Prob #1
Encode113G := proc(x1) local x,n,a,b,c,d,eq1,eq2,eq3,eq4,i,S:
 if(nops(x1) < 5 or nops(x1) > q-5 or q < 11) then
  print("t=4 and 2t+1 <= n <= q-1"):
  RETURN(FAIL):
 fi: 
 x:=[op(x1),a,b,c,d]:
 n:=nops(x):
 eq1 := 0 = add(x[i], i=1..n):
 eq2 := 0 = add(i * x[i], i=1..n):
 eq3 := 0 = add(i^2 * x[i], i=1..n):
 eq4 := 0 = add(i^3 * x[i], i=1..n):
 S := msolve({eq1,eq2,eq3,eq4}, q):
 subs(S, [op(x1), a,b,c,d]):
end:


Sy113G:=proc(x,q) local i,r:
[seq(add(x[i]*i^(r-1),i=1..nops(x)) mod q,r=1..4)]:
end:


Decode113G:=proc(y,q) local S,eq,i,j,L,i1,a,b:
S:=Sy113G(y,q):
if S=[0$4] then
 RETURN(y):
fi:


eq:=(S[2]^2-S[1]*S[3])*i^2+(S[1]*S[4]-S[2]*S[3])*i+ (S[3]^2-S[2]*S[4]) mod q:


if eq=0 then
 i:=S[2]/S[1] mod q:
 a:=S[1]:
  RETURN(y-[0$(i-1),a,0$(nops(y)-i)] mod q):
fi:


#L is the list of roots of eq (mod q)
L:=[]:
for i1 from 0 to q-1 do
 if subs(i=i1,eq) mod q=0 then
  L:=[op(L),i1]:
 fi:
od:
i:=L[1]:
j:=L[2]:


b:=(i*S[1]-S[2])/(i-j) mod q:
a:=S[1]-b mod q:
y-[0$(i-1),a,0$(j-i-1),b,0$(nops(y)-j)] mod q:
end:


#Test code
q:=11: x:=RV(q,6); x1:=Encode113G(x,q);
                    x := [2, 5, 5, 6, 0, 3]


             x1 := [2, 5, 5, 6, 0, 3, 4, 3, 6, 10]


p:=rand(1..nops(x1)): p1:=p(): e1:=[0$p1-1,1,0$nops(x1)-p1]:
while((p2:=p())=p1)do:od: p2: e2:=[0$p2-1,1,0$nops(x1)-p2]: e:=e1+e2;
y:=x1+e1+e2 mod q;
S:=Sy113G(y,q);


              e := [0, 1, 0, 0, 0, 0, 0, 1, 0, 0]


              y := [2, 6, 5, 6, 0, 3, 4, 4, 6, 10]


                       S := [2, 10, 2, 3]


Decode113G(y,q); evalb(%=x1);
                [2, 5, 5, 6, 0, 3, 4, 3, 6, 10]


                              true


q:=13: x:=RV(q,5); x1:=Encode113G(x,q);
                     x := [1, 1, 3, 0, 12]


               x1 := [1, 1, 3, 0, 12, 9, 2, 5, 6]


p:=rand(1..nops(x1)): p1:=p(): e1:=[0$p1-1,1,0$nops(x1)-p1]:
while((p2:=p())=p1)do:od: p2: e2:=[0$p2-1,1,0$nops(x1)-p2]: e:=e1+e2;
y:=x1+e1+e2 mod q;
S:=Sy113G(y,q);


                e := [0, 1, 0, 0, 1, 0, 0, 0, 0]


                y := [1, 2, 3, 0, 0, 9, 2, 5, 6]


                       S := [2, 7, 3, 3]


Decode113G(y,q); evalb(%=x1);
                  [1, 1, 3, 0, 12, 9, 2, 5, 6]


                              true


q:=13: x:=RV(q,9); x1:=Encode113G(x,q);
               x := [5, 9, 4, 4, 0, 12, 1, 7, 9]


                   "t=4 and 2t+1 <= n <= q-1"


                           x1 := FAIL


q:=17: x:=RV(q,9); x1:=Encode113G(x,q);
               x := [1, 4, 0, 2, 1, 1, 11, 3, 3]


        x1 := [1, 4, 0, 2, 1, 1, 11, 3, 3, 10, 13, 0, 2]


p:=rand(1..nops(x1)): p1:=p(): e1:=[0$p1-1,1,0$nops(x1)-p1]:
while((p2:=p())=p1)do:od: p2: e2:=[0$p2-1,1,0$nops(x1)-p2]: e:=e1+e2;
y:=x1+e1+e2 mod q;
S:=Sy113G(y,q);


          e := [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0]


        y := [1, 4, 0, 2, 2, 1, 11, 3, 3, 10, 14, 0, 2]


                      S := [2, 16, 10, 11]


Decode113G(y,q); evalb(%=x1);
           [1, 4, 0, 2, 1, 1, 11, 3, 3, 10, 13, 0, 2]


                              true