Over Z/2Z the polynomial X3+1 factors as X3+1 = (X+1)(X2+X+1). According to the BCH bound the minimal distance of the binary cyclic code generated by g = X2+X+1 is at least
1.
2.
3.
4.
No, if 
is the residue class of
X in Z/2Z[X]/(X3 + 1),
then is always a root of
g.
is the residue class of
X in Z/2Z[X]/(X3 + 1),
then is always a root of
g.
No, note that d is at least c + 1!
Indeed, if
is the residue class of
X in Z/2Z[X]/(X3 + 1),
then both and 2 are roots of
g.
is the residue class of
X in Z/2Z[X]/(X3 + 1),
then both and 2 are roots of
g.
This is impossible, because the length of the code is 3.