Ternary Codes Associated with $O(3,3^r)$ and Power Moments of Kloosterman Sums with Trace Nonzero Square Arguments

TL;DR

This paper constructs ternary linear codes linked to orthogonal groups over finite fields of characteristic three and derives recursive formulas for power moments of Kloosterman sums with specific trace conditions.

Contribution

It introduces new ternary codes associated with $SO(3,q)$ and $O(3,q)$ and establishes recursive formulas for Kloosterman sum moments using these codes.

Findings

Recursive formulas for Kloosterman sum moments derived

Explicit expressions of Gauss sums for orthogonal groups used

Codes constructed for orthogonal groups over finite fields of characteristic three

Abstract

In this paper, we construct two ternary linear codes $C(SO(3,q))$ and $C(O(3,q))$, respectively associated with the orthogonal groups $SO(3,q)$ and $O(3,q)$. Here $q$ is a power of three. Then we obtain two recursive formulas for the power moments of Kloosterman sums with $``$trace nonzero square arguments" 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.

Equations

$$K(\psi ;a)=\sum _{\alpha \in {\mathbb{F}}_q^{\ast }}\psi (\alpha +a{\alpha }^{-1})(a\in {\mathbb{F}}_q^{\ast }).$$

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

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

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

$$T_{0}SK(\psi {)}^h=\sum _{a\in {\mathbb{F}}_q^{\ast },tra=0}K(\psi ;a^2{)}^h,T_{12}SK(\psi {)}^h=\sum _{a\in {\mathbb{F}}_q^{\ast },tra\ne 0}K(\psi ;a^2{)}^h,$$

$$2SK(\psi {)}^h=T_{0}SK(\psi {)}^h+T_{12}SK(\psi {)}^h,$$

$$SK(\psi {)}^h=\sum _{a\in {\mathbb{F}}_q^{\ast },asquare}K(\psi ;a{)}^h,$$

$$\left(\genfrac{}{}{0ex}{}{c}{a,b}\right)=\frac{c!}{a!b!(c-a-b)!},ifa+b\le c,$$

$$\left(\genfrac{}{}{0ex}{}{c}{a,b}\right)=0,ifa+b>c.$$

$$\begin{array}{rl}& ((-1{)}^{h+1}+2^{-h})T_{12}S{K}^h\\ & =-\sum _{j=1}^{h-1}((-1{)}^{j+1}+2^{-j})(\genfrac{}{}{0ex}{}{h}{j})(q^2-1{)}^{h-j}{T}_{12}S{K}^j\\ & +q^{1-h}\sum _{j=0}^{min\{N_1,h\}}(-1{)}^j(C_{1,j}-{\hat{C}}_j)\sum _{t=j}^{h}t!S(h,t)3^{h-t}{2}^{t-h-j}(\genfrac{}{}{0ex}{}{N_1-j}{N_1-t}),\end{array}$$

$$\begin{array}{rl}{C}_{1,j}=& \sum \left(\genfrac{}{}{0ex}{}{q^2}{{\nu }_0,{\mu }_0}\right)\left(\genfrac{}{}{0ex}{}{q^2}{{\nu }_2,{\mu }_2}\right)\\ & \times \prod _{{\beta }^2-2\beta \ne 0square}\left(\genfrac{}{}{0ex}{}{q^2+q}{{\nu }_{\beta },{\mu }_{\beta }}\right)\prod _{{\beta }^2-2\beta nonsquare}\left(\genfrac{}{}{0ex}{}{q^2-q}{{\nu }_{\beta },{\mu }_{\beta }}\right),\end{array}$$

$$\begin{array}{rl}{\hat{C}}_j=& \sum \left(\genfrac{}{}{0ex}{}{q^2}{{\nu }_1,{\mu }_1}\right)\left(\genfrac{}{}{0ex}{}{q^2}{{\nu }_{-1},{\mu }_{-1}}\right)\\ & \times \prod _{{\beta }^2-1\ne 0square}\left(\genfrac{}{}{0ex}{}{q^2+q}{{\nu }_{\beta },{\mu }_{\beta }}\right)\prod _{{\beta }^2-1nonsquare}\left(\genfrac{}{}{0ex}{}{q^2-q}{{\nu }_{\beta },{\mu }_{\beta }}\right).\end{array}$$

