Some novel existence and uniqueness results for the Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions and their application

1.   Introduction

As a generalization of cyclic codes and negacyclic codes, constacyclic codes were first introduced by Berlekamp in 1968 ^[3]. Given a nonzero element $\lambda$ in a finite filed $\mathbb{F}_q$, a linear code $C$ of length $n$ over $\mathbb{F}_q$ is called $\lambda$-constacyclic if $(\lambda c_{n-1}, c_0, \cdots, c_{n-2})\in C$ for every $(c_0, c_1, \cdots, c_{n-1})\in C$. Constacyclic codes over finite fields form a remarkable class of linear codes, as it includes the class of cyclic codes and the class of negacyclic codes as proper subclasses. Constacyclic codes have rich algebraic structure so that they can be efficiently encoded and decoded by means of shift registers. Repeated-root constacyclic codes were a special class of constacyclic codes. Repeated-root constacyclic codes were first studied by Castagnoli et al. ^[4] and van Lint ^[13], and they showed that repeated-root cyclic codes have a concatenated construction and are not asymptotically good.

Recently, repeated-root constacyclic codes have been studied by many authors. To determine the generator polynomials of all constacyclic codes of arbitrary length over finite fields is an important problem. Dinh studied repeated-root constacyclic codes of lengths $2p^s$, $3p^s$, $4p^s$ and $6p^s$ in a series of papers ^[8,9,10,11]. He determined the algebraic structure of these repeated-root constacyclic codes over finite fields in terms of their generator polynomials. In ^[7], Chen et al. introduced an equivalence relation called isometry for the nonzero elements of $\mathbb{F}_q$ to classify constacyclic codes of length $n$ over $\mathbb{F}_q$. They have the same distance structures and the same algebraic structures for belonging to the same equivalence classes induced by isometry. Furthermore, in ^[5], Chen et al. considered a more specified relationship than isometry that enabled us to obtain more explicit description of generator polynomials of all constacyclic codes. According to the equivalence classes, all constacyclic codes of length $\ell p^s$ over $\mathbb{F}_{q^m}$ and their dual are characterized, where $\ell$ is a prime different from $p$ and $s$ is a positive integer. In 2012, Bakshi and Raka ^[1] also determined all $\Lambda$-constacyclic codes of length $2^tp^s$($t\geq 1, s\geq 0$ are integers) over $\mathbb{F}_{p^r}$ using different methods from Chen et al.. In 2015, Chen et al. ^[6] determined the algebraic structure of all constacyclic codes of length $2\ell^m p^s$ over $\mathbb{F}_{p^r}$ and their dual codes in terms of their generator polynomials, where $\ell, p$ are distinct odd primes and $s, m$ are positive integers. In the conclusion of the paper ^[6], they proposed an open problem to study all constacyclic codes of length $k\ell^mp^s$ over $\mathbb{F}_q$, where $p$ is the characteristic of $\mathbb{F}_q$, $\ell$ is an odd prime different from $p$, and $k$ is a prime different from $\ell$ and $p$. Batoul et al. ^[2] investigated the structure of constacyclic codes of length $2^amp^r$ over $\mathbb{F}_{p^s}$ with $a\geq 1$ and $(m, p) = 1$. They also provided certain sufficient conditions under which these codes are equivalent to cyclic codes of length $2^amp^r$ over $\mathbb{F}_{p^s}$. Sharma ^[16] determined all constacyclic codes of length $\ell^t p^s$ over $\mathbb{F}_{p^r}$ and their dual codes, where $\ell, p$ are distinct primes, $\ell$ is odd and $s, t, r$ are positive integers. In 2016, Sharma et al. ^[17] determine generator polynomials of all constacyclic codes of length $4\ell^mp^n$ over the finite field $\mathbb{F}_q$ and their dual codes, where $p, \ell$ are distinct odd primes, $q$ is a power of $p$ and $m, n$ are positive integers. Working in the same direction, Liu et al. obtained generator polynomials of all repeated-root constacyclic codes of length $3\ell p^s$ over $\mathbb{F}_q$ in ^[14], where $\ell$ is an odd prime different from $p$ and 3. In 2017, Liu et al. ^[15] explicitly determine the generator polynomials of all repeated-root constacyclic codes of length $n\ell p^s$ over $\mathbb{F}_q$ and their dual codes, where $\ell$ is an odd prime different from $p$, and $n$ is an odd prime different from both $\ell$ and $p$ such that $n = 2h + 1$ for some prime $h$. In 2019, Wu and Yue et al. ^[19,20] explicitly factorize the polynomial $x^n-\lambda$ for each $\lambda\in\mathbb{F}_q$. As applications, they obtain all repeated-root $\lambda$-constacyclic codes and their dual codes of length $np^s$ over $\mathbb{F}_q$.

In this paper, we answer the question of B. Chen, H. Dinh and Liu. That is we determine all the constacyclic codes of length $p_1p_2^tp^s$ over $\mathbb{F}_q$, where $p$ is the characteristic of $\mathbb{F}_q$, $p_1$ is an odd prime different from $p$, and $p_1$ is a prime different from $p_2$ and $p$. We give the explicit generator polynomials of all the constacyclic codes of length $p_{1}p_{2}^{t}p^{s}$ over $\mathbb{F}_q$ and their dual codes, and determine all self-dual cyclic codes of length $p_1p_2^t p^s$ and their enumeration.

The remainder of this paper is organized as follows. In Section 2 we give a brief background on some basic results which we need in the following parts. In Section 3, we calculate the $q$-cyclotomic cosets modulo $p_{1}p_{2}^{t}$ as a preparation for giving the generator polynomials of constacyclic codes of length $p_1p_2^tp^s$ over $\mathbb{F}_q$. In Section 4, we first describe a general method to obtain the generator polynomials of constacyclic codes, and then with this method and the results of $q$-cyclotomic cosets modulo $p_{1}p_{2}^{t}$ we give the explicit generator polynomials of all the constacyclic codes of length $p_1p_2^tp^s$. And in Section 5, all the self-dual cyclic codes of length $p_1p_2^tp^s$ over $\mathbb{F}_q$ are given. In the last section, as an example we calculate the case of length $5\ell p^{s}$, where $\ell$ is a prime different from $5$ and $p$.

2.   Preliminaries

In this section, we first review some basic results in number theory and finite fields, which we will in the following parts, and then give a brief introduction to the $\lambda$-constacyclic codes. For a positive integer $n$, we denote by $\mathbb{Z}_{n}$ the ring of integers module $n$ throughout this paper. Let $p$ be a prime number, and $q$ be a power of $p$. We denote by $\mathbb{F}_q$ the finite field with $q$ elements, and fix a generator element $\xi$ of the multiplicative group $\mathbb{F}_{q}^{*}$, that is, $\mathbb{F}_{q}^{*} = \langle\xi\rangle$. In this paper, we mainly deal with the repeated-root constacyclic codes of length $p_{1}p_{2}^{t}p^{s}$ over $\mathbb{F}_q$, where $p_{1}$ and $p_{2}$ are two distinct odd prime numbers different from $p$. For any positive integer $d$ and $i = 1, 2$, we write $f_{i, d} = \mathrm{ord}_{p_{i}^{d}}(q)$ for the multiplicative order of $q$ modulo $p_{i}^{d}$, and set $g_{i, d} = \dfrac{\phi(p_{i}^{d})}{f_{i, d}}$, where $\phi$ is the Euler's phi function. When $d = 1$, we write $f_{i} = f_{i, 1}$ and $g_{i} = g_{i, 1}$ for simplicity. For $i = 1, 2$, there are positive integers $u_{i}$ and $w_{i}$ such that $q^{f_{i}} = 1+p_{i}^{u_{i}}w_{i}$ and $p_{i} \nmid w_{i}$. Following the lifting-the-exponent lemma, we immediately have

Lemma 2.1. ^[12] Assume that $r$ is a primitive root of the odd prime $p$ and $(r+tp)^{p-1}$ is not congruent to $1$ modulo $p^{2}$. Then $r+tp$ is a primitive root of $p^{k}$ for each $k\geq 1$.

