Extension of the Helleseth-Zinoviev result on the system of equations from Goethals codes and Kloosterman sums

TL;DR

This paper revises and extends a previous result on the solutions of certain equations related to Goethals codes and Kloosterman sums, correcting an error for even parameters and providing new divisibility properties.

Contribution

It corrects and completes a prior theorem on solution counts and divisibility of Kloosterman sums, offering new insights into their properties.

Findings

Theorem 4 is corrected for even m.

New divisibility modulo 12 of Kloosterman sums is established.

Results differ from previous findings for specific Kloosterman sums.

Abstract

In the paper of Tor Helleseth and Victor Zinoviev (Designs, Codes and Cryptography, \textbf{17}, 269-288(1999)), the number of solutions of the system of equations from $ Z_{4} $-linear Goethals codes $ G_{4} $ was determined and stated in Theorem 4. We found that Theorem 4 is wrong for $ m $ even. In this note, we complete Theorem 4, and give new divisibility modulo 12 of the Kloosterman sums deduced from Theorem 4, which is different from the result of the same authors for $ K(a^{4}+a^{3}) $ modulo 12.

Equations

$$K(a)=\sum _{x\in F^{\ast }}(-1{)}^{\mathrm{Tr}(ax+1/x)}$$

$$\{\begin{array}{ll}x+y+z+u& =1\\ u^2+xy+xz+xu+yz+yu+zu& =b^2\\ x^3+y^3+z^3+u^3& =c\end{array}$$

$${\mu }_2(b,c)=\frac{2}{3}\times \{\begin{array}{ll}{M}_2(b,c)& \text{if}\mathrm{Tr}(c)\ne \mathrm{Tr}(1),\\ M_2(b,c)-1& \text{if}\mathrm{Tr}(c)=\mathrm{Tr}(1).\end{array}$$

$$4M_{2}(b,c)=\sum\limits_{v\in F^{**}}\bigl{(}1+(-1)^{\mathrm{Tr}(\frac{k_{1}}{v})+l}\bigr{)}\bigl{(}1+(-1)^{\mathrm{Tr}(\frac{k_{2}}{v})+l}\bigr{)}$$

$$M_{2}(b,c)=\dfrac{1}{4}\bigl{(}q-3+(-1)^{\mathrm{Tr}(k_{1})}K(k_{1}k_{2})-(-1)^{\mathrm{Tr}(b)}(3+(-1)^{\mathrm{Tr}(k_{1})})\bigr{)}.$$

$$\mu_{2}(b,c)=\frac{1}{6}\bigl{(}q-8+(-1)^{\mathrm{Tr}(b)}(K(k_{1}k_{2})-3)\bigr{)}.$$

$$\mu_{2}(b,c)=\frac{1}{6}\bigl{(}q-2-(-1)^{\mathrm{Tr}(b)}(K(k_{1}k_{2})+3)\bigr{)}.$$

$$K(a^4+a^3)\equiv \{\begin{array}{ll}3& (\mathrm{mod}12),\text{if}m\equiv 1(\mathrm{mod}2),\\ 7& (\mathrm{mod}12),\text{if}m\equiv 0(\mathrm{mod}2).\end{array}$$

$$\mu_{2}(b,c)=\frac{1}{6}\bigl{(}q-5-K(a^{4}+a^{3})\bigr{)}\ \text{for}\ \mathrm{Tr}(c)\in\{0,1\}.$$

$$\mu_{2}(b,c)=\begin{cases}\frac{1}{6}\bigl{(}q-11+K(a^{4}+a^{3})\bigr{)},&\ \text{if}\ \mathrm{Tr}(c)=0,\\ \frac{1}{6}\bigl{(}q+1+K(a^{4}+a^{3})\bigr{)},&\ \text{if}\ \mathrm{Tr}(c)=1.\end{cases}$$

$$K({\xi }^4+{\xi }^3)\equiv \{\begin{array}{ll}3& (\mathrm{mod}12),\text{if}m\equiv 1(\mathrm{mod}2),\\ 7& (\mathrm{mod}12),\text{if}m\equiv 0(\mathrm{mod}2).\end{array}$$

$$K(a)\equiv \{\begin{array}{ll}7& (\mathrm{mod}12),\text{if}\mathrm{Tr}(a)=0,\\ 11& (\mathrm{mod}12),\text{if}\mathrm{Tr}(a)=1.\end{array}$$

Full text

1 Introduction

Let $m$ be a positive integer, $q=2^{m},F:=\mathbb{F}_{q}$ be the finite field of $q$ elements, $F^{*}:=F\setminus\{0\}$, and $F^{**}:=F\setminus\{0,1\}$. The well known Kloosterman sums carlitz ; charpin are defined by

[TABLE]

where $a\in F^{*}$, and $\mathrm{Tr}(\cdot)$ is the trace function of $F$ over $\mathbb{F}_{2}$.

