Non-projective cyclic codes whose check polynomial contains two zeros

TL;DR

This paper proves that a specific family of non-projective cyclic codes with two zeros in their check polynomial cannot be two-weight codes, supporting Vega's conjecture that all two-weight cyclic codes are already known.

Contribution

It establishes that these particular cyclic codes are never two-weight, providing evidence for the conjecture that all two-weight cyclic codes are classified.

Findings

These codes are never two-weight codes.

Supports Vega's conjecture on the classification of two-weight cyclic codes.

Abstract

Let $n\geq 3$ be a positive integer and let $\mathbb{F}_{q^k}$ be the splitting field of $x^n-1$. By $\gamma$ we denote a primitive element of $\mathbb{F}_{q^k}$. Let $C$ be a cyclic code of length $n$ whose check polynomial contains two zeros $\gamma^d$ and $\gamma^{d+D}$, where $de \mid (q-1)$, $e>1$ and $D=(q^k-1)/e$. This family of cyclic codes is not projective. Many authors have studied the weight distribution of these codes for certain parameters. In this paper, we prove that these codes are never two-weight codes. This result would strengthen a conjecture by Vega which states that all two-weight cyclic codes are the "known" ones.

Equations

$$de\mid (q-1),e>1,D=\frac{q^k-1}{e}.$$

$$c_w=\left(\mathrm{Tr}(w),\mathrm{Tr}(w\delta ),\dots ,\mathrm{Tr}(w{\delta }^{n-1})\right),w\in {\mathbb{F}}_{q^k}.$$

$$\sum _{i=0}^n{C}_i\left(\genfrac{}{}{0ex}{}{n-i}{v}\right)=q^{m-v}\sum _{i=0}^n{B}_i\left(\genfrac{}{}{0ex}{}{n-i}{n-v}\right)\text{for}v=0,1,\dots ,n-1.$$

$$de\mid (q-1),e>1,D=\frac{q^k-1}{e}.$$

$$h_d(x)=(x-{\gamma }^d)(x-{\gamma }^{dq})\cdots (x-{\gamma }^{d{q}^{h-1}}),\text{and}$$

$$h_D(x)=(x-{\gamma }^{d+D})(x-{\gamma }^{(d+D)q})\cdots (x-{\gamma }^{(d+D)q^{H-1}}),$$

$$d(q^h-1)\equiv 0(\mathrm{mod}\frac{q^k-1}{q-1})\text{and}(d+D)(q^H-1)\equiv 0(\mathrm{mod}\frac{q^k-1}{q-1}).$$

$$h(x)=h_d(x)h_D(x)$$

$$C_d$$

$$C_D$$

$$C$$

$$n=\lambda \frac{q^k-1}{d}.$$

$$C_d^{\prime }=\{c_u^{\prime }=(\mathrm{Tr}(u),\mathrm{Tr}(u{\beta }^d),\cdots ,\mathrm{Tr}(u{\beta }^{d(n_1-1)})):u\in {\mathbb{F}}_{q^k}\}.$$

$$n_2=\frac{n_1(q-1)}{(q-1,n_1)}=\frac{q^k-1}{((q^k-1)/(q-1),d)}\text{and}g=\left(\frac{q^k-1}{q-1},d\right).$$

$$C_d^{\prime \prime }=\{c_u^{\prime \prime }=(\mathrm{Tr}(u),\mathrm{Tr}(u{\beta }^g),\cdots ,\mathrm{Tr}(u{\beta }^{g(n_2-1)})):u\in {\mathbb{F}}_{q^k}\}.$$

$$\mathrm{wt}(C_d^{\prime \prime })=\frac{d}{g}\mathrm{wt}(C_d^{\prime })=\frac{d}{\lambda g}\mathrm{wt}(C_d).$$