Lemma 2.2. ^[18] Let $n\geq 2$ be an integer, and $\lambda$ be a nonzero element in $\mathbb{F}_q$ with multiplicative order $k = \mathrm{ord}(\lambda)$. The binomial $x^{n}-\lambda$ is irreducible over $\mathbb{F}_q$ if and only if

(1) Every prime divisor of $n$ divides $k$, but not $\frac{q-1}{k}$;

(2) If $4\mid n$, then $4\mid (q-1)$.

Let $\lambda$ be a nonzero element in $\mathbb{F}_q$. A $\lambda$-constacyclic code of length $n$ is a linear code $C$ such that $(c_{0}, c_{1}, \cdots, c_{n-1})\in C$ implies $(\lambda c_{n-1}, c_{0}, \cdots, c_{n-2})\in C$. This definition is a natural generalization of cyclic code and negacyclic code. A $\lambda$-constacyclic code $C$ of length $n$ over $\mathbb{F}_q$ can be regarded as an ideal $(g(x))$ of the quotient ring $\mathbb{F}_q[x]/(x^{n}-\lambda)$, where $g(x)$ is a divisor of $x^{n}-\lambda$. Let $C$ be a $\lambda$-constacyclic code of length $n$ over $\mathbb{F}_q$, then the dual code of code $C$ is given by $C^{\bot} = \{x\in \mathbb{F}_{q}^{n}: x\cdot y = 0, \forall y\in C\}$, where $x\cdot y$ denotes the Euclidean inner product of $x$ and $y$. If $C$ is generated by a polynomial $g(x)$ satisfying $g(x)\mid x^{n}-\lambda$, and $h(x)$ is given by $h(x) = \frac{x^{n}-\lambda}{g(x)}$, then $h(x)$ is called the parity check polynomial of code $C$. It is a classical result that the dual code $C^{\bot}$ is generated by $h(x)^{*}$, where $h(x)^{*} = h(0)^{-1}x^{deg(h(x))}h(x^{-1})$ is the reciprocal polynomial of $h(x)$. The code $C$ is called to be a self-orthogonal if $C\subseteq C^{\bot}$ and a self-dual code if $C = C^{\bot}$. For self-dual cyclic code, a well-known result states that there exist self-dual cyclic codes of length $n$ over $\mathbb{F}_q$ if and only if $n$ is even and the characteristic of $\mathbb{F}_q$ is $p = 2$.

There are $q-1$ classes of constacyclic codes of length $n$ over $\mathbb{F}_q$. However, some of them are turned out to be equivalent in the sense that they have the same structure. To be explicit, two elements $\lambda, \mu\in \mathbb{F}_{q}^{*}$ are called $n$-equivalent in $\mathbb{F}_{q}^{*}$if there exists $a \in \mathbb{F}_{q}^{*}$ such that $a^{n}\lambda = \mu$.

Lemma 2.3. ^[5] For any $\lambda, \mu\in \mathbb{F}_{q}^*$, the following four statements are equivalent:

(1) $\lambda$ and $\mu$ are $n$-equivalent in $\mathbb{F}_{q}^*$.

(2) $\lambda^{-1}\mu\in \langle\xi^{n}\rangle$.

(3) $(\lambda^{-1}\mu)^{d} = 1$, where $d = \frac{q-1}{gcd(n, q-1)}$.

(4) There exists an $a\in \mathbb{F}_{q}^*$ such that

is an $\mathbb{F}_q$-algebra isomorphism. In particular, there are $\gcd(n, q-1)$ $n$-equivalence classes in $\mathbb{F}_{q}^{*}$.

We conclude this section with the introduction of $q$-cyclotomic coset which is important in the computation of constacyclic codes. Let $n$ be a positive integer relatively prime to $n$. For $0\leq s\leq n-1$, the $q$-cyclotomic coset of $s$ modulo $n$ is defined to be

where $n_{s}$ is the least positive integer such that $sq^{n_{s}}\equiv s \pmod n$. It is obvious to see that $n_{s}$ is equal to the multiplicative order of $q$ modulo $\frac{n}{\gcd(s, n)}$. Notice that if $sq^{a} \equiv s^{\prime}q^{b} \pmod n$ for some positive integers $a, b$, then

It follows that for $0 \leq s, s^{\prime} \leq n-1$, $C_{s} \cap C_{s^{\prime}} \neq \emptyset$ if and only if $C_{s} = C_{s^{\prime}}$. Therefore the $q$-cyclotomic cosets give a classification of the element in $\mathbb{Z}_{n}$.

If $\alpha$ is a primitive $n$th root of unit in some extension field of $\mathbb{F}_q$, then the polynomial

is exactly the minimal polynomial of $\alpha^{s}$ over $\mathbb{F}_q$, and

gives the irreducible factorization of $x^{n}-1$ over $\mathbb{F}_q$, where $s$ runs over all representations of distinct $q$-cyclotomic cosets modulo $n$. We call $C_{s}(x)$ the polynomial associated to $C_{s}$.

Let $C_{s} = \{s, sq, \cdots, sq^{n_{s}-1}\}$ be any $q$-cyclotomic coset modulo $n$. The reciprocal coset of $C_{s}$ is defined to be

We say that the coset $C_{s}$ is self-reciprocal if $C_{s} = C_{s}^{*}$. One can check that the polynomial $C_{s}^{*}(x)$ associated to the reciprocal coset $C_{s}^{*}$ is exactly the reciprocal polynomial of $C_{s}(x)$.

3.   $q$-cyclotomic cosets modulo $p_{1}^{t_{1}}p_{2}^{t_{2}}$

The $q$-cyclotomic cosets modulo $p_{1}p_{2}^{t}$ plays an important role in determining all the constacyclic codes of length $p_{1}p_{2}^{t}p^{s}$. In this section we consider a more general case that classifies all the $q$-cyclotomic cosets modulo $p_{1}^{t_{1}}p_{2}^{t_{2}}$, where $p_{1}$ and $p_{2}$ are two distinct odd prime numbers not dividing $q$, and $t_{1}, t_{2}$ are positive integers.

Let $\ell$ be a prime number not dividing $q$, and $\mu$ be a generator of the cyclic group $\mathbb{Z}_{\ell}^{*}$. It is obvious that all the $q$-cyclotomic cosets modulo $\ell$ are given by $C_{0} = \{0\}$ and

For different odd prime numbers $p_{1}$ and $p_{2}$, we claim that there exists an integer $\mu_{1}$ satisfying that:

(1) $\mu_{1}$ is a primitive root modulo $p_{1}^{d}$ for all $d\geq 1$; and

(2) $\mu_{1}\equiv 1 \pmod {p_{2}}$.

We begin with a random primitive root $\eta_{1}^{'}$ modulo $p_{1}$. If $p_{1}^{2}\nmid {\eta_{1}^{'}}^{p_{1}-1}-1$, we let $\eta_{1} = \eta_{1}^{'}$, otherwise we let $\eta_{1} = \eta_{1}^{'}+p_{1}$. It is trivial to see that $\eta_{1}$ satisfies the condition $\gcd(\frac{\eta_{1}^{p_{1}-1}-1}{p_{1}}, p_{1}) = 1$. Let $\mu_{1} = \eta_{1}+(1-\eta_{1})p_{1}^{p_{2}-1}$, then

It follows that

Following Lemma 2.1, $\mu_{1}$ is a primitive root modulo $p_{1}^{d}$ for all $d\geq 1$ such that $\mu_{1}\equiv 1 \pmod {p_{2}}$. By the symmetric argument, we can find an integer $\mu_{2}$ satisfying that

(1) $\mu_{2}$ is a primitive root modulo $p_{2}^{d}$ for all $d\geq 1$; and

(2) $\mu_{2}\equiv 1 \pmod {p_{1}}$.

We fix such a pair of integers $\mu_{1}$ and $\mu_{2}$.