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

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

$$\begin{array}{rl}& ((-1{)}^{h+1}+2^{-h})T_{12}S{K}^h\\ & =-\sum _{j=1}^{h-1}((-1{)}^{j+1}+s^{-j})(\genfrac{}{}{0ex}{}{h}{j})(q^2-1{)}^{h-j}{T}_{12}S{K}^j\\ & +q^{1-h}\sum _{j=0}^{min\{N_2,h\}}(-1{)}^j{C}_{2,j}\sum _{t=j}^{h}t!S(h,t)3^{h-t}{2}^{t-2h-j}(\genfrac{}{}{0ex}{}{N_2-j}{N_2-t})\\ & -q^{1-h}\sum _{j=0}^{min\{N_1,h\}}(-1{)}^j{\hat{C}}_j\sum _{t=j}^{h}t!S(h,t)3^{h-t}{2}^{t-h-j}(\genfrac{}{}{0ex}{}{N_1-j}{N_1-t}),\end{array}$$

$$\begin{array}{rl}{C}_{2,j}=& \sum \prod _{\beta \in {\mathbb{F}}_q}\left(\genfrac{}{}{0ex}{}{n_2(\beta )}{{\nu }_{\beta },{\mu }_{\beta }}\right)(j=0,\cdots ,N_2),\\ & with{n}_2(\beta )=2q^2-2q+q\delta (1,q;\beta -1)+q\delta (1,q;\beta +1).\end{array}$$

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

$$\begin{array}{rl}\delta (1,q;\beta )& =|\{x\in {\mathbb{F}}_q|x^2-\beta x+1=0\}|\\ & =\{\begin{array}{ll}2,& \text{if β2−1=0 is a square,}\\ 1,& \text{if β2−1=0,}\\ 0,& \text{if β2−1 is a nonsquare.}\end{array}\end{array}$$

$$\begin{array}{rl}q& =3^r(r\in {\mathbb{Z}}_{>0}),\\ {\mathbb{F}}_q& =thefinitefieldwithqelements,\\ TrA& =thetraceofAforasquarematrixA,\\ {}^{t}B& =thetransposeofBforanymatrixB.\end{array}$$

$$O(2n+1,q)=\{w\in GL(2n+1,q){|}^{t}wJw=J\},$$

$$J=\left[\begin{array}{ccc}0& 1_n& 0\\ 1_n& 0& 0\\ 0& 0& 1\end{array}\right].$$

$$\left[\begin{array}{ccc}A& B& e\\ C& D& f\\ g& h& i\end{array}\right](A,B,C,Dn\times n,e,fn\times 1,g,h1\times n,i1\times 1)$$

$$\begin{array}{rl}& {}^{t}AC+{}^{t}CA+{}^{t}gg=0,{}^{t}BD+{}^{t}DB+{}^{t}hh=0,\\ & {}^{t}AD+{}^{t}CB+{}^{t}gh=1_n,{}^{t}ef+{}^{t}fe+i^2=1,\\ & {}^{t}Af+{}^{t}Ce+{}^{t}gi=0,{}^{t}Bf+{}^{t}De+{}^{t}hi=0.\end{array}$$

$$\displaystyle\begin{split}P&=P(2n+1,q)\\ &=\left\{\begin{bmatrix}A&0&0\\ 0&{{}^{t}A}^{-1}&0\\ 0&0&i\\ \end{bmatrix}\begin{bmatrix}1_{n}&B&-{{}^{t}h}\\ 0&1_{n}&0\\ 0&h&1\\ \end{bmatrix}\Bigg{|}\begin{array}[]{c}A\in GL(n,q),~{}i=\pm 1\\ B+{{}^{t}B}+{{}^{t}h}h=0\end{array}\right\},\end{split}$$

