  In the paper of Tor Helleseth and Victor Zinoviev (Designs, Codes and
Cryptography, \textbf{17}, 269-288(1999)), the number of solutions of the
system of equations from $ Z_{4} $-linear Goethals codes $ G_{4} $ was
determined and stated in Theorem 4. We found that Theorem 4 is wrong for $ m $
even. In this note, we complete Theorem 4, and give new divisibility modulo 12
of the Kloosterman sums deduced from Theorem 4, which is different from the
result of the same authors for $ K(a^{4}+a^{3}) $ modulo 12.