Theorem 3.1. Let $p_{1}$ and $p_{2}$ be two different odd prime numbers not dividing $q$, and $t_{1}$ and $t_{2}$ be positive integers. Then all the distinct $q$-cyclotomic cosets module $p_{1}^{t_{1}}p_{2}^{t_{2}}$ are given by

for $0 \leq r_{1} \leq t_{1}$, $0\leq r_{2} \leq t_{2}$, $0 \leq k_{1} \leq g_{1, t_{1}-r_{1}} - 1$ and $0 \leq k_{2} \leq g_{2, t_{2}-r_{2}}\cdot \gcd(f_{1, t_{1}-r_{1}}, f_{2, t_{2}-r_{2}}) - 1$, where $c_{r_{1}, r_{2}} = \mathrm{ord}_{p_{1}^{t_{1}-r_{1}}p_{2}^{t_{2}-r_{2}}}(q) = \mathrm{lcm}(f_{1, t_{1}-r_{1}}, f_{2, t_{2}-r_{2}})$.

Proof. First we prove that the given $q$-cyclotomic cosets are all distinct. If $C_{\mu_{1}^{k_{1}}\mu_{2}^{k_{2}}p_{1}^{r_{1}}p_{2}^{r_{2}}} = C_{\mu_{1}^{k_{1}^{\prime}}\mu_{2}^{k_{2}^{\prime}}p_{1}^{r_{1}^{\prime}}p_{2}^{r_{2}^{\prime}}}$ for some $0 \leq r_{1}, r_{1}^{\prime} \leq t_{1}$, $0\leq r_{2}, r_{2}^{\prime} \leq t_{2}$, $0 \leq k_{1}, k_{1}^{\prime} \leq g_{1, t_{1}-r_{1}} - 1$ and $0 \leq k_{2}, k_{2}^{\prime} \leq$$g_{2, t_{2}-r_{2}}\cdot \gcd(f_{1, t_{1}-r_{1}}, f_{2, t_{2}-r_{2}})-1$, then there exists a positive integer $m$ such that

Since $\mu_{1}, \mu_{2}$ and $q$ are relatively prime to $p_{1}^{t_{1}}p_{2}^{t_{2}}$, clearly we have $r_{1} = r_{1}^{\prime}$ and $r_{2} = r_{2}^{\prime}$, and Eq (3.1) can be reduced to

Remembering that $\mu_{1} \equiv 1 \pmod{p_{2}}$ and $\mu_{2} \equiv 1 \pmod{p_{1}}$, then by the Chinese remainder theorem, we have

Equation (3.2) implies that

and therefore $\phi(p_{1}^{t_{1}-r_{1}}) \mid (k_{1}-k_{1}^{\prime})f_{1, t_{1}-r_{1}}$. Since $0 \leq k_{1}, k_{1}^{\prime} \leq g_{1, t_{1}-r_{1}} - 1$, one must have $k_{1} = k_{1}^{\prime}$. Notice that $k_{1} = k_{1}^{\prime}$ indicates that $q^{m} \equiv 1 \pmod{p_{1}^{t_{1}-r_{1}}}$, then $f_{1, t_{1}-r_{1}} \mid m$, which together with Eq (3.3) leads to

Thus $\phi(p_{2}^{t_{2}-r_{2}}) \mid (k_{2}^{\prime}-k_{2})\cdot \dfrac{f_{2, t_{2}-r_{2}}}{\gcd(f_{1, t_{1}-r_{1}}, f_{2, t_{2}-r_{2}})}$. Since $0 \leq k_{2}, k_{2}^{\prime} \leq g_{2, t_{2}-r_{2}}\cdot \gcd(f_{1, t_{1}-r_{1}}, f_{2, t_{2}-r_{2}}) - 1$, we have $k_{2} = k_{2}^{\prime}$.

On the other hand, there are in total

elements in these $q$-cyclotomic cosets, therefore they are all the distinct $q$-cyclotomic cosets module $p_{1}^{t_{1}}p_{2}^{t_{2}}$.

In particular, when $t_{1} = 1$ and $t_{2} = t$, the classification of the $q$-cyclotomic cosets modulo $p_{1}p_{2}^{t}$ is given as follow.

Corollary 3.1. Let the notations be as above. Then all the distinct $q$-cyclotomic cosets modulo $p_{1}p_{2}^{t}$ are

for $0 \leq r \leq t-1$, $0\leq k_{1}\leq g_{1} -1$ and $0 \leq k_{2} \leq g_{2, t-r} \cdot \gcd(f_{1}, f_{2, t-r})$;

for $0\leq k\leq g_{1}-1$; and

for $0\leq r \leq t-1$ and $0\leq k^{\prime} \leq g_{2, t-r}-1$.

Corollary 3.2. Let the notations be as aboved. Then the irreducible factorization of $x^{p_{1}p_{2}^{t}p^{s}}-1$ over $\mathbb{F}_q$ is given by

4.   Constacyclic codes of length $p_{1}p_{2}^{t}p^{s}$ with their dual codes

In this section, we will determine the generator polynomials of all constacyclic codes of length $p_{1}p_{2}^{t}p^{s}$ over $\mathbb{F}_q$ and their dual codes. For $\lambda \in \mathbb{F}_{q}^{*}$, we identify a $\lambda$-constacyclic code of length $p_{1}p_{2}^{t}p^{s}$ with an ideal $(g(x))$ of the quotient ring $\mathbb{F}_q[x]/(x^{p_{1}p_{2}^{t}p^{s}}-\lambda)$, where $g(x)$ is a divisor of $x^{p_{1}p_{2}^{t}p^{s}}-\lambda$. By Lemma 2.3, there are $\gcd(p_{1}p_{2}^{t}, q-1)$ $p_{1}p_{2}^{t}p^{s}$-equivalence classes in $\mathbb{F}_{q}^{*}$, which corresponds to the cosets of $\langle \xi^{p_{1}p_{2}^{t} } \rangle$ in $\mathbb{F}_{q}^{*} = \langle \xi \rangle$.

Before doing the explicit computation, we present a general method to factorize $x^{n} - \lambda$. Let $q = p^{k}$ for $k > 0$, and $n = p^{e}p_{1}^{e_{1}} \cdots p_{m}^{e_{m}}$ be the prime factorization of $n$. Assume that $p_{1}^{e_{1}} \cdots p_{m}^{e_{m}} \mid q-1$, i.e., $v_{p_{i}}(q-1) \geq e_{i}$ for $i = 1, \cdots, m$. In this case we have

For $\lambda\in \langle\xi^{p_{1}^{e_{1}}\cdots p_{m}^{e_{m}}}\rangle\xi^{j\cdot p^{e}}$, where $0\leq j\leq p_{1}^{e_{1}}\cdots p_{m}^{e_{m}}-1$, there exists an element $a\in \mathbb{F}_{q}^*$ such that

We first factorize $x^{n} - \xi^{jp^{e}}$, $0 \leq j \leq p_{1}^{e_{1}}\cdots p_{m}^{e_{m}}-1$. Notice that $j$ can be written as $j = y\cdot p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}$, where $v_{i} = min\{e_{i}, v_{p_{i}}(j)\}$. Then we have

Since $p_{1}^{v_{1}}\cdots p_{m}^{v_{m}} \mid q-1$, $\delta = \xi^{\frac{q-1}{p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}}}$ is a primitive $p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}$-th root of unit. Then

For $0\leq i\leq p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}-1$, $\delta^{i}\xi^{y} = \xi^{y+i\cdot \frac{q-1}{p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}}}$, and then we have

and

For each $p_{i}\mid p_{1}^{e_{1}-v_{1}}\cdots p_{m}^{e_{m}-v_{m}}$, we have that $e_{i} > v_{i}$ and $v_{i} = v_{p_{i}}(j)$, thus $p_{i}\nmid y$. Since $v_{p_{i}}(q-1)\geq e_{i} > v_{i}$, $p_{i}\mid \frac{q-1}{p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}}$, which indicates that $p_{i}\nmid y+i\cdot \frac{q-1}{p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}}$ and $p_{i}\mid \frac{q-1}{y+i\cdot \frac{q-1}{p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}}}$. Moreover if $4\mid p_{1}^{e_{1}-v_{1}}\cdots p_{m}^{e_{m}-v_{m}}$, then $4\mid p_{1}^{e_{1}}\cdots p_{m}^{e_{m}}\mid q-1$. Hence by Lemma 2.2 each $x^{p_{1}^{e_{1}-v_{1}}\cdots p_{m}^{e_{m}-v_{m}}}-\xi^{y}\delta^{i}$ is irreducible over $\mathbb{F}_q$.