$$\displaystyle\begin{split}Q&=Q(2n+1,q)\\ &=\left\{\begin{bmatrix}A&0&0\\ 0&{{}^{t}A}^{-1}&0\\ 0&0&1\\ \end{bmatrix}\begin{bmatrix}1_{n}&B&-{{}^{t}h}\\ 0&1_{n}&0\\ 0&h&1\\ \end{bmatrix}\Bigg{|}\begin{array}[]{c}A\in GL(n,q)\\ B+{{}^{t}B}+{{}^{t}h}h=0\end{array}\right\}.\end{split}$$

$$P(2n+1,q)=Q(2n+1,q)⨿\rho Q(2n+1,q),$$

$$\rho =\left[\begin{array}{ccc}{1}_n& 0& 0\\ 0& 1_n& 0\\ 0& 0& -1\end{array}\right].$$

$${\sigma }_r=\left[\begin{array}{ccccc}0& 0& 1_r& 0& 0\\ 0& 1_{n-r}& 0& 0& 0\\ 1_r& 0& 0& 0& 0\\ 0& 0& 0& 1_{n-r}& 0\\ 0& 0& 0& 0& 1\end{array}\right](0\le r\le n).$$

$$O(2n+1,q)=\coprod _{r=0}^{n}P{\sigma }_{r}P=\coprod _{r=0}^{n}P{\sigma }_{r}Q,$$

$$\begin{array}{rl}O(2n+1,q)& =\coprod _{r=0}^{n}P{\sigma }_r(B_r\setminus Q)\\ & =\coprod _{r=0}^{n}Q{\sigma }_r(B_r\setminus Q)⨿\coprod _{r=0}^n\rho Q{\sigma }_r(B_r\setminus Q),\end{array}$$

Full text

Index terms - power moment, Kloosterman sum, trace nonzero square argument, orthogonal group, Pless power moment identity, weight distribution, Gauss sum.

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

1. Introduction

Let $\psi$ be a nontrivial additive character of the finite field $\mathbb{F}_{q}$ with $q=p^{r}$ elements ($p$ a prime). Then the Kloosterman sum $K(\psi;a)$([11]) is defined by

[TABLE]

The Kloosterman sum was introduced in 1926([10]) to give an estimate for the Fourier coefficients of modular forms.

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

[TABLE]

If $\psi=\lambda$ is the canonical additive character of $\mathbb{F}_{q}$, then $MK(\lambda)^{h}$ will be simply denoted by $MK^{h}$.

Explicit computations on power moments of Kloosterman sums were begun with the paper [16] 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 [16] 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. [12], [15]). $MK^{6}$ can be expressed in terms of the $p$-th eigenvalue for a weight $4$ newform on $\Gamma_{0}(6)$ (cf.[3]). Also, based on numerical evidence, in [1] 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 $q=3^{r}$. Recently, Moisio was able to find explicit expressions of $MK^{h}$, for $h\leq 10$ (cf. [14]). This was done, via Pless power moment identity, by connecting moments of Kloosterman sums and the frequencies of weights in the ternary Melas code of length $q-1$, which were known by the work of Geer, Schoof and Vlugt in [2].

In order to describe our results, we introduce three incomplete power moments of Kloosterman sums. For every nonnegative integer $h$, and $\psi$ as before, we define

[TABLE]

which will be respectively called the $h$-th moment of Kloosterman sums with $``$trace zero square arguments” and those with $``$trace nonzero square arguments.” Then, clearly we have

[TABLE]

where

[TABLE]

which is called the $h$-th moment of Kloosterman sums with $``$square arguments.” If $\psi=\lambda$ is the canonical additive character of $\mathbb{F}_{q}$, then $SK(\lambda)^{h}$, $T_{0}SK(\lambda)^{h}$, and $T_{12}SK(\lambda)^{h}$ will be respectively denoted by $SK^{h}$, $T_{0}SK^{h}$ and $T_{12}SK^{h}$, for brevity.

