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

## TL;DR

This paper corrects and extends a key theorem on solutions of equations related to $Z_4$-linear Goethals codes, deriving new Kloosterman sum identities and simplifying existing formulas through these insights.

## Contribution

It corrects a previous theorem for even $m$, completes its statement, and introduces new Kloosterman sum identities with simpler proofs.

## Key findings

- Corrected Theorem 4 for even $m$
- Derived new Kloosterman sum identities
- Simplified proofs of existing formulas

## Abstract

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.