Recursive formulas generating power moments of multi-dimensional Kloosterman sums and $m$-multiple power moments of Kloosterman sums

TL;DR

This paper develops recursive formulas for calculating power moments of multi-dimensional and m-multiple Kloosterman sums over finite fields of characteristic two, simplifying previous methods using coding theory techniques.

Contribution

It introduces new recursive formulas for Kloosterman sum moments derived from binary linear codes, improving upon earlier approaches with more straightforward recursive relations.

Findings

Derived recursive formulas for multi-dimensional Kloosterman sum moments

Established connections between Kloosterman sums and binary linear codes

Provided simpler recursive formulas compared to previous methods

Abstract

In this paper, we construct two binary linear codes associated with multi-dimensional and $m -$multiple power Kloosterman sums (for any fixed $m$) over the finite field $\mathbb{F}_{q}$. Here $q$ is a power of two. The former codes are dual to a subcode of the binary hyper-Kloosterman code. Then we obtain two recursive formulas for the power moments of multi-dimensional Kloosterman sums and for the $m$-multiple power moments of Kloosterman sums in terms of the frequencies of weights in the respective codes. This is done via Pless power moment identity and yields, in the case of power moments of multi-dimensional Kloosterman sums, much simpler recursive formulas than those associated with finite special linear groups obtained previously.

Equations

$$K_m(\psi ;a)=\sum _{{\alpha }_1,\cdots ,{\alpha }_m\in {\mathbb{F}}_q^{\ast }}\psi ({\alpha }_1+\cdots +{\alpha }_m+a{\alpha }_1^{-1}\cdots {\alpha }_m^{-1})(a\in {\mathbb{F}}_q^{\ast }).$$

$$|K_m(\psi ;a)|\le (m+1)q^{\frac{m}{2}}.$$

$$M{K}_m(\psi {)}^h=\sum _{a\in {\mathbb{F}}_q^{\ast }}{K}_m(\psi ;a{)}^h.$$

$$M{K}^h=q^2{M}_{h-1}-(q-1{)}^{h-1}+2(-1{)}^{h-1}(h\ge 1).$$

$$M_h=|\{({\alpha }_1,\cdots ,{\alpha }_h)\in ({\mathbb{F}}_q^{\ast }{)}^h|\sum _{j=1}^h{\alpha }_i=1=\sum _{j=1}^h{\alpha }_i^{-1}\}|.$$

$$\begin{array}{rl}& M{K}_{n-1}^h=\sum _{l=0}^{h-1}(-1{)}^{h+l+1}(\genfrac{}{}{0ex}{}{h}{l})(q-1{)}^{(n-1)(h-l)}M{K}_{n-1}^l\\ & +q\sum _{j=0}^{min\{(q-1{)}^{n-1},h\}}(-1{)}^{h+j}{C}_{n-1,j}\sum _{t=j}^{h}t!S(h,t)2^{h-t}(\genfrac{}{}{0ex}{}{(q-1{)}^{n-1}-j}{(q-1{)}^{n-1}-t}).\end{array}$$

$$S(h,t)=\frac{1}{t!}\sum _{j=0}^t(-1{)}^{t-j}(\genfrac{}{}{0ex}{}{t}{j})j^h.$$

$$C_{n-1,j}=\sum \prod _{\beta \in {\mathbb{F}}_q}\left(\genfrac{}{}{0ex}{}{\delta (n-1,q;\beta )}{{\nu }_{\beta }}\right),$$

$$\sum _{\beta \in {\mathbb{F}}_q}{\nu }_{\beta }=j,and\sum _{\beta \in {\mathbb{F}}_q}{\nu }_{\beta }\beta =0,and$$

$$\begin{array}{rl}\delta (n-1,q;\beta )& =|\{({\alpha }_1,\cdots ,{\alpha }_{n-1})\in ({\mathbb{F}}_q^{\ast }{)}^{n-1}|\\ & {\alpha }_1+\cdots +{\alpha }_{n-1}+{\alpha }_1^{-1}\cdots {\alpha }_{n-1}^{-1}=\beta \}|\\ & =\{\begin{array}{ll}{q}^{-1}\{(q-1{)}^{n-1}+1\},& \text{if β=0,}\\ K_{n-2}(\lambda ;{\beta }^{-1})+q^{-1}\{(q-1{)}^{n-1}+1\},& \text{if β∈Fq∗}.\end{array}\end{array}$$

