An Infinite Family of Recursive Formulas Generating Power Moments of Kloosterman Sums with Trace One Arguments: O(2n+1,2^r) Case

In this paper, we construct an infinite family of binary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the orthogonal group O(2n+1,q). Here q is a power of two. Then we obtain an infinite family of recursive formulas generating the odd power moments of Kloosterman sums with trace one arguments 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).

For this, we have the Weil bound The Kloosterman sum was introduced in 1926( [9]) to give an estimate for the Fourier coefficients of modular forms. For each nonnegative integer h, by M K(ψ) h we will denote the h-th moment of the Kloosterman sum K(ψ; a). Namely, it is given by If ψ = λ is the canonical additive character of F q , then M K(λ) h will be simply denoted by M K h .
Explicit computations on power moments of Kloosterman sums were begun with the paper [15] of Salié in 1931, where he showed, for any odd prime q, Here M 0 = 0, and, for h ∈ Z >0 , For q = p odd prime, Salié obtained M K 1 , M K 2 , M K 3 , M K 4 in [15] by determining M 1 , M 2 , M 3 . On the other hand, M K 5 can be expressed in terms of the p-th eigenvalue for a weight 3 newform on Γ 0 (15)(cf. [11], [14]). M K 6 can be expressed in terms of the p-th eigenvalue for a weight 4 newform on Γ 0 (6)(cf. [3]). Also, based on numerical evidence, in [2] Evans was led to propose a conjecture which expresses M K 7 in terms of Hecke eigenvalues for a weight 3 newform on Γ 0 (525) with quartic nebentypus of conductor 105. From now on, let us assume that q = 2 r . Carlitz [1] evaluated M K h for h ≤ 4. Recently, Moisio was able to find explicit expressions of M K h , for h ≤ 10 (cf. [13]). This was done, via Pless power moment identity, by connecting moments of Kloosterman sums and the frequencies of weights in the binary Zetterberg code of length q + 1, which were known by the work of Schoof and Vlugt in [16].
In order to describe our results, we introduce two incomplete power moments of Kloosterman sums, namely, the one with the sum over all a in F * q with tr a=0 and the other with the sum over all a in F * q with tr a = 1. For every nonnegative integer h, and ψ as before, we define which will be respectively called the h-th moment of Kloosterman sums with "trace zero arguments" and those with "trace one arguments". Then, clearly we have If ψ = λ is the canonical additive character of F q , then T 0 K(λ) h and T 1 K(λ) h will be respectively denoted by T 0 K h and T 1 K h , for brevity.
In [7], we obtained a recursive formula generating the odd power moments of Kloosterman sums with trace one arguments. This was expressed in terms of the frequencies of weights in the binary linear codes C(O(3, q)) and C(Sp(2, q)), respectively associated with the orthogonal group O(3, q) and the symplectic group Sp(2, q).
In this paper, we will show the main Theorem 1.1 giving an infinite family of recursive formulas generating the odd power moments of Kloosterman sums with trace one arguments. To do that, we construct binary linear codes C(DC(n, q)), associated with the double cosets DC(n, q)=P σ n−1 P , for the maximal parabolic subgroup P =P (2n + 1, q) of the orthogonal group O(2n + 1, q), and express those power moments in terms of the frequencies of weights in the codes C(DC(n, q)) and C( DC(n, q)). Here C( DC(n, q)) is a binary linear code constructed similarly from certain double cosets DC(n, q) in the sympletic group Sp(2n, q). 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 group O(2n + 1, q) [8], we can express the weight of each codeword in the dual of the codes C(DC(n, q)) in terms of Kloosterman sums. Then our formulas will follow immediately from the Pless power moment identity. Theorem 1.1 in the following(cf. (1.6)-(1.8)) is the main result of this paper.
Henceforth, we agree that the binomial coefficient b a = 0 if a > b or a < 0. To simplify notations, we introduce the following ones which will be used throughout this paper at various places.
Theorem 1.1. Let q = 2 r . Assume that n is any odd integer≥ 3, with all q, or n=1, with q ≥ 8. Then, in the notations of (1.4) and (1.5), we have the following. For h=1,3,5,· · · , 6) where N (n, q) = |DC(n, q)| = A(n, q)B(n, q), D j (n, q) = C j (n, q) − C j (n, q), with {C j (n, q)} N (n,q) j=0 respectively the weight distributions of the binary linear codes C(DC(n, q)) and C( DC(n, q)) given by: for j = 0, · · · , N (n, q), (1.8) Here the first sum in (1.6) is 0 if h = 1 and the unspecified sums in (1.7) and (1.8) are over all the sets of nonnegative integers {ν β } β∈Fq satisfying β∈Fq ν β = j and β∈Fq ν β β = 0. In addition, S(h, t) is the Stirling number of the second kind defined by For more details about this section, one is referred to the paper [8]. Throughout this paper, the following notations will be used: Let θ be the nondegenerate quadratic form on the vector space F (2n+1)×1 q of all (2n + 1) × 1 column vectors over F q , given by in GL(2n + 1, q) satisfying the relations: As is well known, there is an isomorphism of groups In particular, for any w ∈ O(2n + 1, q), Let P = P (2n + 1, q) be the maximal parabolic subgroup of O(2n + 1, q) given by The symplectic group Sp(2n, q) over the field F q is defined as: Let P ′ = P ′ (2n, q) be the maximal parabolic subgroup of Sp(2n, q) defined by: Then, with respect to P ′ = P ′ (2n, q), the Bruhat decomposition of Sp(2n, q) is given by Put, for each r with 0 ≤ r ≤ n, Expressing as the disjoint union of right cosets of maximal parabolic subgroups, the double cosets P σ r P and P ′ σ ′ r P ′ can be written respectively as The order of the general linear group GL(n, q) is given by For integers n,r with 0 ≤ r ≤ n, the q-binomial coefficients are defined as: The following results follow either from [8] or from [4] plus the observation that under the isomorphism ι in (2.1) P , A r , σ r are respectively mapped onto P, ′ A ′ r , σ ′ r : (2.5) In particular, with DC(n, q) = P (2n + 1, q)σ n−1 P (2n + 1, q), The following notations will be employed throughout this paper.
Then any nontrivial additive character ψ of F q is given by ψ(x) = λ(ax), for a unique a ∈ F * q .
For any nontrivial additive character ψ of F q and a ∈ F * q , the Kloosterman sum K GL(t,q) (ψ; a) for GL(t, q) is defined as Notice that, for t = 1, K GL(1,q) (ψ; a) denotes the Kloosterman sum K(ψ; a). In [4], it is shown that K GL(t,q) (ψ; a) satisfies the following recursive relation: for integers t ≥ 2, a ∈ F * q , where we understand that K GL(0,q) (ψ, a) = 1.
Theorem 3.1. Let ψ be any nontrivial additive character of F q . Then in the notation of (1.4), we have  The next corollary follows from Theorem 3.1, Proposition 3.2 and a simple change of variables. Corollary 3.3. Let λ be the canonical additive character of F q , and let a ∈ F * q . Then we have (3.5) w∈DC(n,q) λ(aT rw) = λ(a)A(n, q)K(λ; a), f or n = 1, 3, 5, · · · (cf. (1.4)).
Then it is easy to see that w∈P σr P λ(aT rw).
Here we will construct one infinite family of binary linear codes C(DC(n, q)) of length N (n, q) for all positive odd integers n and all q, associated with the double cosets DC(n, q).
Let F + 2 , F + q denote the additive groups of the fields F 2 , F q , respectively. Then we have the following exact sequence of groups: where the first map is the inclusion and the second one is the Artin-Schreier operator in characteristic two given by Θ(x) = x 2 + x. So Θ(F q ) = {α 2 + α|α ∈ F q }, and [F + q : Θ(F q )] = 2.  Theorem 4.3. The map F q → C(DC(n, q)) ⊥ (a → c(a)) is an F 2 -linear isomorphism for each odd integer n ≥ 1and all q, except for n = 1 and q = 4.

