Ternary codes associated with symplectic groups and power moments of Kloosterman sums with square arguments

TL;DR

This paper constructs ternary linear codes linked to symplectic groups and derives recursive formulas for power moments of Kloosterman sums with square arguments, connecting coding theory with exponential sum analysis.

Contribution

It introduces new ternary codes associated with Sp(2,q) and Sp(4,q) and establishes recursive formulas for Kloosterman sum moments using these codes.

Findings

Recursive formulas for Kloosterman sum moments derived

Explicit expressions of Gauss sums for symplectic groups used

Codes linked to symplectic groups constructed

Abstract

In this paper, we construct two ternary linear codes associated with the symplectic groups Sp(2,q) and Sp(4,q). Here q is a power of three. Then we obtain recursive formulas for the power moments of Kloosterman sums with square arguments and for the even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of "Gauss sums" for the symplectic groups Sp(2n,q).