$$\begin{array}{rl}& M{K}^{mh}=\sum _{l=0}^{h-1}(-1{)}^{h+l+1}(\genfrac{}{}{0ex}{}{h}{l})(q-1{)}^{m(h-l)}M{K}^{ml}\\ & +q\sum _{j=0}^{min\{(q-1{)}^m,h\}}(-1{)}^{h+j}{D}_{m,j}\sum _{t=j}^{h}t!S(h,t)2^{h-t}(\genfrac{}{}{0ex}{}{(q-1{)}^m-j}{(q-1{)}^m-t}).\end{array}$$

$$D_{m,j}=\sum \prod _{\beta \in {\mathbb{F}}_q}\left(\genfrac{}{}{0ex}{}{\sigma (m,q;\beta )}{{\nu }_{\beta }}\right),$$

$$\begin{array}{rl}\sigma (m,q;\beta )& =|\{({\alpha }_1,\cdots ,{\alpha }_m)\in ({\mathbb{F}}_q^{\ast }{)}^m|\\ & {\alpha }_1+\cdots +{\alpha }_m+{\alpha }_1^{-1}+\cdots +{\alpha }_m^{-1}=\beta \}|\\ & =\sum \lambda ({\alpha }_1+\cdots +{\alpha }_m)+q^{-1}\{(q-1{)}^m+(-1{)}^{m+1}\},\end{array}$$

$$\begin{array}{rl}& M{K}^h=\sum _{l=0}^{h-1}(-1{)}^{h+l+1}(\genfrac{}{}{0ex}{}{h}{l})(q-1{)}^{h-l}M{K}^l\\ & +q\sum _{j=0}^{min\{(q-1),h\}}(-1{)}^{h+j}{C}_{1,j}\sum _{t=j}^{h}t!S(h,t)2^{h-t}(\genfrac{}{}{0ex}{}{q-1-j}{q-1-t}),\end{array}$$

$$C_{1,j}=\sum \left(\genfrac{}{}{0ex}{}{1}{{\nu }_0}\right)\prod _{tr({\beta }^{-1})=0}\left(\genfrac{}{}{0ex}{}{2}{{\nu }_{\beta }}\right)(j=0,\cdots ,N_1).$$

$$\begin{array}{rl}& M{K}_3^h=\sum _{l=0}^{h-1}(-1{)}^{h+l+1}(\genfrac{}{}{0ex}{}{h}{l})(q-1{)}^{3(h-l)}M{K}_3^l\\ & +q\sum _{j=0}^{min\{(q-1{)}^3,h\}}(-1{)}^{h+j}{C}_{3,j}\sum _{t=j}^{h}t!S(h,t)2^{h-t}(\genfrac{}{}{0ex}{}{(q-1{)}^3-j}{(q-1{)}^3-t}),\end{array}$$

$$C_{3,j}=\sum \left(\genfrac{}{}{0ex}{}{m_0}{{\nu }_0}\right)\prod _{\begin{array}{c}|t|<2\sqrt{q}\\ t\equiv -1(4)\end{array}}\prod _{K(\lambda ;{\beta }^{-1})=t}\left(\genfrac{}{}{0ex}{}{m_t}{{\nu }_{\beta }}\right).$$

$$m_0=q^2-3q+3,$$

$$m_t=t^2+q^2-4q+3,$$

$$\begin{array}{rl}& M{K}^{2h}=\sum _{l=0}^{h-1}(-1{)}^{h+l+1}(\genfrac{}{}{0ex}{}{h}{l})(q-1{)}^{2(h-l)}M{K}^{2l}\\ & +q\sum _{j=0}^{min\{(q-1{)}^2,h\}}(-1{)}^{h+j}{D}_{2,j}\sum _{t=j}^{h}t!S(h,t)2^{h-t}(\genfrac{}{}{0ex}{}{(q-1{)}^2-j}{(q-1{)}^2-t}),\end{array}$$