Notice that $a^{n}\lambda = \xi^{jp^{e}}$, then the irreducible factorization of $x^{n} - \lambda$ follows immediately:

We summerize the above discussions into the following theorem.

Theorem 4.1. Let $p, p_{1}, \cdots, p_{m}$ be distinct prime numbers. Let $q = p^{k}$ and $n = p^{e}p_{1}^{e_{1}}\cdots p_{m}^{e_{m}}$, where $k, e, e_{1}, \cdots, e_{m}$ are positive integers. Suppose that for $1\leq i \leq m$, $v_{p_{i}}(q-1) \geq e_{i}$. Then for any $\lambda \in \mathbb{F}_{q}^{*}$, there exists an element $a \in \mathbb{F}_{q}^{*}$ such that $a^{n}\lambda = \xi^{jp^{e}}$, $0 \leq j \leq p_{1}^{e_{1}}\cdots p_{m}^{e_{m}}$. Furthermore, writing $j$ in the form $j = y\cdot p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}$, where $v_{i} = min\{e_{i}, v_{p_{i}}(j)\}$, then

gives the irreducible factorization of $x^{n}-\lambda$ over $\mathbb{F}_q$.

Now we turn to the case that $p_{1}^{e_{1}}\cdots p_{m}^{e_{m}}\nmid q-1$. Sinve $\gcd(p_{1}^{e_{1}}\cdots p_{m}^{e_{m}}, q) = 1$, thus there exists a least positive integer $d$ such that $p_{1}^{e_{1}}\cdots p_{m}^{e_{m}}\mid q^{d}-1$. By the lifting-the-exponent lemma, if $d'$ is the least positive integer such that $p_{1}\cdots p_{m}\mid q^{d'}-1$, then $d = d'p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}$, where $v_{i} = \max\{e_{i}-v_{p_{i}}(q^{d'}-1), 0\}$.

Let $\lambda$ be a nonzero element in $\mathbb{F}_{q}$. To obtain the irreducible factorization of $x^{n}-\lambda$ over $\mathbb{F}_q$, we first consider the factorization over $\mathbb{F}_{q^{d}}$. By Theorem 4.1, there exists $a \in \mathbb{F}_{q^{d}}$ such that $a^{n}\lambda = \zeta^{jp^{e}}$, $0 \leq j \leq p_{1}^{e_{1}}\cdots p_{m}^{e_{m}} -1$. Writing $j$ as $j = y\cdot p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}$, where $v_{i} = min\{e_{i}, v_{p_{i}}(j)\}$, then

gives the irreducible factorization of $x^{n}-\lambda$ over $\mathbb{F}_{q^{d}}$, where $\delta$ is a primitive $p_{1}^{v_{1}}\cdots p_{m}^{v_{m}}$-th root of unit. Hence each irreducible factor of $x^{n}-\lambda$ over $\mathbb{F}_q$ is of the form

where $z_{i}$ is the least positive integer such that $a^{-q^{z_{i}}p_{1}^{e_{1}-v_{1}}\cdots p_{m}^{e_{m}-v_{m}}}\delta^{i\cdot q^{z_{i}}}\zeta^{y\cdot q^{z_{i}}} = a^{-p_{1}^{e_{1}-v_{1}}\cdots p_{m}^{e_{m}-v_{m}}}\delta^{i}\zeta^{y}$.

Now we determine the generator polynomials of all constacyclic codes of length $p_{1}p_{2}^{t}p^{s}$ and their duals explicitly. We decompose the problem into three cases.

4.1.   $\gcd(q-1, p_{1}p_{2}^{t}p^{s})=1$

As $\gcd(q-1, p_{1}p_{2}^{t}p^{s}) = 1$, all constacyclic codes of length $p_{1}p_{2}^{t}p^{s}$ are equivalent to a cyclic code. By the factorization of $x^{p_{1}p_{2}^{t}p^{s}}-1$ given in Corollary 3.2, we have the following result. For any polynomial

we set $\widehat{F} = a_{n}^{-1}F$ to be the monic polynomial associated to $F$.

Proposition 4.1. Assume that $\gcd(q-1, p_{1}p_{2}^{t}p^{s}) = 1$. Then any nonzero element $\lambda$ in $\mathbb{F}_{q}$ is $p_{1}p_{2}^{t}p^{s}$-equivalent to 1, that is, there is an element $a \in \mathbb{F}_{q}^{*}$ such that $a^{p_{1}p_{2}^{t}p^{s}}\lambda = 1$. Furthermore, the irreducible factorization of $x^{p_{1}p_{2}^{t}p^{s}}-\lambda$ over $\mathbb{F}_q$ is given by

Therefore all the constacyclic codes of length $p_{1}p_{2}^{t}p^{s}$ are

where $0\leq u, v_{\mu_{1}^{k_{1}}\mu_{2}^{k_{2}}p_{2}^{r}}, w_{\mu_{1}^{k}p_{2}^{t}}, x_{\mu_{2}^{k^{\prime}}p_{1}p_{2}^{r}}\leq p^{s}$, with duals

4.2.   $\gcd(q-1, p_{1}p_{2}^{t}p^{s})=p_{1}p_{2}^{t}$

For this case, since $p_{1}p_{2}^{t}\vert q-1$, the following proposition follows straightly from Theorem 4.1.

Theorem 4.2. Assume that $\gcd(q-1, p_{1}p_{2}^{t}p^{s}) = p_{1}p_{2}^{t}$. Then for any $\lambda\in \mathbb{F}_{q}^*$, there exists an element $a\in \mathbb{F}_{q}^*$ such that $a^{p_{1}p_{2}^{t}p^{s}}\lambda = \xi^{j\cdot p^{s}}$, $0\leq j\leq p_{1}p_{2}^{t}-1$. Writing $j$ as $j = y\cdot p_{1}^{v_{1}}p_{2}^{v_{2}}$, where $v_{1} = \min\{1, v_{p_{1}}(j)\}$ and $v_{2} = \min\{t, v_{p_{2}}(j)\}$, then

gives the irreducible factorization of $x^{p_{1}p_{2}^{t}p^{s}}-\lambda$ over $\mathbb{F}_{q}$. Therefore all the $\lambda$-constacyclic codes of length $p_{1}p_{2}^{t}p^{s}$ and their dual codes are given by

and

where $0\leq u_{1}, u_{2}, \cdots, u_{n^{v_{1}}\ell^{v_{2}}}\leq p^{s}$.

4.3.   $\gcd(q-1, p_{1}p_{2}^{t}p^{s})=p_{2}^{r}$ for some $0 < r \leq t$

In this case, for any $d\geq 1$ we have $f_{2, d} = p_{2}^{max\{0, d-r\}}$, and $f = \mathrm{lcm}(f_{1}, f_{2, t})$ is the least positive integer such that $q^{f}\equiv 1 \pmod {p_1p_{2}^{t}}$. By the bais results of finite fields, there is a primitive element $\zeta$ in $\mathbb{F}_{q^{f}}^{*}$ such that $\xi = \zeta^{\frac{q^{f}-1}{q-1}} = \zeta^{1+q+\cdots+q^{f-1}}$. Then we have

and

By the assumption that $p_{1}p_{2}^{t}\mid q^{f}-1$ and $v_{p_{1}}(q-1) = 0$, $v_{p_{2}}(q-1) = r$, we have that $p_{1}p_{2}^{t-r}\mid (1+q+\cdots+q^{f-1})$. Therefore $\xi^{p_{2}^{r}} = \zeta^{p_{2}^{r}(1+q+\cdots+q^{f-1})}\in \langle\zeta^{p_{1}p_{2}^{t}}\rangle$. Furthermore, for $0\leq j\leq p_{2}^{r}-1$, there exists some $0\leq j^{'}\leq p_{1}p_{2}^{t}-1$ such that $jp^{s}(1+q+\cdots+q^{f-1})\equiv j^{'}p^{s} \pmod {p_{1}p_{2}^{t}}$, that is, $\xi^{jp^{s}}\in \langle\zeta^{p_{1}p_{2}^{t}}\rangle\zeta^{j^{'}p^{s}}$. Hence we have the following theorem.

