Ternary Codes Associated with O^-(2n,q) and Power Moments of Kloosterman Sums with Square Arguments

TL;DR

This paper constructs ternary linear codes linked to orthogonal groups over finite fields of characteristic three and derives recursive formulas for power moments of Kloosterman sums with square arguments using these codes.

Contribution

It introduces new ternary codes associated with orthogonal groups and provides recursive formulas for Kloosterman sum moments based on these codes.

Findings

Recursive formulas for power moments of Kloosterman sums derived.

Explicit expressions of Gauss sums for orthogonal groups utilized.

Codes constructed for groups O^-(2,q), SO^-(2,q), and SO^-(4,q).

Abstract

In this paper, we construct three ternary linear codes associated with the orthogonal group O^-(2,q) and the special orthogonal groups SO^-(2,q) and SO^-(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 orthogonal and special orthogonal groups O^-(2n,q) and SO^-(2n,q).