Infinite Families of Recursive Formulas Generating Power Moments of Kloosterman Sums: Symplectic Case

TL;DR

This paper develops infinite families of recursive formulas for power moments of Kloosterman sums using binary linear codes associated with symplectic groups over fields of characteristic two.

Contribution

It introduces new infinite families of codes linked to symplectic groups and derives recursive formulas for Kloosterman sum moments using these codes.

Findings

Recursive formulas for power moments of Kloosterman sums

Construction of binary linear codes from symplectic group cosets

Explicit expressions for exponential sums over double cosets

Abstract

In this paper, we construct two infinite families of binary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the symplectic group Sp(2n,q) Here q is a power of two. Then we obtain an infinite family of recursive formulas for the power moments of Kloosterman sums and those of 2-dimensional Kloosterman sums 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 exponential sums over those double cosets related to the evaluations of "Gauss sums" for the symplectic groups Sp(2n,q).