$${\bar{C}}_d=\{{\bar{c}}_u=({\mathrm{Tr}}_p(u),{\mathrm{Tr}}_p(u{\beta }^g)...,{\mathrm{Tr}}_p(u{\beta }^{g(n_2-1)})):u\in {\mathbb{F}}_{q^k}\}.$$

$$\text{wt}({\bar{C}}_d)=\frac{q(p-1)}{p(q-1)}\text{wt}(C_d^{\prime \prime }).$$

$$\mathrm{wt}(C_d)=\frac{\lambda gp(q-1)}{dq(p-1)}\mathrm{wt}({\bar{C}}_d).$$

$$q=p^t,g=\left(\frac{q^k-1}{q-1},d\right),h={\text{ord}}_g(p),s=\frac{kt}{h}.$$

$$w_1=\frac{\lambda (q-1)p^{s\theta }(p^{s(h-\theta )}-ϵm)}{dq},w_2=\frac{\lambda (q-1)p^{s\theta }(p^{s(h-\theta )}-ϵm+ϵg)}{dq},$$

$$\theta (g,p)=\frac{1}{p-1}\min \{S_p\left(\frac{j(p^h-1)}{g}\right):1\le j\le g-1\},$$

$$B_2=\left(\frac{\lambda f(q-1)}{de}-1\right)(q-1).$$

$$B_2=N(q-1).$$

$$\text{lcm}\left(e,\frac{q^k-1}{(q-1)d^{\prime }}\right)=\frac{(q^k-1)e}{(q-1)d^{\prime }{f}^{\prime }},$$

$$d^{\prime }{f}^{\prime }=\left(\frac{q^k-1}{q-1},e{d}^{\prime }\right)=\left(\frac{q^k-1}{q-1},\frac{q^k-1}{q-1}e,de\right)=\left(\frac{q^k-1}{q-1},de\right)=f.$$

$$de=q-1\text{and}\lambda =f=1.$$

$$w_1=\frac{n(q-1)q^{k-1}}{q^k-1}=\frac{q-1}{d}{q}^{k-1}.$$

$$C_2=\left(\frac{q-1}{d}-1\right)(q-1).$$

$$(q^k-1){\left(\frac{q-1}{d}\right)}^2{q}^k={\left(\frac{(q^k-1)(q-1)}{d}\right)}^2+\frac{(q^k-1)(q-1)}{d}+2(q-1)(\frac{q-1}{d}-1),$$

$$w_1=\mu q^{k-1},\text{where}\mu =\frac{\lambda (q-1)}{d}\mid (q-1).$$

Full text

1 Introduction

A linear code is called projective if its dual code has weight at least $3$. We call a linear code non-projective if its dual code contains a word of weight at most $2$. A cyclic code is irreducible if its check polynomial is irreducible. More details about cyclic codes can be found in [3]. The class of two-weight cyclic codes has been studied intensively by many authors [1, 2, 4, 7, 8, 9, 10, 12].

Two-weight irreducible cyclic codes were completely classified by Schmidt and White, see [7]. They gave necessary and sufficient conditions for the existence of these codes. Moreover, the nonzero weights are also explicitly described. It remains of interest to classify all two-weight cyclic codes which are not irreducible. In this direction, Wolfmann [11] proved that if a two-weight projective cyclic code is not irreducible, then it is the direct sum of two one-weight irreducible cyclic subcodes of the same dimension. Later, Vega [8] and Feng [2] complete the classification by giving necessary and sufficient conditions for these codes to be direct sum of two one-weight irreducible cyclic subcodes of the same dimension. Nevertheless, the non-projective case remains open.

The authors in [1], [4], [10], [12] studied the weight distributions of cyclic codes of various parameters. All these codes are not projective codes and not two-weight codes. The studied parameters belong to a bigger family of codes whose description was given by Feng in the concluding remarks in [2]. It is the purpose of this paper to prove that these codes are non-projective and never two-weight.

