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In the paper of Tor Helleseth and Victor Zinoviev (Designs, Codes andCryptography, \textbf{17}, 269-288(1999)), the number of solutions of thesystem of equations from $ Z_{4} $-linear Goethals codes $ G_{4} $ wasdetermined 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 12of the Kloosterman sums deduced from Theorem 4, which is different from theresult of the same authors for $ K(a^{4}+a^{3}) $ modulo 12.

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