Theorem 4.3. Assume that $\gcd(q-1, p_{1}p_{2}^{t}p^{s}) = p_{2}^{r}$, $0 < r \leq t$. For any $0\leq j\leq p_{2}^{r}-1$, there exists an element $a\in \mathbb{F}_{q^{f}}^{*}$ such that $a^{p_{1}p_{2}^{t}p^{s}}\xi^{j\cdot p^{s}} = \zeta^{j'\cdot p^{s}}$. Moreover, each irreducible factor of $x^{p_{1}p_{2}^{t}}-\xi^{j}$ over $\mathbb{F}_{q}$ is of the form

where $j' = y'p_{1}^{v_{1}}p_{2}^{v_{2}}$, $v_{1} = \min\{1, v_{p_{1}}(j')\}$, $v_{2} = \min\{t, v_{p_{2}}(j')\}$, and $z_{i}$ is the least positive integer such that $a^{-q^{z_{i}}p_{1}^{1-v_{1}}p_{2}^{t-v_{2}}}\delta^{iq^{z_{i}}}\zeta^{y'q^{z_{i}}} = a^{p_{1}^{1-v_{1}}p_{2}^{t-v_{2}}}\delta^{i}\zeta^{y'}$.

For any $0\leq i, i^{'}\leq p_{1}^{v_{1}}p_{2}^{v_{2}}-1$, we define a relation $\sim$ to be such that $i\sim i^{'}$ if and only if $a^{-q^{m}p_{1}^{1-v_{1}}p_{2}^{t-v_{2}}}\delta^{iq^{m}}\zeta^{y'q^{m}} = a^{p_{1}^{1-v_{1}}p_{2}^{t-v_{2}}}\delta^{i^{'}}\zeta^{y'}$ for some nonnegative integers $m$. It is obvious to see that $\sim$ is an equivalence relation. Assume that $S$ is a complete system of equivalence class representatives of $\{0, 1, \cdots, p_{1}^{v_{1}}p_{•}^{v_{2}}-1\}$ relative to this relation $\sim$. For any $i\in S$ we denote the irreducible polynomial

by $M_{i}(x)$. Then we have the following corollary.

Corollary 4.1. Assume that $\gcd(q-1, p_{1}p_{2}^{t}p^{s}) = p_{2}^{r}$. For any $0\leq j\leq p_{2}^{r}-1$, there exists an element $a\in \mathbb{F}_{q^{f}}^{*}$ such that $a^{p_{1}p_{2}^{t}p^{s}}\xi^{j\cdot p^{s}} = \zeta^{j'\cdot p^{s}}$. Then

gives the irreducible factorization of $x^{p_{1}p_{2}^{t}p^{s}}-\xi^{j p^{s}}$ over $\mathbb{F}_q$. Furthermore we have that

and

where $0\leq u_{i}\leq p^{s}$, for $i\in S$.

4.4.   $\gcd(q-1, p_{1}p_{2}^{t}p^{s})=p_{1}p_{2}^{r}$ for some $0 < r < t$

The same argument as in the last section can be applied in this situation, only noticing that the least positive integer $f$ such that $q^{f}\equiv 1 \pmod {p_1p_2^t}$ is $f = f_{2, t} = p_2^{\max\{0, t-r\}}$. We find a primitive element $\zeta$ in $\mathbb{F}_{q^{f}}^{*}$ such that $\xi = \zeta^{\frac{q^{f}-1}{q-1}} = \zeta^{1+q+\cdots+q^{f-1}}$, then

and

By the assumption that $p_{1}p_{2}^{t}\mid q^{f}-1$ and $v_{p_{2}}(q-1) = r$, we have that $p_{2}^{t-r}\mid (1+q+\cdots+q^{f-1})$, and $\xi^{p_{1}p_{2}^{r}} = \zeta^{p_{1}p_{2}^{r}(1+q+\cdots+q^{f-1})}\in \langle\zeta^{p_{1}p_{2}^{t}}\rangle$. Furthermore, for $0\leq j\leq p_{1}p_{2}^{r}-1$, there exists some $0\leq j^{'}\leq p_{1}p_{2}^{t}-1$ such that $jp^{s}(1+q+\cdots+q^{f-1})\equiv j^{'}p^{s} \pmod {p_1p_2^t}$, that is, $\xi^{jp^{s}}\in \langle\zeta^{p_{1}p_{2}^{t}}\rangle\zeta^{j^{'}p^{s}}$. Hence we have the following theorem.

Theorem 4.4. Assume that $\gcd(q-1, p_{1}p_{2}^{t}p^{s}) = p_{1}p_{2}^{r}$ for $0 < r < t$, then for any $0\leq j\leq p_{2}^{r}-1$, there exists an element $a\in \mathbb{F}_{q^{f}}^{*}$ such that $a^{p_{1}p_{2}^{t}p^{s}}\xi^{j\cdot p^{s}} = \zeta^{j'\cdot p^{s}}$. Moreover, each irreducible factor of $x^{p_{1}p_{2}^{t}}-\xi^{j}$ over $\mathbb{F}_{q}$ is of the form

where $j' = y'p_{1}^{v_{1}}p_{2}^{v_{2}}$, $v_{1} = \min\{1, v_{p_{1}}(j')\}$, $v_{2} = \min\{t, v_{p_{2}}(j')\}$, and $z_{i}$ is the least positive integer such that $a^{-q^{z_{i}}p_{1}^{1-v_{1}}p_{2}^{t-v_{2}}}\delta^{iq^{z_{i}}}\zeta^{y'q^{z_{i}}} = a^{p_{1}^{1-v_{1}}p_{2}^{t-v_{2}}}\delta^{i}\zeta^{y'}$.

For any $0\leq i, i^{'}\leq p_{1}^{v_{1}}p_{2}^{v_{2}}-1$, we define a relation $\sim$ to be such that $i\sim i^{'}$ if and only if $a^{-q^{m}p_{1}^{1-v_{1}}p_{2}^{t-v_{2}}}\delta^{iq^{m}}\zeta^{y'q^{m}} = a^{p_{1}^{1-v_{1}}p_{2}^{t-v_{2}}}\delta^{i^{'}}\zeta^{y'}$ for some nonnegative integers $m$. It is obvious to see that $\sim$ is an equivalence relation. Assume that $S$ is a complete system of equivalence class representatives of $\{0, 1, \cdots, p_{1}^{v_{1}}p_2^{v_{2}}-1\}$ relative to this relation $\sim$. For any $i\in S$ we denote the irreducible polynomial

by $M_{i}(x)$. Then we have the following corollary.

Corollary 4.2. Assume that $\gcd(q-1, p_{1}p_{2}^{t}p^{s}) = p_{1}p_{2}^{r}$ for $0 < r < t$. For any $0\leq j\leq p_{2}^{r}-1$, there exists an element $a\in \mathbb{F}_{q^{f}}^{*}$ such that $a^{p_{1}p_{2}^{t}p^{s}}\xi^{j\cdot p^{s}} = \zeta^{j'\cdot p^{s}}$. Then

gives the irreducible factorization of $x^{p_{1}p_{2}^{t}p^{s}}-\xi^{j p^{s}}$ over $\mathbb{F}_q$. Furthermore we have that

and

where $0\leq u_{i}\leq p^{s}$, for $i\in S$.

5.   All self-dual cyclic codes of length $p_1p_2^tp^{s}$ over $\mathbb{F}_q$

Based on the results in the last section, we now give all the self-dual cyclic codes of length $p_{1}p_{2}^{t}p^{s}$ over $\mathbb{F}_q$ and their enumeration. It is a well-known conclusion that self-dual cyclic codes of length $N$ over $\mathbb{F}_q$ exists if and only if $N$ is even and the characteristic of $\mathbb{F}_q$ is $2$. Therefore we only consider the case of self-dual cyclic codes of length $p_{1}p_{2}^{t}\cdot 2^{s}$ over $\mathbb{F}_{2^{k}}$.