Theorem 1.1**.**

Let $n\geq 3$ be a positive integer. Let $q$ be a prime power and let $\mathbb{F}_{q^{k}}$ be the splitting field of $x^{n}-1$. Let $\gamma$ denote a primitive element of $\mathbb{F}_{q^{k}}$. Let $C$ be the cyclic code of length $n$ over $\mathbb{F}_{q}$ whose check polynomial is the minimal polynomial over $\mathbb{F}_{q}$ containing two zeros $\gamma^{d}$ and $\gamma^{d+D}$ in which $\gamma$ is a primitive element of $\mathbb{F}_{q^{k}}$ in which

[TABLE]

Then the code $C$ is non-projective and $C$ is not a two-weight code.

2 Structure of the Code C

In this section, we study the structure of the code $C$ described in Theorem 1.1 and provide necessary tools for the proof of Theorem 1.1. First, we fix some notations and state basic definitions of cyclic codes.

Let $m$ and $n$ be coprime integers. By $\operatorname{{\rm ord}}_{n}(m)$ we denote the smallest positive integer $k$ such that $m^{k}\equiv 1\pmod{n}$.

Definition 2.1**.**

*Let $h(x)$ be an irreducible divisor of $x^{n}-1$ over $\mathbb{F}_{q}$, where $(q,n)=1$. The cyclic code $W$ of length $n$ over $\mathbb{F}_{q}$ with check polynomial $h(x)$ is called an irreducible cyclic code.

Moreover, let $\mathbb{F}_{q^{k}}$ be the splitting field of $x^{n}-1$ over $\mathbb{F}_{q}$ (note that $k=\operatorname{{\rm ord}}_{n}(q)$). Let $\alpha$ be a root of $f(x)$ and put $\delta=\alpha^{-1}$. By $\operatorname{{\rm Tr}}$ we denote the trace of $\mathbb{F}_{q^{k}}$ over $\mathbb{F}_{q}$. Then the code $W$ consists of the following words.*

[TABLE]

The main tools used in the proof of Theorem 1.1 is MacWilliams identities [5] and the results by Schmidt and White [7]. While MacWilliams gives relation between the weights of a linear code, Schmidt and White give an explicit description for the weights of a two-weight irreducible cyclic codes. The following result is taken from [5, Lemma 2.2].

Result 2.2**.**

Let $W$ be an $[n,m]$ linear code over $\mathbb{F}_{q}$. Let $W^{\perp}$ denote the dual code of $W$. For each $i=0,\dots,n$, let $C_{i}(B_{i})$ denote the number of words in $W(W^{\perp})$ which have weight $i$. Then

[TABLE]

Let $w_{1},\dots,w_{N}$ be all the nonzero weights in the code $W$ and let $A_{i}$ be the numbers of words of weight $w_{i}$ in $W$. Letting $v=0,1,2$ in (1), we obtain the following three identities which will be useful later.

Result 2.3**.**

Under the above notations, we have

• (1)

$\sum_{i=1}^{N}A_{i}=q^{m}-1$.

• (2)

$\sum_{i=1}^{N}w_{i}A_{i}=(n(q-1)-B_{1})q^{m-1}$.

• (3)

$\sum_{i=1}^{N}w_{i}^{2}A_{i}=[n^{2}(q-1)^{2}+n(q-1)-B_{1}(q+2(n-1)(q-1))+2B_{2}]q^{m-2}$.

Next, we give a description for the code $C$ in Theorem 1.1. From now on, we always fix a prime power $q$ and positive integers $n,k,d,e,D$ with the properties $n\geq 3$, $k=\operatorname{{\rm ord}}_{n}(q)$ and

[TABLE]

Fix $\gamma$ as a primitive element of $\mathbb{F}_{q^{k}}$. By $C$ we denote the cyclic code of length $n$ whose check polynomial is the minimal polynomial over $\mathbb{F}_{q}$ containing two zeros $\gamma^{d}$ and $\gamma^{d+D}$.