$$\begin{array}{rl}& D_{2,j}=\sum \left(\genfrac{}{}{0ex}{}{2q-3}{{\nu }_0}\right)\prod _{\beta \in {\mathbb{F}}_q^{\ast }}\left(\genfrac{}{}{0ex}{}{K(\lambda ;{\beta }^{-1})+q-3}{{\nu }_{\beta }}\right)\\ & =\sum \left(\genfrac{}{}{0ex}{}{2q-3}{{\nu }_0}\right)\prod _{\begin{array}{c}|t|<2\sqrt{q}\\ t\equiv -1(4)\end{array}}\prod _{K(\lambda ;{\beta }^{-1})=t}\left(\genfrac{}{}{0ex}{}{t+q-3}{{\nu }_{\beta }}\right),with\end{array}$$

$$\begin{array}{rl}& M{K}_2^h=\sum _{l=0}^{h-1}(-1{)}^{h+l+1}(\genfrac{}{}{0ex}{}{h}{l})(q^2-3q+1{)}^{(h-l)}M{K}_2^l\\ & +q\sum _{j=0}^{min\{(q-1{)}^2,h\}}(-1{)}^{h+j}{D}_{2,j}\sum _{t=j}^{h}t!S(h,t)2^{h-t}(\genfrac{}{}{0ex}{}{(q-1{)}^2-j}{(q-1{)}^2-t}),\end{array}$$

$$R=\{t\in \mathbb{Z}||t|<2\sqrt{q},t\equiv -1(mod4)\}.$$

$$K_2(\lambda ;a)=K(\lambda ;a{)}^2-q.$$

$$\begin{array}{rl}q& =2^r(r\in {\mathbb{Z}}_{>0}),\\ {\mathbb{F}}_q& =thefinitefieldwithqelements,\\ tr(x)& =x+x^2+\cdots +x^{2^{r-1}}thetracefunction{\mathbb{F}}_q\to {\mathbb{F}}_2,\\ \lambda (x)& =(-1{)}^{tr(x)}thecanonicaladditivecharacterof{\mathbb{F}}_q.\end{array}$$

$$v_{n-1}=(\cdots ,{\alpha }_1+\cdots +{\alpha }_{n-1}+{\alpha }_1^{-1}\cdots {\alpha }_{n-1}^{-1},\cdots ),$$

$$\delta (n-1,q;\beta )=|\{({\alpha }_1,\cdots ,{\alpha }_{n-1})\in ({\mathbb{F}}_q^{\ast }{)}^{n-1}|{\alpha }_1+\cdots ,+{\alpha }_{n-1}+{\alpha }_1^{-1}\cdots {\alpha }_{n-1}^{-1}=\beta \}|$$

$$\delta (n-1,q;0)=q^{-1}\{(q-1{)}^{n-1}+1\},$$

$$\delta (n-1,q;\beta )=K_{n-2}(\lambda ;{\beta }^{-1})+q^{-1}\{(q-1{)}^{n-1}+1\},$$

$$\delta (1,q;\beta )=\{\begin{array}{ll}2,& \text{if tr(β−1)=0},\\ 1,& \text{if β=0,}\\ 0,& \text{if tr(β−1)=1.}\end{array}$$

Full text

Index terms-recursive formula, multi-dimensional Kloosterman sum, Kloosterman sum, Pless power moment identity, weight distribution.

MSC 2000: 11T23, 20G40, 94B05.

1. Introduction and Notations

Let $\psi$ be a nontrivial additive character of the finite field $\mathbb{F}_{q}$ with $q=p^{r}$ elements ($p$ a prime), and let $m$ be a positive integer. Then the $m$-dimensional Kloosterman sum $K_{m}(\psi;a)$([10]) is defined by

[TABLE]

For this, we have the Deligne bound

[TABLE]

In particular, if $m=1$, then $K_{1}(\psi;a)$ is simply denoted by $K(\psi;a)$, and is called the Kloosterman sum. The Kloosterman sum was introduced in 1926 [8] to give an estimate for the Fourier coefficients of modular forms. It has also been studied to solve various problems in coding theory and cryptography over finite fields of characteristic two.

For each nonnegative integer $h$, by $MK_{m}(\psi)^{h}$ we will denote the $h$-th moment of the $m$-dimensional Kloosterman sum $K_{m}(\psi;a)$. Namely, it is given by

[TABLE]

If $\psi=\lambda$ is the canonical additive character of $\mathbb{F}_{q}$, then $MK_{m}(\lambda)^{h}$ will be simply denoted by $MK_{m}^{h}$. If futher $m=1$, for brevity $MK_{1}^{h}$ will be indicated by $MK^{h}$. The power moments of Kloosterman sums can be used, for example, to give an estimate for the Kloosterman sums.

