Codes Associated with Orthogonal Groups and Power Moments of Kloosterman Sums

TL;DR

This paper constructs specific binary linear codes linked to orthogonal groups over finite fields and derives recursive formulas for power moments of Kloosterman sums, improving computational efficiency over previous methods.

Contribution

It introduces new codes associated with orthogonal groups and provides more efficient recursive formulas for Kloosterman sum moments compared to prior approaches.

Findings

Recursive formulas for Kloosterman sum moments derived

Codes constructed for orthogonal groups over fields of characteristic two

Enhanced computational efficiency over previous methods

Abstract

In this paper, we construct three binary linear codes $C(SO^{-}(2,q))$, $C(O^{-}(2,q))$, $C(SO^{-}(4,q))$, respectively associated with the orthogonal groups $SO^{-}(2,q)$, $O^{-}(2,q)$, $SO^{-}(4,q)$, with $q$ powers of two. Then we obtain recursive formulas for the power moments of Kloosterman and 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 Gauss sums for the orthogonal groups. We emphasize that, when the recursive formulas for the power moments of Kloosterman sums are compared, the present one is computationally more effective than the previous one constructed from the special linear group $SL(2,q)$. We illustrate our results with some examples.