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-- The construction of the polynomial system of {6,22,17} --
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We give the method constructing the polynomial system of {6,22,17} 
using a computer algebra system Asir.

Notation: 
The brackets including number, e.g. [0], are prompts of Asir.
Capital letters, e.g. X,Y,G, are variables of Asir.
";" represents the command terminator with screen output.
"$" represents the commnad terminator without screen output.
"//" represents comments. You must not input these lines.
"-->" represents outputs on Asir.

Before starting Asir, we must copy the file "delta.rr" to the folder
"bin" in the folder installed Asir.

[0] load("./delta.rr");
// The required functions are loaded by this command.

[14] X=t^6+a2*t^4+a3*t^3+a4*t^2+a5*t+a6$
[15] Y=t^22+b1*t^21+b2*t^20+b3*t^19+b4*t^18+b5*t^17+b6*t^16+b7*t^15+b8*t^14
+b9*t^13+b10*t^12+b11*t^11+b12*t^10+b13*t^9+b14*t^8+b15*t^7+b16*t^6+b17*t^5
+b18*t^4+b19*t^3+b20*t^2+b21*t+b22$
[16] G=y^3-x^11+(c01+c11*x+c21*x^2+c31*x^3+c41*x^4+c51*x^5+c61*x^6+c71*x^7)*y
+c00+c10*x+c20*x^2+c30*x^3+c40*x^4+c50*x^5+c60*x^6+c70*x^7+c80*x^8+c90*x^9$
//We input polynomials. X corresponds to $x$, Y to $y$ and G to $g_2$
//on page 6--7 in our preprint. 

[17] CL=coeflist(subst(G,x,X,y,Y),t,17)$
//subst(G,x,X,y,Y) substitutes X to x, Y to y for Polynomial G. 
//coeflist(P,t,17) outputs the list of coefficients of terms in polynomial P 
//with t-degree greater than 17.

[18] PL=eliminate(CL,b1)$
//We get the polynomial system PL eliminated the variable b1 
//using the linear polynomial w.r.t. b1.

[19] PL=eliminate_list(PL,[c71,c61,c90,c51,c80,c41,c70,c31,c60,c21,c50,
c11,c40,c01,c30])$
//We eliminate the variables c71,c61,c90,c51,c80,c41,c70,c31,c60,c21,c50,
//c11,c40,c01 and c30 successively.
//This function was made in order to save used memory.

//We execute the following commands to eliminate variables b3,b4,b5,b6,b7,b9,
//b10,b11,b13,b15,b16,b17,b19,b21 and b22.
 PL=eliminate(PL,b3)$
 PL=eliminate(PL,b4)$
 PL=eliminate(PL,b5)$
 PL=eliminate(PL,b6)$
 PL=eliminate(PL,b7)$
 PL=eliminate(PL,b9)$
 PL=eliminate(PL,b10)$
 PL=eliminate(PL,b11)$
 PL=eliminate(PL,b13)$
 PL=eliminate(PL,b15)$
 PL=eliminate(PL,b16)$
 PL=eliminate(PL,b17)$
 PL=eliminate(PL,b19)$
 PL=eliminate(PL,b21)$
 PL=eliminate(PL,b22)$
 vars(PL);
-->[a2,a3,a4,a5,a6,b2,b8,b12,b14,b18,b20]
//As a result, we get the polynomial system with 11 variables and 18 polys.

[36] PL=list_ptozp(list_subst_list(PL,[[a2,a],[a3,b],[a4,c],[a5,d],[a6,e],
[b2,f],[b8,k],[b12,o],[b14,q],[b18,u],[b20,w]]))$
//We replace a2 with a, a3 with b, a4 with c, a5 with d, a6 with e, 
//b2 with f, b8 with k, b12 with o, b14 with q, b18 with u, b20 with w.

[37] CP=car(reverse(PL))$
//We get the check polynomial CP of {6,22,17}, which is the coefficient 
//of term with t-degree=17 in G.

[38] bsave(CP,"./6-22-17_chkpoly");
//We save the polynomial C as a binary data at the file "6-22-17_chkpoly".

[39] PL=reverse(cdr(reverse(PL)))$
//We omit the last element CP from the list PL.
//We get the polynomial system of {6,22,17}.

[40] bsave(PL,"./6-22-17_poly_17");
//We save the polynomial system PL as a binary data at the file 
//"6-22-17_poly_17".

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