Hyperplane sections of fermat varieties in P³ in char. 2 and some applications to cyclic codes

We consider the cyclic codes C₃⁽ᵗ⁾ of length 2³−1 generated by m₁(X)mnt(X) where mᵢ(X) is the minimal polynomial of a primitive element of GF(2³), and ask when these codes have minimum distance ≥ 5. Words of weight ≤ 4 in these codes are directly related to rational points in GF(2³) on the curves corresponding to the polynomials Xᵗ+Yᵗ+Zᵗ+(X+Y+Z)ᵗ over the algebraic closure of GF(2). Study of the singularities and absolutely irreducible components of these polynomials leads to results on the minimum distance of the codes.