Note that there is no integer $i$ such that $0\leq i\leq k-1$ and $d+D\equiv dq^{i}\pmod{q^{k}-1}$. Otherwise, the congruence $d+(q^{k}-1)/e\equiv dq^{i}\pmod{q^{k}-1}$ implies $q^{i}\equiv 1\pmod{(q^{k}-1)/(de)}$, so $i=0$ and $D\equiv 0\pmod{q^{k}-1}$, impossible. Hence, the minimal polynomials (over $\mathbb{F}_{q}$) $h_{d}(x)$ and $h_{D}(x)$ of $\gamma^{d}$ and $\gamma^{d+D}$ have no common zero. These polynomials are

[TABLE]

[TABLE]

where $h$ and $H$ are the smallest positive integers such that

[TABLE]

As $d<q-1$, we have $h=k$. Moreover note that $(q^{k}-1,d+D)=d\left(\frac{q^{k}-1}{de}e,1+\frac{q^{k}-1}{de}\right)=d\left(e,1+\frac{q^{k}-1}{de}\right)$ divides $de$, so $(d+D,(q^{k}-1)/(q-1))\leq de\leq q-1$. Hence we also have $H=k$. Therefore, the polynomial

[TABLE]

is a polynomial of degree $2k$ and $C$ is an $[n,2k]$ linear code.

We have proved the following lemma.

Lemma 2.4**.**

Let $C_{d}$ and $C_{D}$ be the cyclic irreducible codes whose check polynomial are $h_{d}(x)$ and $h_{D}(x)$ described as above. Then both $C_{d}$ and $C_{D}$ have dimension $k$. Moreover, the code $C$ has dimension $2k$ with check polynomial $h(x)=h_{d}(x)h_{D}(x)$. Denote $\beta=\gamma^{-1}$. The codes $C_{d},C_{D}$ and $C$ can be explicitly described as follows.

[TABLE]

The existence of the code $C$ of length $n$ implies that $\beta^{dn}=1$, so $(q^{k}-1)\mid dn$. As $q^{k}-1\equiv 0\pmod{n}$, there exists a divisor $\lambda$ of $d$ such that

[TABLE]

By Lemma 3.2, both $C_{d}$ and $C_{D}$ are two-weight codes if $C$ is two-weight. For the time being, we assume the validity of this result, that is, the codes $C$, $C_{d}$ and $C_{D}$ are all two-weight codes.

By $\operatorname{{\rm wt}}(W)$ we denote the set of weights of the code $W$. The following results in [7] allow us to focus on two-weight codes over $\mathbb{F}_{p}$.

Result 2.5**.**

Put $n_{1}=(q^{k}-1)/d=n/\lambda$. The following code $C^{\prime}_{d}$ is a two-weight code of length $n_{1}$ and $\operatorname{{\rm wt}}(C_{d})=\lambda\operatorname{{\rm wt}}(C^{\prime}_{d})$.

[TABLE]

Define

[TABLE]

The following code $C^{\prime\prime}_{d}$ is an irreducible cyclic code of length $n_{2}$.

[TABLE]

Moreover, the code $C^{\prime\prime}_{d}$ is a two-weight code and

[TABLE]

Result 2.6**.**

Let $\operatorname{{\rm Tr}}_{p}$ denote the trace of $\mathbb{F}_{q^{k}}$ over $\mathbb{F}_{p}$ and let $\bar{C}_{d}$ denote the following irreducible cyclic code over $\mathbb{F}_{p}$.

[TABLE]

Then the code $\bar{C}_{d}$ is two-weight and

[TABLE]

Combining (3) and (4), we obtain

[TABLE]

Using Result 2.6 and [7, Corollary 3.2], we can describe the two weights of $C_{d}$ in the following result.

Result 2.7**.**

Denote

[TABLE]

The following are two weights of the code $C_{d}$.