Let $b,c\in F$. The problem of finding the coset weight distribution of $Z_{4}$-linear Goethals codes $G_{4}$, is transformed into solving the following nonlinear system of equations over $F$ tor-victor :

[TABLE]

where $x,y,z$ and $u$ are pairwise distinct elements of $F$. The number of solutions of (2) (see p.284 of tor-victor ), denoted by $\mu_{2}(b,c)$, is given by

[TABLE]

The $M_{2}(b,c)$ in (3) is defined by

[TABLE]

where $k_{1}=b^{2}+c+1,k_{2}=b^{2}+b+c+\sqrt{c}$, and $l=\mathrm{Tr}(b)$. The explicit evaluation of $M_{2}(b,c)$ is given by the following equation (see (33) of tor-victor ):

[TABLE]

We found that Theorem 4 of tor-victor , which gives the explicit evaluation of $\mu_{2}(b,c)$, is wrong for $m$ even. It is obvious that the authors of tor-victor forgot to take account of the fact, that $\mathrm{Tr}(1)=1$ for $m$ odd and 0 for $m$ even, in the last step of the proof of Theorem 4. The following is the correct version of Theorem 4 of tor-victor :

Theorem 1

Let $\mu_{2}(b,c)$ be the number of different 4-tuples $(x,y,z,u)$, where $x,y,z,u$ are pairwise distinct elements of $F$, which are solutions to the system (2) over $F$, where $b,c$ are arbitrary elements of $F$.

1. (1)

If $m$ is odd and $\mathrm{Tr}(c)=1$ or $m$ is even and $\mathrm{Tr}(c)=0$, then

[TABLE] 2. (2)

If $m$ is odd and $\mathrm{Tr}(c)=0$ or $m$ is even and $\mathrm{Tr}(c)=1$, then

[TABLE]

Where $k_{1}=b^{2}+c+1$ and $k_{2}=b^{2}+b+c+\sqrt{c}$.

From Theorem 4 of tor-victor , the divisibility modulo 12 of the Kloosterman sums $K(a^{4}+a^{3})$ is obtained and stated in Theorem 5 of tor-victor . In Section 2, we will establish new divisibility modulo 12 of $K(a^{4}+a^{3})$.

2 New divisibility modulo 12 of $K(a^{4}+a^{3})$

Before stating the new results on $K(a^{4}+a^{3})$ modulo 12, we need several auxiliary results.

Lemma 1** (carlitz )**

Let $a\in F^{*}$. Then, $K(a)=K(a^{2})$.

Lemma 2** (lidl-nieder )**

The quadratic equation $x^{2}+ax+b=0$, where $a\in F^{*}$ and $b\in F$, has two distinct roots in $F$, if and only if $\mathrm{Tr}(b/a^{2})=0$.

Lemma 3** (lidl-nieder )**

Let $a,b\in F$. Then,

1. (1)

$\mathrm{Tr}(a)=\mathrm{Tr}(a^{2}),\mathrm{Tr}(a+b)=\mathrm{Tr}(a)+\mathrm{Tr}(b)$. 2. (2)

$\mathrm{Tr}(1)=1$* if $m$ is odd, and $\mathrm{Tr}(1)=0$ if $m$ is even.*

The following theorem shows that $K(a^{4}+a^{3})$ modulo 12 may have different values than Theorem 5 of tor-victor if $a$ is under certain restriction:

Theorem 2

Let $a=b^{2}+b+1$ where $b\in F$. If $a\neq 0$, then

[TABLE]

