//First a couple of preliminary functions

/*Gives the dimension of the p-ary code of the affine geometry
design of points and t-dimensional flats in AG_d(p)*/
 
grm:=func<d,t,p|
&+[&+[(-1)^k *Binomial(d,k)*Binomial(i-k*p+d-1,i-k*p):
k in [0..(i div p)]]
:i in [0..(d-t)*(p-1)]]>;

//Input p=prime,t and d 
pgrm:=func<d,t,p|&+[ grm(j,t,p) : j in [t..d]]>;

AGrank:=function(m,r,p,t);
/*Want rank_p(AG_{m,rr}(F_q)) where q=p^t*/

if t eq 1 then
 return grm(m,r,p);
else
q := p^t;
n := (q^m)-1;
np :=  n div (q-1);
 "q = ", q;


wts := [qwt(u,q) : u in [1..n]];
rr:=r;
k := ( m-rr)*(q-1);

we:=&meet[ {i : i in [1..q^m -2]
|wts[(p^j)*i mod  n] le k }: j in [0..t-1]];

print "dim(R_F_q/F_p(",k,",",m,")=", #(we)+1;
print "=", 
p, "-rank of AG_{", m, ",", rr, "}(F_", q, ")";

return #(we)+1;
end if;
end function;


PGrank:=function(m,r,p,t);
/*Want Rank_p(PG_{m,r}(F_q)) where q=p^t*/

if t eq 1 then
 return pgrm(m,r,p);
else
m:=m+1;rr:=r+1;
q := p^t;
n := (q^m)-1;
np :=  n div (q-1);
wts := [qwt(u,q) : u in [1..n]];
k := ( m-rr)*(q-1);
   we:=&meet[ {i : i in [1..q^m -2]
   |wts[(p^j)*i mod  n] le k }: j in [0..t-1]];
wee:={ i: i in we | (i mod (q-1)) eq 0};

 "dim(R_F_q/F_p^(q-1)(",k,",",m,")=",  #(wee)+1;
print "=",
 p, "-rank of PG_{", m-1, ",", rr-1, "}(F_", q, ")";
end if;

return #wee+1;

end function;