Explicit computations on power moments of Kloosterman sums were begun with the paper [17] of Sali$\acute{e}$ in 1931, where he showed, for any odd prime $q$,

[TABLE]

Here $M_{0}=0$, and, for $h\in\mathbb{Z}_{>0}$,

[TABLE]

For $q=p$ odd prime, Sali$\acute{e}$ obtained $MK^{1}$, $MK^{2}$, $MK^{3}$, $MK^{4}$ in that same paper by determining $M_{1}$, $M_{2}$, $M_{3}$. On the other hand, $MK^{5}$ can be expressed in terms of the $p$-th eigenvalue for a weight $3$ newform on $\Gamma_{0}(15)$ (cf. [11], [16]). $MK^{6}$ can be expressed in terms of the $p$-th eigenvalue for a weight $4$ newform on $\Gamma_{0}(6)$ (cf.[4]). Also, based on numerical evidence, in [3] Evans was led to propose a conjecture which expresses $MK^{7}$ in terms of Hecke eigenvalues for a weight $3$ newform on $\Gamma_{0}(525)$ with quartic nebentypus of conductor 105.

From now on, let us assume that $q=2^{r}$. Carlitz [1] evaluated $MK^{h}$ for $h\leq 4$. Recently, Moisio was able to find explicit expressions of $MK^{h}$, for $h\leq 10$ (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 [18].

Also, Moisio considered binary hyper-Kloosterman codes ${C}(r,m)$ and determined the weight distributions of ${C}(r,m)$ and ${C}^{\bot}(r,m)$, for $r=2$ and all $m\geq 2$, and for all $r\geq 2$ and $m=3$ (cf. [14]). In [15], these results were further extended to the case of $r=3,4$ and all $m\geq 2$.

In this paper, along the line of [6] we construct two binary linear codes ${C}_{n-1}$ and ${D}_{m}$, respectively connected with multi-dimensional and $m$-multiple power Kloosterman sums (for any fixed $m$) over the finite field $\mathbb{F}_{q}$. Here $q$ is a power of two. The code ${C}_{n-1}^{\bot}$ is a subcode of the hyper-Kloosterman code ${C}(r,n)$, which is mentioned above. Then we obtain two recursive formulas for the power moments of multi-dimensional Kloosterman sums and the $m$-multiple power moments of Kloosterman sums in terms of the frequencies of weights in the respective codes. This is done via Pless power moment identity and yields, in the case of power moments of multi-dimensional Kloosterman sums, much simpler recursive formulas than those obtained previously in [5].

Theorem 1.1**.**

$(1)$* Let $n=2^{s}$, $q=2^{r}$. For $r\geq 3$, and $h=1,2,\cdots,$*

[TABLE]

Here $S(h,t)$ indicates the Stirling number of the second kind given by

[TABLE]

In addition, $\{C_{n-1,j}\}_{j=0}^{(q-1)^{n-1}}$ denotes the weight distribution of the binary linear code $C_{n-1}$, given by

[TABLE]

where the sum runs over all the sets of integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}(0\leq\nu_{\beta}\leq\delta(n-1,q;\beta))$ satisfying

[TABLE]

[TABLE]

*Here we understand that $K_{0}(\lambda;\beta^{-1})=\lambda(\beta^{-1})$.

(2) Let $q=2^{r}$. For $r\geq 3$, and $m,h=1,2,\cdots,$

[TABLE]

Here $\{D_{m,j}\}_{j=0}^{(q-1)^{m}}$ is the weight distribution of the binary linear code $D_{m}$, given by

[TABLE]

where the sum runs over all the sets of integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}(0\leq\nu_{\beta}\leq\sigma(m,q;\beta))$ satisfying (1.5), and

[TABLE]

*with the sum running over all $\alpha_{1},\cdots,\alpha_{m}\in\mathbb{F}_{q}^{*}$, satisfying $\alpha_{1}^{-1}+\cdots+\alpha_{m}^{-1}=\beta$.

(1) and (2) of the following are respectively $n=2$ and $n=4$ cases of Theorem 1.1 (1) (cf. (3.3), (3.4)), and (3) and (4) are equivalent and $n=2$ case of Theorem 1.1 (2) ((cf. (5.4), (5.8)).