We derived recursive formulas generating the odd power moments of Kloosterman sums with trace one arguments in [7] and [8]. To do that we constructed binary linear codes associated with $O(3,2^{r})$ and with double cosets with respect to certain maximal parabolic subgroup of $O(2n+1,2^{r})$.

In this paper, we will show the main Theorem 1.1 giving recursive formulas for the power moments of Kloosterman sums with $``$trace nonzero square arguments.” To do that, we construct ternary linear codes $C(SO(3,q))$ and $C(O(3,q))$, respectively associated with the orthogonal groups $SO(3,q)$ and $O(3,q)$, and express those power moments in terms of the frequencies of weights in the codes. Then, thanks to our previous results on the explicit expressions of $``$Gauss sums” for the orthogonal group $O(2n+1,q)$ [6], we can express the weight of each codeword in the duals of the codes in terms of Kloosterman sums. Then our formulas will follow immediately from the Pless power moment identity.

Henceforth, we agree that, for nonnegative integers $a,b,c$,

[TABLE]

and

[TABLE]

Theorem 1.1**.**

Let $q=3^{r}$. Then we have the following.

$(1)$* For $h=1,2,3,\cdots,$ *

[TABLE]

where $N_{1}=|SO(3,q)|=q(q^{2}-1)$, and $\{C_{1,j}\}_{j=0}^{N_{1}}$ and $\{\hat{C}_{j}\}_{j=0}^{N_{1}}$ are respectively the weight distributions of $C(SO(3,q))$ and $C(Sp(2,q))$ given by: for $j=0,\cdots,N_{1}$,

[TABLE]

[TABLE]

Here the first sum in (1.6) is 0 if $h=1$ and the unspecified sums in (1.7) and (1.8) run over all the sets of nonnegative integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ and $\{\mu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ satisfying

[TABLE]

In addition, $S(h,t)$ is the Stirling number of the second kind defined by

[TABLE]

$(2)$* For $h=1,2,3,\cdots,$*

[TABLE]

where $N_{2}=|O(3,q)|=2q(q^{2}-1)$, and $\{C_{2,j}\}_{j=0}^{N_{1}}$ is the weight distribution of $C(O(3,q))$ given by: for $j=0,\cdots,N_{2}$,

[TABLE]

Here the first sum in (1.10) is 0 if $h=1$, the unspecified sum in (1.11) runs over all the sets of nonnegative integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ and $\{\mu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ satisfying

[TABLE]

$S(h,t)$* indicates the Stirling number of the second as in (1.9), $\hat{C}_{j}$’s are as in (1.8), and*

[TABLE]

2. $O(2n+1,q)$

For more details about the results of this section, one is referred to the paper [6]. Throughout this paper, the following notations will be used:

[TABLE]

The orthogonal group $O(2n+1,q)$ is defined as:

[TABLE]

where

[TABLE]

It consists of the matrices

[TABLE]

in $GL(2n+1,q)$ satisfying the relations:

[TABLE]

Let $P(2n+1,q)$ be the maximal parabolic subgroup of $O(2n+1,q)$ given by

[TABLE]

and let $Q=Q(2n+1,q)$ be the subgroup of $P(2n+1,q)$ of index 2 defined by

[TABLE]

Then we see that

[TABLE]

with

[TABLE]

Let $\sigma_{r}$ denote the following matrix in $O(2n+1,q)$

[TABLE]

Then the Bruhat decomposition of $O(2n+1,q)$ with respect to $P=P(2n+1,q)$ is given by

[TABLE]

which can further be modified as

[TABLE]

with

[TABLE]

The special orthogonal group $SO(2n+1,q)$ is defined as

[TABLE]

Then we see from (2.1) that

[TABLE]

The sympletic group $Sp(2n,q)$ is defined as:

[TABLE]

with

[TABLE]

As is well-known or mentioned in [4] and [6],

[TABLE]

[TABLE]

For integers $n$, $r$ with $0\leq r\leq n$, the $q$-binomial coefficients are defined as:

[TABLE]

It is shown in [6] that

[TABLE]

3. Gauss sums for $O(2n+1,q)$

The following notations will be employed throughout this paper.

