Codes associated with orthogonal groups and power moments of Kloosterman sums

zbMATH Open1296.11163arXiv0808.3003MaRDI QIDQ2869650

Dae San Kim

Publication date: 3 January 2014

Published in: Journal of Combinatorics and Number Theory (Search for Journal in Brave)

Abstract: In this paper, we construct three binary linear codes $C(S{O}^{-}(2,q))$, $C(O^{-}(2,q))$, $C(S{O}^{-}(4,q))$, respectively associated with the orthogonal groups $S{O}^{-}(2,q)$, $O^{-}(2,q)$, $S{O}^{-}(4,q)$, 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.

Full work available at URL: https://arxiv.org/abs/0808.3003

zbMATH Keywords

orthogonal groupweight distributionKloosterman sum

Mathematics Subject Classification ID

Linear algebraic groups over finite fields (20G40) Linear codes (general theory) (94B05) Exponential sums (11T23)

Related Items (3)

Codes Associated with and Power Moments of Kloosterman Sums ⋮ Title not available (Why is that?) ⋮ CODES ASSOCIATED WITH Sp(4,q) AND EVEN-POWER MOMENTS OF KLOOSTERMAN SUMS

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