Corollary 1.2**.**

$(1)$* Let $q=2^{r}$. For $r\geq 3$, and $h=1,2,\cdots,$*

[TABLE]

where $\{{C}_{1,j}\}_{j=0}^{q-1}$ is the weight distribution of the binary linear code ${C}_{1}$, with

[TABLE]

*Here the sum is over all the sets of nonnegative integers $\{\nu_{0}\}\bigcup\{\nu_{\beta}\}_{tr(\beta^{-1})=0}$ satisfying $\nu_{0}+\sum_{tr(\beta^{-1})=0}\nu_{\beta}=j$ and $\sum_{tr(\beta^{-1})=0}\nu_{\beta}\beta=0$.

$(2)$* Let $q=2^{r}$. For $r\geq 3$, and $h=1,2,\cdots,$*

[TABLE]

where $\{{C}_{3,j}\}_{j=0}^{(q-1)^{3}}$ is the weight distribution of the binary linear code ${C}_{3}$, with

[TABLE]

Here the sum runs over all the sets of nonnegative integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ satisfying (1.5),

[TABLE]

and

[TABLE]

*for every integer $t$ satisfying $|t|<2\sqrt{q}$ and $t\equiv-1(4)$ .

$(3)$* Let $q=2^{r}$. For $r\geq 3$, and $h=1,2,\cdots,$*

[TABLE]

where $\{D_{2,j}\}_{j=0}^{(q-1)^{2}}$ is the weight distribution of the binary linear code ${D}_{2}$, with

[TABLE]

*the sum running over all the sets of nonnegative integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ satisfying (1.5).

$(4)$* Let $q=2^{r}$. For $r\geq 3$, and $h=1,2,\cdots,$*

[TABLE]

*where $D_{2,j}(0\leq j\leq(q-1)^{2})$’s are just as in (1.12).

The next two theorems will be of use later.

Theorem 1.3**.**

$($[9]$)$* Let $q=2^{r}$, with $r\geq 2$. Then the range $R$ of $K(\lambda;a)$, as $a$ varies over $\mathbb{F}_{q}^{*}$, is given by*

[TABLE]

*In addition, each value $t\in R$ is attained exactly $H(t^{2}-q)$ times, where $H(d)$ is the Kronecker class number of $d$.

Theorem 1.4**.**

$($[2]$)$* For the canonical additive character $\lambda$ of $\mathbb{F}_{q}$, and $a\in\mathbb{F}_{q}^{*}$,*

[TABLE]

Before we proceed further, we will fix the notations that will be used throughout this paper:

[TABLE]

Note that any nontrivial additive character $\psi$ of $\mathbb{F}_{q}$ is given by $\psi(x)=\lambda(ax)$, for a unique $a\in\mathbb{F}_{q}^{*}$.

2. Construction of codes associated with multi-dimensional Kloosterman sums

We will construct binary linear codes ${C}_{n-1}$ of length $N_{1}=(q-1)^{n-1}$, connected with the $(n-1)$-dimensional Kloosterman sums. Here $n=2^{s}$, with $s\in\mathbb{Z}_{>0}$.

Let

[TABLE]

where $\alpha_{1},\alpha_{2},\cdots,\alpha_{n-1}$ run respectively over all elements of $\mathbb{F}_{q}^{*}$. Here we do not specify the ordering of the components of $v_{n-1}$, but we assume that some ordering is fixed.

Proposition 2.1**.**

$($[5]**, Proposition 11$)$ For each $\beta\in\mathbb{F}_{q}$, let

[TABLE]

*(Note that $\delta(n-1,q;\beta)$ is the number of components with those equal to $\beta$ in the vector $v_{n-1}$(cf.(2.1))).

Then*

[TABLE]

and, for $\beta\in\mathbb{F}_{q}^{*}$,

[TABLE]

*where $K_{0}(\lambda;\beta^{-1})=\lambda(\beta^{-1})$ by convention.

Corollary 2.2**.**

$(1)$**

[TABLE]

$(2)$**

[TABLE]

The binary linear code ${C}_{n-1}$ is defined as

[TABLE]

where the dot denotes the usual inner product in $\mathbb{F}_{q}^{N_{1}}$.

The following Delsarte’s theorem is well-known.

