  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.