Proof

Set $k_{2}=b^{2}+\sqrt{c}$, i.e., $b^{2}+b+c+\sqrt{c}=b^{2}+\sqrt{c}$. We obtain $b=c$, and $k_{1}=b^{2}+b+1$. Further, we have $\mathrm{Tr}(b)=\mathrm{Tr}(c),\mathrm{Tr}(k_{1})=\mathrm{Tr}(b^{2}+b+1)=\mathrm{Tr}(1)$, and $k_{1}^{2}k_{2}^{2}=b^{8}+b^{6}+b^{4}+b^{5}+b^{3}+b$. Let $a=b^{2}+b+1$. It is easy to see that $k_{1}^{2}k_{2}^{2}=b^{8}+b^{6}+b^{4}+b^{5}+b^{3}+b=a^{4}+a^{3}$ in $F$. By Lemma 1, $K(k_{1}k_{2})=K((k_{1}k_{2})^{2})=K(a^{4}+a^{3})$. Substituting $\mathrm{Tr}(k_{1})=\mathrm{Tr}(1)$ and $\mathrm{Tr}(b)=\mathrm{Tr}(c)$ into (5), using (3), and taking account that $\mathrm{Tr}(1)=1$ if $m$ is odd and 0 if $m$ is even, we obtain that

1. (1)

Case that $m$ is odd:

[TABLE] 2. (2)

Case that $m$ is even:

[TABLE]

The number $6\mu_{2}(b,c)$ is a multiple of 12 tor-victor . It can be verified that $(2^{m}-5)\equiv 3\pmod{12}$ if $m$ is odd, and $-(2^{m}-11)\equiv-(2^{m}+1)\equiv 7\pmod{12}$ if $m$ is even. The theorem follows from above argument.

∎

Based on computer investigation, we propose the following conjecture:

Conjecture 1

Let $a=b^{2}+b+1$ where $b\in F$, and $\xi=\sum_{i=0}^{n}a^{2i}+a$ with $n\geq 0$. If $\xi\neq 0$, then

[TABLE]

To illustrate Conjecture 1, we present the following theorem:

Theorem 3

Let $\xi=b^{4}+b+1$ where $b\in F$. If $\xi\neq 0$, then

[TABLE]

Proof

Let $a=b^{2}+b+1$ where $b\in F$. Remark that $\xi=a^{2}+a+1$. Set $k_{2}=b^{2}+b+c+\sqrt{c}=c+\sqrt{b}$. Then, $c=b^{4}+b^{2}+b$ and $k_{1}=b^{4}+b+1$. Further, we obtain that $\mathrm{Tr}(c)=\mathrm{Tr}(b^{4}+b^{2}+b)=\mathrm{Tr}(b),\mathrm{Tr}(k_{1})=\mathrm{Tr}(b^{4}+b+1)=\mathrm{Tr}(1)$, and $k_{1}^{2}k_{2}^{2}=(b^{8}+b^{2}+1)(b^{8}+b^{4}+b^{2}+b)=\xi^{4}+\xi^{3}$ in $F$. The rest of the proof is analogous to Theorem 2, we omit the details.

∎

If $a\in F$ satisfies a certain condition, the binomial $a^{4}+a^{3}$ of $K(a^{4}+a^{3})$ from Theorem 5 of tor-victor may be reduced into a monomial of degree one, which is illustrated by the following theorem:

Theorem 4

Let $a\in F^{**}$ be a third root of unity in $F^{*}$. Then,

[TABLE]

Proof

Set $k_{1}=b^{2}+c+1=b+\sqrt{c}$, then $k_{2}=b^{2}+b+c+\sqrt{c}=1$. The condition $k_{1}=b^{2}+c+1=b+\sqrt{c}$ holds if and only if $\xi^{2}+\xi+1=0$ where $\xi=(b+\sqrt{c})\in F^{**}$, i.e., $\xi$ is a third root of unity in $F^{*}$. Further, we have $K(k_{1}k_{2})=K(\xi)$. The rest of the proof is similar to Theorem 2, we omit the details.

∎

3 Conclusion

In this note, we pointed out that Theorem 4 of tor-victor is wrong for $m$ even, and completed it by taking account of the fact that $\mathrm{Tr}(1)=1$ for $m$ odd and 0 for $m$ even in the proof of the theorem. We established some new divisibility modulo 12 of the Kloosterman sums $K(a^{4}+a^{3})$ too.