[TABLE]

where $\epsilon=\pm 1$ and $m$ is a positive integer with following properties

• (i)

$m\mid(g-1)$,

• (ii)

$mp^{s\theta}\equiv\epsilon\pmod{g}$, where $\epsilon=\pm 1$,

• (iii)

$m(g-m)=(g-1)p^{s(h-2\theta)}$,

and $\theta=\theta(g,p)$ is an integer defined by

[TABLE]

where $S_{p}(x)$ denotes the sum of the $p$-digits of $x$.

The last result in this section is taken from [11, Theorem 12].

Result 2.8**.**

Let $n$ be a positive integer and let $q$ be a prime power such that $(n,q)=1$. Let $C$ be a two-weight projective cyclic code of length $n$ over $\mathbb{F}_{q}$. Assume that $C$ is not an irreducible code. Then $C$ is the direct sum of two one-weight irreducible cyclic subcodes of the same dimension and of the same unique nonzero weight $w_{1}$. Moreover, all irreducible cyclic subcodes of $C$ have the same weight $w_{1}$.

3 Proof of Theorem 1.1

Lemma 3.1**.**

Define $f=((q^{k}-1)/(q-1),de)$. The number $B_{2}$ of words in the dual code $C^{\perp}$ of $C$ having weight $2$ is

[TABLE]

Moreover, the code $C$ is not a projective code.

Proof.

Note that there is no word in $C^{\perp}$ or weight $1$, as such a word induces a nonzeoro polynomial $ax^{m}$, $0\leq m\leq n-1$, which contains two zeros $\gamma^{d}$ and $\gamma^{d+D}$, impossible. Therefore, the code $C$ is projective if and only if $B_{2}\neq 0$.

The number of words in $C^{\perp}$ having weight $2$ is equal to the number of pairs $(a_{m},b_{m})\in\mathbb{F}_{q}^{*}\times\mathbb{F}_{q}$ such that $1\leq m\leq n-1$ and the polynomial $a_{m}x^{m}-b_{m}$ contains two zeros $\gamma^{d}$ and $\gamma^{d+D}$. Let $N$ be the number of integers $m$ such that $1\leq m\leq n-1$ and there exists a polynomial $x^{m}-c_{m}\in\mathbb{F}_{q}[x]$ which contains two zeros $\gamma^{d}$ and $\gamma^{d+D}$. By the linearity of $C$, we have

[TABLE]

Note that $x^{m}-c_{m}$ has zeros $\gamma^{d}$ and $\gamma^{d+D}$ if and only if $\gamma^{dm}=c_{m}\in\mathbb{F}_{q}^{*}$ and $\gamma^{Dm}=1$. Hence $(q^{k}-1)\mid Dm$ and $(q^{k}-1)/(q-1)\mid dm$. The first condition implies $e\mid m$. Put $d^{\prime}=(\frac{q^{k}-1}{q-1},d)$. The second condition implies $\frac{q^{k}-1}{(q-1)d^{\prime}}\mid m$. Thus $m$ is divisible by the following number

[TABLE]

where $f^{\prime}=(\frac{q^{k}-1}{(q-1)d^{\prime}},e)$. We have

[TABLE]

Therefore, $m$ is a multiple of $\frac{(q^{k}-1)e}{(q-1)f}=n\frac{de}{\lambda f(q-1)}$. The number $N$ of integers $1\leq m\leq n-1$ which has this property is $N=\lambda f(q-1)/(de)-1$. Combining with (8), we prove (7).

