBasisP1(divf);
1/f in B;
#
A:=[[1,2],[3,4]];
h:=MoebiusTransformation(A,R1);
var:=IndeterminatesOfPolynomialRing(R1);
Value(h,var,Z(11)^0*[7]);

A:=Z(11)^0*[[1,2],[3,4]];
ActionMoebiusTransformationOnDivisorDefinedP1(A,div1);
A:=Z(11)^0*[[1,2],[1,4]];
ActionMoebiusTransformationOnDivisorDefinedP1(A,div1);

agp:=DivisorAutomorphismGroup(div1); time;
#<matrix group with 10 generators>
#8036
Size(agp);
#10
A:=Z(11)^0*[[1,2],[3,4]];
ActionMoebiusTransformationOnFunction(A,f,R2);


basis:=RiemannRochSpaceBasisP1(div1);
Length(basis);

div3:=DivisorOnAffineCurve([1,1,1,1,2],
   [Z(11)^2,Z(11)^3,Z(11)^6,Z(11),Z(11)^9],crvP1);

div4:=DivisorOnAffineCurve([1,2,3,-4],[Z(11)^2,Z(11)^3,Z(11)^6,Z(11)^9],crvP1);

div5:=DivisorOnAffineCurve([1,2,3,-5],
      [Z(11)^2,Z(11)^3,Z(11)^6,Z(11)^9],crvP1);

z:=List([1..11],i->1);; w:=div5.support;;
C2:=GeneralizedSrivastavaCode(pts,w,z,F);
G2:=GeneratorMat(C2);;PutStandardForm(G2);;Display(G2);

g:=a^4+a^3+1;
C3:=GoppaCode(g,pts);
G3:=GeneratorMat(C3);;PutStandardForm(G3);;Display(G3);
C1;C2;C3;
C2=C3;C1=C3;C1=C2;
CoveringRadius(C1);
CoveringRadius(C2);
CoveringRadius(C3);

div:=DivisorOnAffineCurve([1,1,1,1,1],
        [Z(11)^2,Z(11)^3,Z(11)^6,Z(11),Z(11)^9],crvP1);
G:=DivisorAutomorphismGroup(div); time;
IdGroup(G);
g:=Random(G);
rho:=MatrixRepresentationOnRiemannRochSpaceP1(g,div);
Display(rho);

div:=DivisorOnAffineCurve([1,1,1,1,1],
       [Z(11)^4,Z(11)^5,Z(11)^7,Z(11)^8,Z(11)^10],crvP1);
Gdiv:=DivisorAutomorphismGroup(div); time;
H:=Intersection(G,Gdiv);


q:=4;
F:=GF(q^2);
a:=X(F); 
R1:=PolynomialRing(F,[a]);;
var1:=IndeterminatesOfPolynomialRing(R1);;
b:=X(F); 
R2:=PolynomialRing(F,[a,b]);
var2:=IndeterminatesOfPolynomialRing(R2);;
#R1:=PolynomialRing(F,["a"]);;
#R2:=PolynomialRing(R1,["b"]);;
crv:=AffineCurve(b^q+b-a^(q+1),R2);

OnePointRiemannRochSpaceAtInfinity(-1,crv);
OnePointRiemannRochSpaceAtInfinity(0,crv);
OnePointRiemannRochSpaceAtInfinity(3,crv);
OnePointRiemannRochSpaceAtInfinity(4,crv);
OnePointRiemannRochSpaceAtInfinity(7,crv);
OnePointRiemannRochSpaceAtInfinity(8,crv);
OnePointRiemannRochSpaceAtInfinity(11,crv);
OnePointRiemannRochSpaceAtInfinity(12,crv);


alpha:=Random(F);
K:=CurveFiberFF2(alpha,crv);
beta:=Random(K);
Pts:=PointsOnAffineCurveFF(crv,F);;
Length(Pts);


##########################################################
##
## To use EvaluationCode, need below but then
## Pts:=Difference(Pts,[ [ 0*Z(2^4)^0, 0*Z(2)^0 ] ]);
##
q:=4;
F:=GF(q^2);
R:=PolynomialRing(F,2);;
vars:=IndeterminatesOfPolynomialRing(R);
x:=vars[1];
y:=vars[2];
crv:=AffineCurve(y^q+y-x^(q+1),R);
L:=[ Z(2)^0, x, x^2*y^-1 ];
Pts:=PointsOnAffineCurveFF(crv,F);;
P:=Difference(Pts,[[ 0*Z(2^4)^0, 0*Z(2)^0 ]]);;
C2:=EvaluationCode(P,L,R);
MinimumDistance(C2);



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q:=3;
R:=PolynomialRing(F,2);;
vars:=IndeterminatesOfPolynomialRing(R);
x:=vars[1];
y:=vars[2];
crv:=AffineCurve(y^q+y-x^(q+1),R);
Pts:=PointsOnAffineCurveFF(crv,F);;
P:=Difference(Pts,[[ 0*Z(3^2)^0, 0*Z(3)^0 ]]);;
L42:=[ Z(3)^0, y, x, x^2*y^-1 ];
C2:=EvaluationCode(P,L42,R);
MinimumDistance(C2);

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