New Kloosterman sum identities from the Helleseth-Zinoviev result on $ Z_{4}$-linear Goethals codes
This work addresses a specific error in coding theory, leading to incremental advances in mathematical identities for cryptography and number theory.
The authors corrected an error in a prior theorem on Z4-linear Goethals codes for even m, and derived new Kloosterman sum identities from it, while simplifying proofs of existing formulas.
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.