Theorem 2.3**.**

$($[12]$)$**

Let $B$ be a linear code over $\mathbb{F}_{q}$. Then

[TABLE]

In view of this theorem, the dual ${C}_{n-1}^{\bot}$ of ${C}_{n-1}$ is given by

[TABLE]

Lemma 2.4**.**

$(q-1)^{n-1}>nq^{\frac{n-1}{2}}$, for all $n=2^{s}~{}(s\in\mathbb{Z}_{>0})$, and $q=2^{r}\geq 8$.

Proof.

This can be proved, for example, by induction on $s$. ∎

Proposition 2.5**.**

For $q=2^{r}$, with $r\geq 3$, the map $\mathbb{F}_{q}\rightarrow{C}_{n-1}^{\bot}(a\mapsto c(a))$ is an $\mathbb{F}_{2}$-linear isomorphism.

Proof.

The map is clearly $\mathbb{F}_{2}$-linear and onto. Let $a$ be in the kernel of the map. Then $tr(a(\alpha_{1}+\cdots+\alpha_{n-1}+\alpha_{1}^{-1}\cdots\alpha_{n-1}^{-1}))=0$, for all $\alpha_{1},\cdots,\alpha_{n-1}\in\mathbb{F}_{q}^{*}$. Suppose that $a\neq 0$. Then, on the one hand,

[TABLE]

On the other hand, (2.6) is equal to $K_{n-1}(\lambda;a)$ (cf. proof of Proposition 11 in [5]), and so from Deligne’s estimate in (1.1) we get

[TABLE]

But this is impossible for $q\geq 8$, in view of Lemma 2.4. ∎

3. Recursive formulas for power moments of multi-dimensional Kloostermann sums

We are now ready to derive, via Pless power moment identity, a recursive formula for the power moments of multi-dimensional Kloosterman sums in terms of the frequencies of weights in ${C}_{n-1}$.

Theorem 3.1**.**

$($Pless power moment identity, [12]$)$ Let $B$ be an $q$-ary [$n,k$] code, and let $B_{i}$(resp. $B_{i}^{\bot}$) denote the number of codewords of weight $i$ in $B$(resp. in $B^{\bot}$). Then, for $h=0,1,2,\cdots,$

[TABLE]

*where $S(h,t)$ is the Stirling number of the second kind defined in(1.3).

For the following lemma, observe that $(n,q-1)=1$.

Lemma 3.2**.**

The map $a\mapsto a^{n}:\mathbb{F}_{q}^{*}\rightarrow\mathbb{F}_{q}^{*}$ is a bijection.

Lemma 3.3**.**

For $a\in\mathbb{F}_{q}^{*}$, the Hamming weight $w(c(a))$(cf. (2.5)) of $c(a)$ can be expressed as follows:

[TABLE]

Proof.

[TABLE]

([10], Theorem 2.23(v))

[TABLE]

∎

Denote for the moment $v_{n-1}$ in (2.1) by $v_{n-1}=(g_{1},g_{2},\cdots,g_{N_{1}})$. Let $u=(u_{1},\cdots,u_{N_{1}})\in\mathbb{F}_{2}^{N_{1}}$, with $\nu_{\beta}$ $1$’s in the coordinate places where $g_{l}=\beta$, for each $\beta\in\mathbb{F}_{q}$. Then we see from the definition of the code ${C}_{n-1}$ (cf. (2.4)) that $u$ is a codeword with weight $j$ if and only if $\sum_{\beta\in\mathbb{F}_{q}}\nu_{\beta}=j$ and $\sum_{\beta\in\mathbb{F}_{q}}\nu_{\beta}\beta=0$ (an identity in $\mathbb{F}_{q}$). As there are $\prod_{\beta\in\mathbb{F}_{q}}{\binom{\delta(n-1,q;\beta)}{\nu_{\beta}}}$ (cf. Proposition 2.1) many such codewords with weight $j$, we obtain the following result.