Now, assume that $C$ is projective. We have $B_{2}=0$, which implies

[TABLE]

By Result 2.8, the irreducible subcode $C_{d}$ of $C$ have a unique non-zero weight $w_{1}$. The identities $(1)$ and $(2)$ from Result 2.3 imply

[TABLE]

Note that none of words in the dual code $C_{d}^{\perp}$ of $C_{d}$ has weight $1$, as $\gamma^{d}$ cannot be zero of any nonzero polynomial $ax^{m}\in\mathbb{F}_{q}[x]$. Let $C_{2}$ be the number of words in $C_{d}^{\perp}$ having weight $2$. Let $M$ be the number of integers $r$ such that $1\leq r\leq n-1$ and there exists a polynomial $x^{r}-c_{r}\in\mathbb{F}_{q}[x]$ which contains a zero $\gamma^{d}$. By similar reasoning as before, we obtain $C_{2}=M(q-1)$ and $(q^{k}-1)/(q-1)\mid rd$. As $f=((q^{k}-1)/(q-1),de)=1$, we have $(q^{k}-1)/(q-1)\mid r$. The number of integers $1\leq r\leq n-1$ which is a multiple of $(q^{k}-1)/(q-1)=nd/(q-1)$ is $(q-1)/d-1$. Thus

[TABLE]

By the identity $(3)$ from Result 2.3, we obtain

[TABLE]

which implies $(q^{k}-1)(q-1)/d$ divides $2(q-1)((q-1)/d-1)$. This is possible only when $k=1$ and $(q-1)/d\mid 2$. We obtain $n=(q-1)/d<3$, a contradiction. ∎

Since $C_{d}$ and $C_{D}$ are subcodes of $C$, they have at most two weights. In the next lemma, we prove that they cannot be one-weight codes.

Lemma 3.2**.**

Under the same notations as above, suppose that the code $C$ is two-weight. Then both $C_{d}$ and $C_{D}$ are two-weight codes.

Proof.

We prove by contradiction. Suppose that either $C_{d}$ or $C_{D}$ is one-weight. Assume that is $C_{d}$. Note that there is no word in the dual code of $C_{d}$ having weight $1$. Let $w_{1}=\text{wt}(C_{d})$. By the equation $(2)$ of Result 2.3, we obtain $(q^{k}-1)w_{1}=n(q-1)q^{k-1}$. Hence

[TABLE]

Note that $w_{1}$ is also one weight of $C$. Next, we apply the MacWilliams identities again to find the other weight $w_{2}$ of $C$. Recall that $A_{1}$ and $A_{2}$ be the numbers of words in $C$ of weights $w_{1}$ and $w_{2}$. Moreover, the numbers $B_{1}$ and $B_{2}$ denote the numbers of words in $C^{\perp}$ of weights $1$ and $2$. Note that $B_{1}=0$ and the value of $B_{2}$ is given in (7). By Result 2.3, we have the following identities for the $[n,2k]$ cyclic code $C$.

• (1)

$A_{1}+A_{2}=q^{2k}-1$.

• (2)

$A_{1}w_{1}+A_{2}w_{2}=n(q-1)q^{2k-1}$.

• (3)

$A_{1}w_{1}^{2}+A_{2}w_{2}^{2}=\left(n^{2}(q-1)^{2}+n(q-1)+2\left(\frac{\lambda f(q-1)}{de}-1\right)(q-1)\right)q^{2k-2}$.

As $(A_{1}w_{1}+A_{2}w_{2})(w_{1}+w_{2})-(A_{1}+A_{2})w_{1}w_{2}=A_{1}w_{1}^{2}+A_{2}w_{2}^{2}$, we obtain

[TABLE]

Note that $w_{1}=\mu q^{k-1}$ with $\mu\mid(q-1)$, by (10). The equation (11) implies that $w_{2}=\alpha q^{k-1}$ for some $\alpha\in\mathbb{Z}^{+}$. In (11) using $(q^{k}-1)\mu/(q-1)=n$, we obtain

[TABLE]

which implies $n\mid(2\lambda f(q-1)/(de)-2)$. By Lemma 3.1, the number $2\lambda f(q-1)/(de)-2$ is nonzero, as $B_{2}\neq 0$. Thus

[TABLE]

as $f=((q^{k}-1)/(q-1),de)\leq de$. Since $d\leq(q-1)/e\leq(q-1)/2$, we have

[TABLE]

which implies $k=1$. In this case, we have $f=((q^{k}-1)/(q-1),de)=1$ and the inequality $n<2\lambda f(q-1)/(de)$ implies

[TABLE]

so $e\leq de<2$, a contradiction.

