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A.8.2 AG codes
The library brnoeth.lib provides an implementation of the Brill-Noether
algorithm for solving the Riemann-Roch problem and applications to Algebraic
Geometry codes. The procedures can be applied to plane (singular) curves
defined over a prime field of positive characteristic.
| LIB "brnoeth.lib";
ring s=2,(x,y),lp; // characteristic 2
poly f=x3y+y3+x; // the Klein quartic
list KLEIN=Adj_div(f); // compute the conductor
==> Computing affine singular points ...
==> Computing all points at infinity ...
==> Computing affine singular places ...
==> Computing singular places at infinity ...
==> Computing non-singular places at infinity ...
==> Adjunction divisor computed successfully
==>
==> The genus of the curve is 3
KLEIN=NSplaces(1..3,KLEIN); // computes places up to degree 3
==> Computing non-singular affine places of degree 1 ...
==> Computing non-singular affine places of degree 2 ...
==> Computing non-singular affine places of degree 3 ...
KLEIN=extcurve(3,KLEIN); // construct Klein quartic over F_8
==>
==> Total number of rational places : NrRatPl = 24
==>
KLEIN[3]; // display places (degree, number)
==> [1]:
==> 1,1
==> [2]:
==> 1,2
==> [3]:
==> 1,3
==> [4]:
==> 2,1
==> [5]:
==> 3,1
==> [6]:
==> 3,2
==> [7]:
==> 3,3
==> [8]:
==> 3,4
==> [9]:
==> 3,5
==> [10]:
==> 3,6
==> [11]:
==> 3,7
// We define a divisor G of degree 14=6*1+4*2:
intvec G=6,0,0,4,0,0,0,0,0,0,0; // 6 * place #1 + 4 * place #4
// We compute an evaluation code which evaluates at all rational places
// outside the support of G (place #4 is not rational)
intvec D=2..24;
// in D, the number i refers to the i-th element of the list POINTS in
// the ring KLEIN[1][5].
def RR=KLEIN[1][5];
setring RR; POINTS[1]; // the place in the support of G (not in supp(D))
==> [1]:
==> 0
==> [2]:
==> 1
==> [3]:
==> 0
setring s;
def RR=KLEIN[1][4];
==> // ** redefining RR (def RR=KLEIN[1][4];) ./examples/AG_codes.sing:18
setring RR;
matrix C=AGcode_L(G,D,KLEIN); // generator matrix for the evaluation AG code
==> Forms of degree 5 :
==> 21
==>
==> Vector basis successfully computed
==>
nrows(C);
==> 12
ncols(C);
==> 23
//
// We can also compute a generator matrix for the residual AG code
matrix CO=AGcode_Omega(G,D,KLEIN);
==> Forms of degree 5 :
==> 21
==>
==> Vector basis successfully computed
==>
//
// Preparation for decoding:
// We need a divisor of degree at least 6 whose support is disjoint with the
// support of D:
intvec F=6; // F = 6*point #1
// in F, the i-th entry refers to the i-th element of the list POINTS in
// the ring KLEIN[1][5]
list K=prepSV(G,D,F,KLEIN);
==> Forms of degree 5 :
==> 21
==>
==> Vector basis successfully computed
==>
==> Forms of degree 4 :
==> 15
==>
==> Vector basis successfully computed
==>
==> Forms of degree 4 :
==> 15
==>
==> Vector basis successfully computed
==>
K[size(K)][1]; // error-correcting capacity
==> 3
//
// Encoding and Decoding:
matrix word[1][11]; // a word of length 11 is encoded
word = 1,1,1,1,1,1,1,1,1,1,1;
def y=word*CO; // the code word (length: 23)
matrix disturb[1][23];
disturb[1,1]=1;
disturb[1,10]=a;
disturb[1,12]=1+a;
y=y+disturb; // disturb the code word (3 errors)
def yy=decodeSV(y,K); // error correction
yy-y; // display the error
==> _[1,1]=1
==> _[1,2]=0
==> _[1,3]=0
==> _[1,4]=0
==> _[1,5]=0
==> _[1,6]=0
==> _[1,7]=0
==> _[1,8]=0
==> _[1,9]=0
==> _[1,10]=(a)
==> _[1,11]=0
==> _[1,12]=(a+1)
==> _[1,13]=0
==> _[1,14]=0
==> _[1,15]=0
==> _[1,16]=0
==> _[1,17]=0
==> _[1,18]=0
==> _[1,19]=0
==> _[1,20]=0
==> _[1,21]=0
==> _[1,22]=0
==> _[1,23]=0
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