| #### D.10.1.7 AGcode_Omega Procedure from library `brnoeth.lib` (see brnoeth_lib). Usage: AGcode_Omega( G, D, EC ); G,D intvec, EC a list Return: a generator matrix for the residual AG code defined by the divisors G and D. Note: The procedure must be called within the ring EC[1][4], where EC is the output of `extcurve(d)` (or within the ring EC[1][2] if d=1). The entry i in the intvec D refers to the i-th rational place in EC[1][5] (i.e., to POINTS[i], etc., see extcurve). The intvec G represents a rational divisor (see BrillNoether for more details). The code computes the residues of a vector space basis of 1064#1064 at the rational places given by D. Warnings: G should satisfy 1063#1063, which is not checked by the algorithm. G and D should have disjoint supports (checked by the algorithm). Example:
    LIB "brnoeth.lib";
    int plevel=printlevel;
    printlevel=-1;
    ring s=2,(x,y),lp;
    list HC=Adj_div(x3+y2+y);
    ==> The genus of the curve is 1
    HC=NSplaces(1..2,HC);
    HC=extcurve(2,HC);
    ==> Total number of rational places : NrRatPl = 9
    def ER=HC[1][4];
    setring ER;
    intvec G=5;      // the rational divisor G = 5*HC[3][1]
    intvec D=2..9;   // D = sum of the rational places no. 2..9 over F_4
    // let us construct the corresponding residual AG code :
    matrix C=AGcode_Omega(G,D,HC);
    ==> Vector basis successfully computed
    // here is a linear code of type [8,3,>=5] over F_4
    print(C);
    ==> 1,    0,    (a),(a+1),1,1,0,0,
    ==> (a+1),(a),  1,  0,    1,0,1,0,
    ==> (a+1),(a+1),(a),(a),  1,0,0,1
    printlevel=plevel;
See also: AGcode_L; Adj_div; BrillNoether; extcurve.

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