∎

Proof of Theorem 1.1

Proof.

We prove by contradiction. Suppose that $C$ is two-weight. Let $w_{1}$ and $w_{2}$ denote the two nonzero weights of $C$. By Lemma 3.2, both $C_{d}$ and $C_{D}$ are also two-weight. The equation (11) implies that $q^{2k-2}\mid w_{1}w_{2}$. We show that the values of $w_{1}$ and $w_{2}$ defined in (6) cannot satisfy this condition. Recall that

[TABLE]

where $\epsilon=\pm 1$ and $m$ is a positive integer with following properties

• (i)

$m\mid(g-1)$,

• (ii)

$mp^{s\theta}\equiv\epsilon\pmod{g}$, where $\epsilon=\pm 1$,

• (iii)

$m(g-m)=(g-1)p^{s(h-2\theta)}$,

and $\theta=\theta(g,p)$ is defined by

[TABLE]

Since $q^{2k-2}\mid w_{1}w_{2}$, we have $q^{2k}=p^{2kt}\mid p^{2s\theta}(p^{s(h-\theta)}-\epsilon m)(p^{s(h-\theta)}-\epsilon m+\epsilon g)$. Note that $kt=sh$, so $p^{2s(h-\theta)}$ divides $(p^{s(h-\theta)}-\epsilon m)(p^{s(h-\theta)}-\epsilon m+\epsilon g)$. The difference between $(p^{s(h-\theta)}-\epsilon m+\epsilon g)$ and $(p^{s(h-\theta)}-\epsilon m)$ is $\epsilon g$, a divisor of $(q-1)$ and not divisible by $p$. Thus, only one of the numbers $(p^{s(h-\theta)}-\epsilon m)$ or $(p^{s(h-\theta)}+\epsilon(g-m))$ is divisible by $p^{2s(h-\theta)}$.

Case 1. $(p^{s(h-\theta)}-\epsilon m)$ is divisible by $p^{2s(h-\theta)}$.

Write $m=ap^{s(h-\theta)},a\in\mathbb{Z}^{+}$. By (iii), we have $g-1=ap^{s\theta}(g-m)$. Note that $m\mid(g-1)$ and $p^{s\theta}\geq p\geq 2$, so $m=g-1$ and $g=1+ap^{s\theta}$. The equation (iii) again implies $h=2\theta$. Note that $h=\text{ord}_{g}(p)$, so $g=1+ap^{s\theta}$ divides $p^{h}-1=p^{2\theta}-1$. We obtain $s=1$ and $a=1$. The condition (ii) implies $\epsilon=1$. We obtain $p^{s(h-\theta)}-\epsilon m=0$ and thus $w_{1}=0$, a contradiction.

Case 2. $(p^{s(h-\theta)}+\epsilon(g-m))$ is divisible by $p^{2s(h-\theta)}$.

Write $g-m=(ap^{s(h-\theta)}-\epsilon)p^{s(h-\theta)},a\in\mathbb{Z}^{+}$. By (iii), we have

[TABLE]

Note that $g\mid(p^{h}-1)$ and $\theta\leq h-1$, so

[TABLE]

We obtain $a=m=s=\epsilon=1$ and $g-1=p^{h}-p^{\theta}$. Replacing $m=1$ into (iii), we obtain $g-1=(p^{h-\theta}-1)p^{h-\theta}$. Thus, $h=2\theta$. The condition (ii) implies $p^{\theta}\equiv 1\pmod{g}$, contradicting with $\text{ord}_{g}(p)=h=2\theta$. ∎