Extension of the Helleseth-Zinoviev result on the system of equations from Goethals codes and Kloosterman sums

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.