Proposition 3.4**.**

Let $\{{C}_{n-1,j}\}_{j=0}^{N_{1}}$ be the weight distribution of ${C}_{n-1}$, where ${C}_{n-1,j}$ denotes the frequency of the codewords with weight $j$ in ${C}_{n-1}$. Then

[TABLE]

where the sum runs over all the sets of integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}~{}(0\leq\nu_{\beta}\leq\delta(n-1,q;\beta))$ satisfying

[TABLE]

Corollary 3.5**.**

$(1)$* Let $\{{C}_{1,j}\}_{j=0}^{q-1}$ be the weight distribution of ${C}_{1}$. Then*

[TABLE]

*where the sum is over all the sets of nonnegative integers $\{\nu_{0}\}\cup\{\nu_{\beta}\}_{tr(\beta^{-1})=0}$ satisfying $\nu_{0}+\sum_{tr(\beta^{-1})=0}\nu_{\beta}=j$ and $\sum_{tr(\beta^{-1})=0}\nu_{\beta}\beta=0$ (cf.(2.2)).

$(2)$* Let $\{{C}_{3,j}\}_{j=0}^{(q-1)^{3}}$ be the weight distribution of ${C}_{3}$. Then*

[TABLE]

where the sum runs over all the sets of integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ satisfying

[TABLE]

[TABLE]

and

[TABLE]

*for every integer $t$ satisfying $|t|<2\sqrt{q}$ and $t\equiv-1(4)$ (cf. Theorem 1.3, (2.3)).

Remark 3.6*.*

This shows that the weight distribution of ${C}_{1}$ is the same as that of ${C}(SO^{+}(2,q))$ (cf. [7]).

From now on, we will assume that $r\geq 3$, and hence every codeword in ${C}_{n-1}^{\bot}$ can be written as $c(a)$, for a unique $a\in\mathbb{F}_{q}$ (cf. Proposition 2.5).

We now apply the Pless power moment identity in (3.1) to ${C}_{n-1}^{\bot}$, in order to obtain the result in Theorem 1.1 (1) about a recursive formula. Then the left hand side of that identity in (3.1) is equal to

[TABLE]

with $w(c(a))$ given by (3.2). So (3.5) is

[TABLE]

On the other hand, noting that $dim_{\mathbb{F}_{2}}{C}_{n-1}=r$ (cf. Proposition 2.5) the right hand side of the Pless moment identity(cf. (3.1)) becomes

[TABLE]

Our result in (1.2) follows now by equating (3.6) and (3.7).

Remark 3.7*.*

A recursive formula for the power moments of multi-dimensional Kloosterman sums was obtained in [5] by constructing binary linear codes ${C}(SL(n,q))$ and utilizing explicit expressions of Gauss sums for the finite special linear group $SL(n,q)$. However, our result in (1.2) is better than that in (1) of [5]. Because our formula here is much simpler than the one there. Indeed, the length of the code ${C}_{n-1}$ here is $N_{1}=(q-1)^{n-1}$, whereas that of ${C}(SL(n,q))$ there is $N=q^{\binom{n}{2}}\prod_{j=2}^{n}(q^{j}-1)$, both of which appear in their respective expressions of recursive formulas.

4. Construction of codes associated with powers of

Kloosterman sums

We will construct binary linear codes $D_{m}$ of length $N_{2}=(q-1)^{m}$, connected with the $m$-th powers of (the ordinary) Kloosterman sums. Here $m\in\mathbb{Z}_{>0}$.

Let

[TABLE]

where $\alpha_{1},\alpha_{2},\cdots,\alpha_{m}$ run respectively over all elements of $\mathbb{F}_{q}^{*}$. Here we do not specify the ordering of the components of $w_{m}$, but we assume that some ordering is fixed.

Theorem 4.1**.**

$($[7]$)$* Let $\lambda$ be the canonical additive character of $~{}\mathbb{F}_{q}$, and let $\beta\in\mathbb{F}_{q}^{*}$. Then*

[TABLE]

Proposition 4.2**.**

For each $\beta\in\mathbb{F}_{q}$, let

[TABLE]

(Note that $\sigma(m,q;\beta)$ is the number of components with those equal to $\beta$ in the vector $w_{m}$ (cf. (4.1)). Then

$(1)$**

[TABLE]

*where the sum in (4.3) runs over all $\alpha_{1},\cdots,\alpha_{m}\in\mathbb{F}_{q}^{*}$, satisfying $\alpha_{1}^{-1}+\cdots+\alpha_{m}^{-1}=\beta$.

$(2)$**

[TABLE]

Proof.

(1) can be proved just as Proposition 2.1(cf. [5], Proposition 11). The details are left to the reader.