Power moments of Kloosterman sums with trace one arguments
Here we will be able to find, via Pless power moment identity, an infinite family of recursive formulas generating the odd power moments of Kloosterman sums with trace one arguments over all F q in terms of the frequencies of weights in C(DC(n, q)) and C( DC(n, q)), respectively.
Theorem 5.1. (Pless power moment identity, [12]) Let B be an q-ary [n, k] code, and let B i (resp. B ⊥ i ) denote the number of codewords of weight i in B(resp. in B ⊥ ). Then, for h = 0, 1, 2, · · · , where S(h, t) is the Stirling number of the second kind defined in (1.9).
Proof. Under the replacements ν β → n(β) − ν β , for each β ∈ F q , the first equation in The formula appearing in the next theorem and stated in (1.7) follows from the formula in (5.3), using the explicit value of n(β) in (3.9).
Here the sum is over all the sets of nonnegative integers {ν β } β∈Fq satisfying β∈Fq ν β = j and β∈Fq ν β β = 0. In addition, S(h, t) is the Stirling number of the second kind as in (1.9).
From now on, we will assume that n is any odd integer ≥ 3, with all q, or n = 1, with q ≥ 8. Under these assumptions, each codeword in C(DC(n, q)) ⊥ can be written as c(a), for a unique a ∈ F q (cf. Theorem 4.3, (4.3)) and Theorem 5.6 in the above can be applied.  On the other hand, the right hand side of the identity in (5.1) is given by: (5.7) q min{N (n,q),h} j=0 (−1) j C j (n, q) h t=j t!S(h, t)2 −t N (n, q) − j N (n, q) − t .