[TABLE]

Then any nontrivial additive character $\psi$ of $\mathbb{F}_{q}$ is given by $\psi(x)=\lambda(ax)$, for a unique $a\in\mathbb{F}_{q}^{*}$. Also, since $\lambda(a)$ for any $a\in\mathbb{F}_{q}$ is a 3th root of 1, we have

[TABLE]

For any nontrivial additive character $\psi$ of $\mathbb{F}_{q}$ and $a\in\mathbb{F}_{q}^{*}$, the Kloosterman sum $K_{GL(t,q)}(\psi;a)$ for $GL(t,q)$ is defined as

[TABLE]

Observe that, for $t=1$, $K_{GL(1,q)}(\psi;a)$ denotes the Kloosterman sum $K(\psi;a)$.

In [4], it is shown that $K_{GL(t,q)}(\psi;a)$ satisfies the following recursive relation: for integers $t\geq 2$, $a\in\mathbb{F}_{q}^{*}$,

[TABLE]

where we understand that $K_{GL(0,q)}(\psi;a)=1$.

Proposition 3.1**.**

$($[6]$)$* Let $\psi$ be a nontrivial additive character of $\mathbb{F}_{q}$. For each positive integer $r$, let $\Omega_{r}$ be the set of all $r\times r$ nonsingular symmetric matrices over $\mathbb{F}_{q}$. Then we have*

[TABLE]

From [4] and [6], the Gauss sums for $SO(2n+1,q)$ and $O(2n+1,q)$ are respectively equal to $\psi(1)$ times that for $Sp(2n,q)$ and $\psi(1)+\psi(-1)$ times that for $Sp(2n,q)$. Indeed, using the decomposition in (2.2), for any nontrivial additive character $\psi$ of $\mathbb{F}_{q}$, it is shown that

[TABLE]

Similarly, from the decomposition in (2.1) it is shown in [6] that

[TABLE]

For our purposes, we only need the following expressions of Gauss sums for $SO(3,q)$ and $O(3,q)$. So we state them separately as a theorem. Also, for the ease of notations, we introduce

[TABLE]

Theorem 3.2**.**

Let $\psi$ be any nontrivial additive character of $\mathbb{F}_{q}$. Then we have

[TABLE]

The next corollary follows from Theorem 3.1 and by simple change of variables.

Corollary 3.3**.**

Let $\lambda$ be the canonical additive character of $\mathbb{F}_{q}$, and let $a\in\mathbb{F}_{q}^{*}$. Then we have

[TABLE]

[TABLE]

Proposition 3.4**.**

$($[5]**, $($5.3-5)$$) Let $\lambda$ be the canonical additive character of $F_{q}$, $m\in\mathbb{Z}_{\geq 0}$, $\beta\in\mathbb{F}_{q}$. Then

[TABLE]

where, for $m\geq 1$,

[TABLE]

and

[TABLE]

Remark 3.5*.*

Here one notes that

[TABLE]

Let $G(q)$ be one of finite classical groups over $\mathbb{F}_{q}$. Then we put, for each $\beta\in\mathbb{F}_{q}$,

[TABLE]

Then it is easy to see that

[TABLE]

For brevity, we write

[TABLE]

Using (3.3)-(3.5), and (4.1), one derives the following.

Proposition 3.6**.**

With the notations in (3.6), (3.7), and (3.9), we have:

[TABLE]

[TABLE]

Corollary 3.7**.**

$Tr:G_{1}(q)\rightarrow\mathbb{F}_{q}$, and $Tr:G_{2}(q)\rightarrow\mathbb{F}_{q}$ are surjective.