Let $x^{p_{1}p_{2}^{t}2^{s}}-1 = (x^{p_{1}p_{2}^{t}}-1)^{2^{s}} = f_{1}(x)^{2^{s}}\cdots f_{n}(x)^{2^{s}}h_{1}(x)^{2^{s}}\cdots h_{m}(x)^{2^{s}}h_{1}^{*}(x)^{2^{s}}\cdots h_{m}^{*}(x)^{2^{s}}$ be the irreducible factorization of $x^{p_{1}p_{2}^{t}2^{s}}-1$ over $\mathbb{F}_q$, where each $f_{i}(x)$ is a monic irreducible self-reciprocal polynomial for $1\leq i\leq n$, and $h_{j}^{*}(x)$ is the reciprocal polynomial of $h_{j}(x)$ for each $1\leq j\leq m$. Now, given a cyclic code $C = (g(x))$ of length $p_{1}p_{2}^{t}2^{s}$, it can be written in the form

where $0\leq \tau_{i}, \delta_{j}, \sigma_{j}\leq 2^{s}$ for any $1\leq i\leq n$ and $1\leq j\leq m$. Then the reciprocal polynomial $h^{*}(x)$ of the parity check polynomial $h(x)$ of $C$ is

Therefore it is obvious to see that the following theorem holds.

Theorem 5.1. With the above notations, we have that $C$ is self-dual if and only if $2\tau_{i} = 2^{s}$ for $1\leq i\leq n$, and $\delta_{j}+\sigma_{j} = 2^{s}$ for $1\leq j\leq m$.

Recall the irreducible factorization of $x^{p_{1}p_{2}^{t}p^{s}}-1$ given in Corollary 3.2. Now we determine for each irreducible factor its reciprocal polynomial.

Lemma 5.1. Let the notations be defined as Corollary 3.1. Then one of the following holds.

(1) If both $f_{1}$ and $f_{2}$ are odd, then we have that

(2) If $f_{1}$ is odd and $f_{2}$ is even, then we have that

(3) If $f_{1}$ is even and $f_{2}$ is odd, then we have that

(4) If both $f_{1}$ and $f_{2}$ are even, then we have when $v_{2}(f_{1})\neq v_{2}(f_{2})$,

when $v_{2}(f_{1}) = v_{2}(f_{2})$,

Proof. First it is trivial that the reciprocal of $C_{0}$ is always itself. For $C_{\mu_{1}^{k}p_{2}^{t}}$, notice that $C_{\mu_{1}^{k}p_{2}^{t}}^{*} = C_{\mu_{1}^{k}p_{2}^{t}}$ if and only if the congruence equation $-\mu_{1}^{k}p_{2}^{t}\equiv -\mu_{1}^{k}p_{2}^{t}q^{x} \pmod {p_1p_{2}^{t}}$ is solvable. Since the equation is equivalent to $-1\equiv q^{x} \pmod {p_2^{r}}$, then the condition holds if and only if $f_{1} = \mathrm{ord}_{p_{1}}(q)$ is even. In the similar way we can check that $C_{\mu_{2}^{k^{\prime}}p_{1}p_{2}^{r}}^{*} = C_{\mu_{2}^{k^{\prime}}p_{1}p_{2}^{r}}$ if and only if $f_{2, t-r} = f_{2}p_{2}^{max\{0, t-r\}}$ is even. Notice that by assumption $p_{2}$ is odd, therefore the condition holds if and only if $f_{2}$ is even. For $C_{\mu_{1}^{k_{1}}\mu_{2}^{k_{2}}p_{2}^{r}}$, consider the congruence equation $-\mu_{1}^{k_{1}}\mu_{2}^{k_{2}}p_{2}^{r}\equiv \mu_{1}^{k_{1}}\mu_{2}^{k_{2}}p_{2}^{r}q^{x} \pmod {p_1p_{2}^{t}}$. It is equivalent to that $-1\equiv q^{x} \pmod {p_1}$ and $-1\equiv q^{x} \pmod {p_2^{t-r}}$ holds simultaneously. This requires not only both $f_{1}$ and $f_{2}$ are even, but also $\gcd(f_{1}, f_{2, t-r})\mid\dfrac{f_{1}-f_{2, t-r}}{2}$. And it is trivial to check that the last condition holds if and only if $v_{2}(f_{1}) = v_{2}(f_{2, t-r}) = v_{2}(f_{2})$.

Based on the above lemma, we now determine all the self-dual cyclic codes of length $p_{1}p_{2}^{t}$ and their enumeration.

Theorem 5.2.

(1) If both $f_{1}$ and $f_{2}$ are odd, then there exist $(2^{s}+1)^{\frac{p_{1}p_{2}^{t}-1}{2}}$ self-dual cyclic codes of length $p_{1}p_{2}^{t}$ over $\mathbb{F}_{2^{k}}$, which are given by

(2) If $f_{1}$ is odd and $f_{2}$ is even, then there exist $(2^{s}+1)^{\frac{p_{1}(p_{2}^{t}-1)}{2}}$ self-dual cyclic codes of length $p_{1}p_{2}^{t}$ over $\mathbb{F}_{2^{k}}$, which are given by

(3) If $f_{1}$ is even and $f_{2}$ is odd, then there exist $(2^{s}+1)^{\frac{p_{2}^{t}(p_{1}-1)}{2}}$ self-dual cyclic codes of length $p_{1}p_{2}^{t}$ over $\mathbb{F}_{2^{m}}$, which are given by

(4) If both $f_{1}$ and $f_{2}$ are even, then we have when $v_{2}(f_{1})\neq v_{2}(f_{2})$, there exist $(2^{s}+1)^{\frac{(p_{1}-1)(p_{2}^{t}-1)}{2}}$ self-dual cyclic codes of length $p_{1}p_{2}^{t}$ over $\mathbb{F}_{2^{m}}$, which are given by

When $v_{2}(f_{1}) = v_{2}(f_{2})$, there exist only one self-dual cyclic codes of length $p_{1}p_{2}^{t}$ over $\mathbb{F}_{2^{m}}$, which is given by

6.   Constacyclic codes of length $5\ell p^{s}$ over $\mathbb{F}_{q}$

In this section, we illustrate the above process with the example of constacyclic codes of length $5\ell p^{s}$, where $\ell$ is a prime number different from $5$ and $p$. We determine all the constacyclic codes of length $5\ell p^{s}$ and their dual codes over $\mathbb{F}_{q}$, and then all the self-dual codes of length $5\ell p^{s}$ are also given.

First we determine all the $q$-cyclotomic cosets modulo $5\ell$. Let $f = \mathrm{ord}_{\ell}(q)$, and $e = \dfrac{\ell-1}{f}$. Then we have:

(1) $\mathrm{ord}_{5\ell}(q) = f$, when $q\equiv 1 \pmod 5$.

(2) $\mathrm{ord}_{5\ell}(q) = f$, when $q\equiv 4 \pmod 5$ with $f$ even.

(3) $\mathrm{ord}_{5\ell}(q) = 2f$, when $q\equiv 4 \pmod 5$ with $f$ odd.

(4) $\mathrm{ord}_{5\ell}(q) = f$, when $q\equiv 2$ or $q\equiv 3 \pmod 5$ with $4\mid f$.

(5) $\mathrm{ord}_{5\ell}(q) = 2f$, when $q\equiv 2$ or $q\equiv 3 \pmod 5$ with $2\mid f$ but $4\nmid f$.

(6) $\mathrm{ord}_{5\ell}(q) = 4f$, when $q\equiv 2$ or $q\equiv 3 \pmod 5$ with $f$ odd.

As the discussion given in the Section 3, we can find a primitive root $\mu$ modulo $\ell^{t}$ for all $t\geq 1$ such that $\mu\equiv 1 \pmod 5$. The following lemma give more explicit formula for the $q$-cyclotomic cosets modulo $5\ell$.

Lemma 6.1.

(1) If $q\equiv 1 \pmod 5$, then we have that all the distinct $q$-cyclotomic cosets modulo $5\ell$ are given by $C_{0} = \{0\}$, $C_{\ell} = \{\ell\}$, $C_{2\ell} = \{2\ell\}$, $C_{-\ell} = \{-\ell\}$, $C_{-2\ell} = \{-2\ell\}$, and $C_{a\mu^{k}} = \{a\mu^{k}, a\mu^{k}q, \cdots, a\mu^{k}q^{f-1}\}$ for $a\in R = \{1, 2, -1, -2, 5\}$ and $0\leq k\leq e-1$.

