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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 present a series of newKloosterman sum identities deduced from Theorem 4. Moreover, we show thatseveral previously established formulas on the Kloosterman sum identities canbe rediscovered from Theorem 4 with much simpler proofs.

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