Proof.

This is immediate from the above Proposition 3.6. ∎

4. Construction of codes

Let

[TABLE]

Here we will construct ternary linear codes $C(G_{1}(q))$ of length $N_{1}$ and $C(G_{2}(q))$ of length $N_{2}$, respectively associated with the orthogonal groups $G_{1}(q)$ and $G_{2}(q)$. By abuse of notations, let $g_{1},g_{2},\cdots,g_{N_{i}}$ be a fixed ordering of the elements in the group $G_{i}(q)$, for $i=1,2$.

Also, we put

[TABLE]

Then the ternary linear code is defined as

[TABLE]

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

The following theorem of Delsarte is well-known.

Theorem 4.1**.**

$($[13]$)$* Let B be a linear code over $\mathbb{F}_{q}$. Then*

[TABLE]

In view of this theorem, the dual $C(G_{i}(q))^{\bot}$ is given by

[TABLE]

Proposition 4.2**.**

For every $q=3^{r}$, the map $\mathbb{F}_{q}\rightarrow C(G_{i}(q))^{\bot}(a\mapsto c_{i}(a))$ is an $\mathbb{F}_{3}$-linear isomorphism, for $i=1,2$.

Proof.

The maps are clearly $\mathbb{F}_{3}$-linear and surjective. Let $a$ be in the kernel of either of the map. Then, in view of Corollary 3.7, $tr(a\beta)=0$, for all $\beta\in\mathbb{F}_{q}$. Since the trace function $\mathbb{F}_{q}\rightarrow\mathbb{F}_{2}$ is surjective, $a=0$. ∎

5. Power moments of Kloosterman sums with trace nonzero square arguments

In this section, we will be able to find, via Pless power moment identity, recursive formulas for the power moments of Kloosterman sums with trace nonzero square arguments in terms of the frequencies of weights in $C(SO(3,q))$ and $C(O(3,q))$.

Theorem 5.1**.**

$($Pless power moment identity, [13]$)$ 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 (1.9).

Lemma 5.2**.**

Let $c_{i}(a)=(tr(aTrg_{1}),\cdots,tr(aTrg_{N_{i}}))\in C(G_{i}(q))^{\bot}$, for $a\in\mathbb{F}_{q}^{*}$, and $i=1,2$. Then the Hamming weight $w(c_{i}(a))$ can be expressed as follows:

[TABLE]

Proof.

For $i=1,2,$

[TABLE]

Our results now follow from (3.1), (3.3), (3.4) and (4.1). ∎

Fix $i(i=1,2)$, and let $u=(u_{1},\cdots,u_{N_{i}})\in\mathbb{F}_{3}^{N_{i}}$, with $\nu_{\beta}$ 1’s and $\mu_{\beta}$ 2’s in the coordinate places where $Tr(g_{j})=\beta$, for each $\beta\in\mathbb{F}_{q}$. Then we see from the definition of the code $C(G_{i}(q))$(cf. (4.2)) that $u$ is a codeword with weight $j$ if and only if $\sum_{\beta\in\mathbb{F}_{q}}\nu_{\beta}+\sum_{\beta\in\mathbb{F}_{q}}\mu_{\beta}=j$ and $\sum_{\beta\in\mathbb{F}_{q}}\nu_{\beta}\beta=\sum_{\beta\in\mathbb{F}_{q}}\mu_{\beta}\beta$(an identity in $\mathbb{F}_{q}$). Note that there are $\prod_{\beta\in\mathbb{F}_{q}}{\binom{n_{i}(\beta)}{\nu_{\beta},\mu_{\beta}}}$(cf. (1.4), (1.5)) many such codewords with weight $j$. Now, we get the following formulas in (5.3)-(5.4), by using the explicit values of $n_{i}(\beta)$ in (3.10),(3.11)(cf. (3.6), (3.7)).

