New Kloosterman sum identities from the Helleseth-Zinoviev result on $ Z_{4}$-linear Goethals codes

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 present a series of new Kloosterman sum identities deduced from Theorem 4. Moreover, we show that several previously established formulas on the Kloosterman sum identities can be rediscovered from Theorem 4 with much simpler proofs.