# Extension of the Helleseth-Zinoviev result on the system of equations from Goethals codes and Kloosterman sums

## TL;DR

This paper revises and extends a previous result on the solutions of certain equations related to Goethals codes and Kloosterman sums, correcting an error for even parameters and providing new divisibility properties.

## Contribution

It corrects and completes a prior theorem on solution counts and divisibility of Kloosterman sums, offering new insights into their properties.

## Key findings

- Theorem 4 is corrected for even m.
- New divisibility modulo 12 of Kloosterman sums is established.
- Results differ from previous findings for specific Kloosterman sums.

## 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 give new divisibility modulo 12 of the Kloosterman sums deduced from Theorem 4, which is different from the result of the same authors for $ K(a^{4}+a^{3}) $ modulo 12.