(2) If $q\equiv 4 \pmod 5$ and $f$ is even, we have that all the distinct $q$-cyclotomic cosets modulo $5\ell$ are given by $C_{0} = \{0\}$, $C_{\ell} = \{\ell, \ell q\}$, $C_{2\ell} = \{2\ell, 2\ell q\}$, $C_{\mu^{k'}} = \{\mu^{k'}, \mu^{k'}q, \cdots, \mu^{k'}q^{f-1}\}$, $C_{2\mu^{k'}} = \{2\mu^{k'}, 2\mu^{k'}q, \cdots, 2\mu^{k'}q^{f-1}\}$ for $0\leq k'\leq 2e-1$, and $C_{5\mu^{k}} = \{5\mu^{k}, 5\mu^{k}q, \cdots, 5\mu^{k}q^{f-1}\}$ for $0\leq k\leq e-1$.

(3) If $q\equiv 4 \pmod 5$ and $f$ is odd, we have that all the distinct $q$-cyclotomic cosets modulo $5\ell$ are given by $C_{0} = \{0\}$, $C_{\ell} = \{\ell, \ell q\}$, $C_{2\ell} = \{2\ell, 2\ell q\}$, $C_{\mu^{k}} = \{\mu^{k}, \mu^{k}q, \cdots, \mu^{k}q^{2f-1}\}$, $C_{2\mu^{k}} = \{2\mu^{k}, 2\mu^{k}q, \cdots, 2\mu^{k}q^{2f-1}\}$, and $C_{5\mu^{k}} = \{5\mu^{k}, 5\mu^{k}q, \cdots, 5\mu^{k}q^{f-1}\}$ for $0\leq k\leq e-1$.

(4) If $q\equiv 2$ or $3 \pmod 5$ and $4\mid f$, we have that all the distinct $q$-cyclotomic cosets modulo $5\ell$ are given by $C_{0} = \{0\}$, $C_{\ell} = \{\ell, \ell q, \ell q^{2}, \ell q^{3}\}$, $C_{\mu^{k'}} = \{\mu_{k'}, \mu^{k'}q, \cdots, \mu^{k'}q^{f-1}\}$ for $0\leq k'\leq 4e-1$, and $C_{5\mu^{k}} = \{5\mu^{k}, 5\mu^{k}q, \cdots, 5\mu^{k}q^{f-1}\}$ for $0\leq k\leq e-1$.

(5) If $q\equiv 2$ or $3 \pmod 5$ and $2\mid f$ but $4\nmid f$, we have that all the distinct $q$-cyclotomic cosets modulo $5\ell$ are given by $C_{0} = \{0\}$, $C_{\ell} = \{\ell, \ell q, \ell q^{2}, lq^{3}\}$, $C_{\mu^{k'}} = \{\mu_{k'}, \mu^{k'}q, \cdots, \mu^{k'}q^{2f-1}\}$ for $0\leq k'\leq 2e-1$, and $C_{5\mu^{k}} = \{5\mu^{k}, 5\mu^{k}q, \cdots, 5\mu^{k}q^{f-1}\}$ for $0\leq k\leq e-1$.

(6) If $q\equiv 2$ or $3 \pmod 5$ and $f$ is odd, we have that all the distinct $q$-cyclotomic cosets modulo $5\ell$ are given by $C_{0} = \{0\}$, $C_{\ell} = \{\ell, \ell q, \ell q^{2}, \ell q^{3}\}$, $C_{\mu^{k}} = \{\mu_{k}, \mu^{k}q, \cdots, \mu^{k}q^{4f-1}\}$, and $C_{5\mu^{k}} = \{5\mu^{k}, 5\mu^{k}q, \cdots, 5\mu^{k}q^{f-1}\}$ for $0\leq k\leq e-1$.

Proof. The methods to prove the above 6 situations are similar, and we will give the proof of the second situation as a instance. First since $\mu$ is a fixed primitive root modulo $l$ such that $\mu\equiv 1 \pmod 5$, it is trivial to verify that $C_{0}$, $C_{\ell}$, $C_{2\ell}$, $C_{\mu^{k'}}$, $C_{2\mu^{k'}}$ for $0\leq k'\leq 2e-1$ and $C_{5\mu^{k}}$ for $0\leq k\leq e-1$ are $q$-cyclotomic cosets modulo $5\ell$. And then we claim that all these cosets are all distinct. If we have that $a_{1}\mu^{k_{1}}\equiv a_{2}\mu^{k_{2}}q^{j}$, where $a_{1}$, $a_{2}$, $k_{1}$, $k_{2}$ and $j$ satisfy the definitions in (2). Since

we have that either $a_{1} = a_{2}$ or $a_{1}\neq a_{2}$ and both $a_{1}$ and $a_{2}$ are not equal to $5$. We divide the proof into 2 subcases.

Subcase 1. If $a_{1} = a_{2}$, we have that $\mu^{k_{1}-k_{2}}\equiv q^{j}\pmod {\ell}$ and $\mu^{(k_{1}-k_{2})f}\equiv 1\pmod {\ell}$, therefore $\phi(\ell)\mid (k_{1}-k_{2})f$ and $\dfrac{\phi(\ell)}{f}\mid (k_{1}-k_{2})$, which indicates that $k_{1} = k_{2}$.

Subcase 2. If $a_{1}\neq a_{2}$ and none of them is equal to $5$, we have that $a_{1}a_{2}^{-1}\equiv \mu^{k_{2}-k_{1}}q^{j}\pmod {5\ell}$, but notice that $a_{1}a_{2}^{-1}\equiv \pm2 \pmod 5$ and $\mu^{k_{2}-k_{1}}q^{j}\equiv \pm1 \pmod 5$, which is a contradiction. Hence the given cosets are all distinct, and we only need to prove they are all the $q$-cyclotomic cosets to complete the proof.

Notice that

Therefore the conclusion holds.

Theorem 6.1. The irreducible factorization of $x^{5\ell }-1$ over $\mathbb{F}_q$ is given as follows.

(1) If $q\equiv 1 \pmod 5$, then

where $R = {1, 2, 3, 4, 5}$.

(2) If $q\equiv 4 \pmod 5$ and $f$ is even, then

(3) If $q\equiv 4 \pmod 5$ and $f$ is odd, then

(4) If $q\equiv 2$ or $3 \pmod 5$ and $4\mid f$, then

(5) If $q\equiv 2$ or $3 \pmod 5$ and $2\mid f$ but $4\nmid f$, then

(6) If $q\equiv 2$ or $3 \pmod 5$ and $f$ is odd, then

With the irreducible factorization of $x^{5\ell }-1$, we can straightly follow the process given in Section 4 to calculate all the constacyclic codes of length $5\ell p^{s}$ over $\mathbb{F}_{q}$. We list the result as follow.

Theorem 6.2. Assume that $\gcd(q-1, 5\ell p^{s}) = 1$, then $\lambda$-constacyclic codes $C$ of length $5\ell p^{s}$ over $\mathbb{F}_q$ are equivalent to the cyclic codes, i.e., for any $\lambda\in \mathbb{F}_{q}^*$, there exists a unique element $a\in \mathbb{F}_{q}^*$ such that $a^{5\ell p^{s}}\lambda = 1$. Furthermore, the irreducible factorization of $x^{5\ell p^{s}}-\lambda$ over $\mathbb{F}_q$ is given by

(1) If $q\equiv 4 \pmod 5$ and $f$ is even, then

Therefore we have that

and

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \tau_{k^{'}}, \nu_{k^{'}}, \rho_{k}\leq p^{s}$, for any $k^{'} = 0, 1, \cdots, 2e-1$, and $k = 0, 1, \cdots, e-1$.

(2) If $q\equiv 4 \pmod 5$ and $f$ is odd, then

Therefore we have that

and

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \tau_{k}, \nu_{k}, \rho_{k}\leq p^{s}$, for $k = 0, 1, \cdots, e-1$.

(3) If $q\equiv 2$ or $3 \pmod 5$ and $4\mid f$, then

Therefore we have that

and

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \tau_{k^{'}}, \nu_{k}\leq p^{s}$, for $k^{'} = 0, 1, \cdots, 4e-1$, and $k = 0, 1, \cdots, e-1$.

