Article ID: MTJPAM-D-19-00002

Title: RECURSIVE FORMULAS GENERATING POWER MOMENTS OF TERNARY KLOOSTERMAN SUMS WITH TRACE NONZERO SQUARE ARGUMENTS: O(2n + 1, 3r) CASE


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-19-00002; Volume 1 / Issue 2 / Year 2019, Pages 21-41

Document Type: Research Paper

Author(s): Dae San Kim a

aDepartment of Mathematics, Sogang University, Seoul 121-742, Korea

Received: 4 July 2019, Accepted: 4 December 2019, Available online: 28 December 2019.

Corresponding Author: Dae San Kim (Email address: dskim@sogang.ac.kr)

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Abstract

In this paper, we construct four infinite families of ternary linear codes associated with double cosets in O(2n + 1, q) with respect to certain maximal parabolic subgroup of the special orthogonal group SO(2n + 1, q). Here q is a power of three. Then we obtain two infinite families of recursive formulas, the one generating the power moments of Kloosterman sums with “trace nonzero square arguments” and the other generating the even power moments of those. Both of these families are expressed in terms of the frequencies of weights in the codes associated with those double cosets in O(2n + 1, q) and in the codes associated with similar double cosets in the symplectic group Sp(2n, q). 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 orthogonal group O(2n + 1, q).

Keywords: power moment, Kloosterman sum, trace nonzero square argument, orthogonal group, symplectic group, double cosets, maximal parabolic subgroup, Pless power moment identity, weight distribution, Gauss sum

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