Codes Associated with and Power Moments of Kloosterman Sums

Abstract

We shall construct three binary linear codes C.SO C .2; q//, C.O C .2; q//, and C.SO C .4; q//, respectively associated with the orthogonal groups SO C .2; q/, O C .2; q/, and SO C .4; q/, with q a power 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 the Pless power moment identity and by utilizing the explicit expressions of Gauss sums for the orthogonal groups.

Infinite families of recursive formulas generating power moments of Kloosterman sums: O −(2n,2r) case

“…Proposition 5 ( [9]): For n = 2 s (s ∈ Z ≥0 ), and ψ a nontrivial additive character of F q , K(ψ; a n ) = K(ψ; a).…”

“…The formulas appearing in the next theorem and stated in (10) and (14) follow by applying the formula in (51) to each C(DC ± i (n, q)), using the explicit values of N DC ± i (n,q) (β) in (39) and (40), and taking Theorem 18 into consideration.…”

“…In particular, when n = 2, this gives a recursive formula for the power moments of Kloosterman sums. Also, in order to get recursive formulas for the power moments of Kloosterman and 2-dimensional Kloosterman sums, we constructed in [10] three binary linear codes C(SO + (2, q)), C(O + (2, q)), C(SO + (4, q)), respectively associated with SO + (2, q), O + (2, q), SO + (4, q), and in [11] three binary linear codes C(SO − (2, q)), C(O − (2, q)), C(SO − (4, q)), respectively associated with SO − (2, q), O − (2, q), SO − (4, q). All of these were done via Pless power moment identity and by utilizing our previous results on explicit expressions of Gauss sums for the stated finite classical groups.…”

“…To do that, we construct four infinite families of binary linear codes C(DC + 1 (n, q)) (n = 2, 4, · · · ), C(DC − 1 (n, q)) (n = 1, 3, · · · ), both associated with P + σ + n−1 P + , and C(DC + 2 (n, q)) (n = 2, 4, · · · ), C(DC − 2 (n, q)) (n = 3, 5, · · · ), both associated with P + σ + n−2 P + , with respect to the maximal parabolic subgroup P + = P + (2n, q) of the orthogonal group O + (2n, q), and express those power moments in terms of the frequencies of weights in each code. Then, thanks to our previous results on the explicit expressions of exponential sums over those double cosets related to the evaluations of "Gauss sums" for the orthogonal groups O + (2n, q) [9], we can express the weight of each codeword in the duals of the codes in terms of Kloosterman or 2-dimensional Kloosterman sums. Then our formulas will follow immediately from the Pless power moment identity.…”

Quadratic Form Gauss Sums

“…In addition, Gauss sums have been generalized over various algebraic structures, including various types of groups [46], [64], [76], [77] [82] ; rings [36], [105], [119]; and fields [14], [57], [84], [90]. Regardless of the generalization, the Gauss sum can be used for additive or multiplicative problems.…”