Hyperplane sections of fermat varieties in P 3 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 m_i(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.