Theorem 5.3**.**

Let $q=3^{r}$ be as before, and let $\{C_{i,j}\}_{j=0}^{N_{i}}$ be the weight distribution of $C(G_{i}(q))$, for $i=1,2.$ Then

$(1)$**

[TABLE]

with

[TABLE]

$(2)$**

[TABLE]

with

[TABLE]

Here in both (5.3) and (5.4) the unspecified sums run over all the sets of nonnegative integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ and $\{\mu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ satisfying

[TABLE]

and, for every $\beta\in\mathbb{F}_{q}$, $\delta(1,q;\beta)$ is as in (3.7).

The recursive formula in the following theorem follows from the study of ternary linear codes associated with the symplectic group $Sp(2,q)=SL(2,q)$. It is slightly modified from its original version, which makes it more usable in below.

Theorem 5.4**.**

$($[9]$)$* For $h=1,2,3,\cdots$,*

[TABLE]

where $N_{1}=q(q^{2}-1)=|Sp(2,q)|=|SO(3,q)|$, $S(h,t)$ indicates the Stirling number of the second kind as in (1.9), and $\{\hat{C}_{j}\}_{j=0}^{N_{1}}$ denotes the weight distribution of the ternary linear code $C(Sp(2,q))$, given by

[TABLE]

$(j=0,\cdots,N_{1})$*.

Here the sum is over all the sets of nonnegative integers $\{\nu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ and $\{\mu_{\beta}\}_{\beta\in\mathbb{F}_{q}}$ satisfying $\sum_{\beta\in\mathbb{F}_{q}}\nu_{\beta}+\sum_{\beta\in\mathbb{F}_{q}}\mu_{\beta}=j$ and $\sum_{\beta\in\mathbb{F}_{q}}\nu_{\beta}\beta=\sum_{\beta\in\mathbb{F}_{q}}\mu_{\beta}\beta$.*

We are now ready to apply the Pless power moment identity in (5.1) to $C(G_{i}(q))^{\bot}$ for $i=1,2$, in order to obtain the result in Theorem 1.1(cf. (1.6)-(1.8), (1.10)-(1.12)) about recursive formulas. We do this for $i=1,2$ at the same time.

The left hand side of that identity in (5.1) is equal to

[TABLE]

with the $w(c_{i}(a))$ given by (5.2).

In below, $``$the sum over $tra=0$ (resp. $tra\neq 0$)” will mean $``$the sum over all $a\in\mathbb{F}_{q}^{*}$ with $tra=0$(resp. $tra\neq 0$).”

(5.6) is given by

[TABLE]

[TABLE]

On the other hand, the right hand side of (5.1) is

[TABLE]

Here one has to note that $dim_{\mathbb{F}_{2}}C(SO(3,q))=dim_{\mathbb{F}_{2}}C(O(3,q))=r$(cf. Prop. 4.2) and to separate the term corresponding to $l=h$ of the second sum in (5.7). Our main results in Theorem 1.1 now follow by equating (5.7) and (5.8).

Corollary 5.5**.**

Let $q=3^{r}$. Then we have the following.

1. (1)

$SK={\frac{1}{2}}\{(-1)^{r}q+1\}$, 2. (2)

$T_{0}SK={\frac{1}{3}}(-1)^{r}q+1$, 3. (3)

$T_{12}SK={\frac{2}{3}}(-1)^{r}q$.

Proof.

From either (1.6) or (1.10), we get (3). (1) follows from our previous result ([9], (4)) or can be derived directly as follows.

[TABLE]

[TABLE]

[TABLE]

In (5.9), we note that $\alpha+\alpha^{-1}=0$ has a solution in $\mathbb{F}_{q}^{*}$ if and only if -1 is a square in $\mathbb{F}_{q}$ if and only if $r$ is even, in which case there are two distinct solutions. Finally, (2) follows from the relation (1.2) with $h=1$ and $\psi=\lambda$. ∎