(2) If $m=2$, from (4.3)

[TABLE]

where $\alpha_{1}$ and $\alpha_{2}$ run over all elements in $\mathbb{F}_{q}^{*}$, satisfying $\alpha_{1}^{-1}+\alpha_{2}^{-1}=\beta$.

If $\beta=0$, then the result is clear. Assume now that $\beta\neq 0$. Then the sum in (4.5) is

[TABLE]

∎

The binary linear code $D_{m}$ is defined as

[TABLE]

where the dot denotes the usual inner product in $\mathbb{F}_{q}^{N_{2}}$.

Remark 4.3*.*

Clearly, the binary linear codes ${C}_{1}$ and $D_{1}$ coincide.

In view of Theorem 2.3, the dual $D_{m}^{\bot}$ of $D_{m}$ is given by

[TABLE]

Lemma 4.4**.**

$(q-1)^{m}>2^{m}q^{\frac{m}{2}}$, for all $m\in\mathbb{Z}_{>0}$ and $q=2^{r}\geq 8$.

Proof.

This can be shown, for example, by induction on $m$. ∎

Proposition 4.5**.**

For $q=2^{r}$, with $r\geq 3$, the map $\mathbb{F}_{q}\rightarrow D_{m}^{\bot}(a\mapsto d(a))$ is an $\mathbb{F}_{2}$-linear isomorphism.

Proof.

The map is clearly $\mathbb{F}_{2}$-linear and onto. Let $a$ be in the kernel of the map. Then $tr(a(\alpha_{1}+\cdots+\alpha_{m}+\alpha_{1}^{-1}+\cdots+\alpha_{m}^{-1}))=0$, for all $\alpha_{1},\cdots,\alpha_{m}\in\mathbb{F}_{q}^{*}$. Suppose that $a\neq 0$. Then, on the one hand,

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On the other hand, (4.7) is equal to $K(\lambda;a)^{m}$, and so from Weil’s estimate (i.e. (1.1) with $m=1$) we get

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But this is impossible for $q\geq 8$, in view of Lemma 4.4 ∎

5. Recursive formulas for $m$-multiple power moments of Kloostermann sums

We are now ready to derive, via Pless power moment identity, a recursive formula for the $m$-multiple power moments of Kloosterman sums in terms of the frequencies of weights in $D_{m}$.

Lemma 5.1**.**

For $a\in\mathbb{F}_{q}^{*}$, the Hamming weight $w(d(a))$ of $d(a)$ (cf. (4.6)) can be expressed as follows:

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Proof.

This can be shown exactly as the proof of Lemma 3.3. ∎

Corollary 5.2**.**

For $m=2$,

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The same argument leading to Proposition 3.4 shows the next proposition.

Proposition 5.3**.**

Let $\{D_{m,j}\}_{j=0}^{N_{2}}$ be the weight distribution of $D_{m}$, where $D_{m,j}$ denotes the frequency of the codewords with weight $j$ in $D_{m}$. Then

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where the sum runs over all the sets of integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}~{}(0\leq\nu_{\beta}\leq\sigma(m,q;\beta))$, satisfying

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Corollary 5.4**.**

Let $\{D_{2,j}\}_{j=0}^{(q-1)^{2}}$ be the weight distribution of $D_{2}$, and let $q=2^{r}$, with $r\geq 2$. Then, in view of Theorem 1.3 and (4.4), we have

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where the sum runs over all the sets of nonnegative integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ satisfying

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From now on, we will assume that $r\geq 3$, and hence every codeword in $D_{m}^{\bot}$ can be written as $d(a)$, for a unique $a\in\mathbb{F}_{q}$(cf. Proposition 4.5).

We now apply the Pless power moment identity in (3.1) to $D_{m}^{\bot}$, in order to obtain the result in Theorem 1.1 (1) about a recursive formula. Then the left hand side of that identity in (3.1) is equal to

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with $w(d(a))$ given by (5.1). So (5.5) is seen to be equal to

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On the other hand, noting that $dim_{\mathbb{F}_{2}}D_{m}=r$(cf. Proposition 4.5) the right hand side of the Pless moment identity(cf. (3.1)) becomes

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Our result in (1.6) follows now by equating (5.6) and (5.7).

Remark 5.5*.*

If $m=2$, from the alternative expression of $w(d(a))$ in (5.2) we see that (5.5) can also be given as

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