(4) If $q\equiv 2$ or $3 \pmod 5$ and $2\mid f$ but $4\nmid f$, then

Therefore we have that

and

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \tau_{k^{'}}, \nu_{k}\leq p^{s}$, for $k^{'} = 0, 1, \cdots, 2e-1$, and $k = 0, 1, \cdots, e-1$.

(5) If $q\equiv 2$ or $3 \pmod 5$ and $f$ is odd, then

Therefore we have that

and

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \tau_{k}, \nu_{k}\leq p^{s}$, for $k = 0, 1, \cdots, e-1$.

Theorem 6.3. Assume that $\gcd(q-1, 5\ell p^{s}) = 5\ell$, then $\mathbb{F}_{q}^* = \langle\xi\rangle = \langle\xi^{5\ell }\rangle\cup \langle\xi^{5\ell }\rangle\xi^{p^{s}}\cup \cdots\cup \langle\xi^{5\ell }\rangle\xi^{p^{s}(5\ell -1)}$. For any $\lambda\in \mathbb{F}_{q}^*$, there exists an element $a\in \mathbb{F}_{q}^*$ such that $a^{5\ell p^{s}}\lambda = \xi^{j\cdot p^{s}}$, where $0\leq j\leq 5\ell -1$. Then $j$ can be written as $j = y\cdot 5^{v_{1}}\ell^{v_{2}}$, where $v_{1} = \min\{1, v_{5}(j)\}$ and $v_{2} = \min\{1, v_{\ell}(j)\}$. And

gives the irreducible factorization of $x^{5\ell p^{s}}-\lambda$ over $\mathbb{F}_{q}$. Moreover, all the $\lambda$-constacyclic codes of length $5lp^{s}$ and their dual codes are given by

and

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \cdots, \varepsilon_{5^{v_{1}}\ell^{v_{2}}}\leq p^{s}$.

Theorem 6.4. Assume that $\gcd(q-1, 5\ell p^{s}) = 5$, then for any $0\leq j\leq 4$, there exists an element $a\in \mathbb{F}_{q^{f}}*$ such that $a^{5\ell p^{s}}\xi^{j\cdot p^{s}} = \zeta^{j'\cdot p^{s}}$. Moreover, each irreducible factor of $x^{5\ell}-\xi^{j}$ over $\mathbb{F}_{q}$ is of the form

where $j' = y'5^{v_{1}}\ell^{v_{2}}$, $v_{1} = \min\{1, v_{5}(j')\}$, $v_{2} = \min\{1, v_{\ell}(j')\}$, and $z_{i}$ is the least positive integer such that $a^{-q^{z_{i}}5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{iq^{z_{i}}}\zeta^{y'q^{z_{i}}} = a^{5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{i}\zeta^{y'}$.

For any $0\leq i, i^{'}\leq 5^{v_{1}}\ell^{v_{2}}-1$, we define a relation $\sim$ to be such that $i\sim i^{'}$ if and only if $a^{-q^{m}5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{iq^{m}}\zeta^{y'q^{m}} = a^{5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{i^{'}}\zeta^{y'}$ for some nonnegative integers $m$. It is obvious to see that $\sim$ is an equivalence relation. Assume that $S$ is a complete system of equivalence class representatives of $\{0, 1, \cdots, 5^{v_{1}}\ell^{v_{2}}-1\}$ relative to this relation $\sim$. For any $i\in S$ we denote the irreducible polynomial

by $M_{i}(x)$, and denote

by $M_{i}^{'}(x)$. Then we have the following corollary.

Corollary 6.1. Assume that $\gcd(q-1, 5\ell p^{s}) = 5$. For any $0\leq j\leq 4$, there exists an element $a\in \mathbb{F}_{q^{f}}*$ such that $a^{5\ell p^{s}}\xi^{j\cdot p^{s}} = \zeta^{j'\cdot p^{s}}$. Then

gives the irreducible factorization of $x^{5\ell p^{s}}-\xi^{j p^{s}}$ over $\mathbb{F}_q$. Furthermore we have that

and

where $0\leq \varepsilon_{i}\leq p^{s}$, for $i\in X$.

Theorem 6.5. Assume that $\gcd(q-1, 5\ell p^{s}) = \ell$, then

(1) If $q\equiv 4 \pmod 5$, for any $0\leq j\leq \ell-1$, the following equations

have a unique solution $j'$ up to modulo $5\ell$. Moreover, each irreducible facotor of $x^{5\ell}-\xi^{j}$ over $\mathbb{F}_{q}$ is of the form

where $j' = y'5^{v_{1}}\ell^{v_{2}}$, $v_{1} = \min\{1, v_{5}(j')\}$, $v_{2} = \min\{1, v_{\ell}(j')\}$, and $z_{i}$ is the least positive integer such that $a^{-q^{z_{i}}5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{iq^{z_{i}}}\zeta^{y'q^{z_{i}}} = a^{5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{i}\zeta^{y'}$.

(2) If $q\equiv 2, 3 \pmod 5$, for any $0\leq j\leq \ell-1$, the following equations

have a unique solution $j'$ up to modulo $5\ell$. Moreover, each irreducible facotor of $x^{5\ell}-\xi^{j}$ over $\mathbb{F}_{q}$ is of the form

where $j' = y'5^{v_{1}}\ell^{v_{2}}$, $v_{1} = min{1, v_{5}(j')}$, $v_{2} = min{1, v_{\ell}(j')}$, and $z_{i}$ is the least positive integer such that $a^{-q^{z_{i}}5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{iq^{z_{i}}}\zeta^{y'q^{z_{i}}} = a^{5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{i}\zeta^{y'}$.

For any $0\leq i, i^{'}\leq 5^{v_{1}}\ell^{v_{2}}-1$, we define a relation $\sim$ to be such that $i\sim i^{'}$ if and only if $a^{-q^{m}5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{iq^{m}}\zeta^{y'q^{m}} = a^{5^{1-v_{1}}\ell^{1-v_{2}}}\delta^{i^{'}}\zeta^{y'}$ for some nonnegative integer $m$. It is obvious to see that $\sim$ is an equivalence relation. Assume that $S$ is a complete system of equivalence class representatives of $\{0, 1, \cdots, 5^{v_{1}}\ell^{v_{2}}-1\}$ relative to this relation $\sim$. For any $i\in S$ we denote the irreducible polynomial

by $M_{i}(x)$, and denote

by $M_{i}^{'}(x)$.

Corollary 6.2. Assume that $\gcd(q-1, 5\ell p^{s}) = \ell$, then

(1) If $q\equiv 4 \pmod 5$, and $j, j^{'}$ is defined as in the first case of Theorem 5.1, then

gives the irreducible factorization of $x^{5\ell p^{s}}-\xi^{j p^{s}}$ over $\mathbb{F}_q$. Furthermore we have that

and

where $0\leq \varepsilon_{i}\leq p^{s}$, for $i\in X$.

(2) If $q\equiv 2, 3 \pmod 5$, and $j, j^{'}$ is defined as in the second case of Theorem 5.1, then

gives the irreducible factorization of $x^{5\ell p^{s}}-\xi^{j p^{s}}$ over $\mathbb{F}_q$. Furthermore we have that

and

where $0\leq \varepsilon_{i}\leq p^{s}$, for $i\in X$.

Finally we give all the self-dual constacylic codes of length $5\ell p^{s}$ as the end of this section. Since self-dual cyclic codes of length $N$ over $\mathbb{F}_q$ exists if and only if $N$ is even and the characteristic of $\mathbb{F}_q$ is $p = 2$, as in the general case, we only consider the case of self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$.

Lemma 6.2. Assume that $q\equiv 1 \pmod 5$. For the $q$-cyclotomic cosets, one of the following holds.

(1) If $f = \mathrm{ord}_{\ell}(q)$ is even, we have that

where $0\leq k\leq e-1$.

(2) If $f = \mathrm{ord}_{\ell}(q)$ is odd, we have that

where $\{C_{5\mu^{k}}\} = \{C_{5\mu^{k^{'}}}\}\bigcup \{C_{-5\mu^{k^{'}}}\}$, and $0\leq k\leq e-1$, $0\leq k^{'}\leq \dfrac{e}{2}-1$.

Proof.

(1) By the definition of reciprocal coset, it is clear that $C_{0}^{*} = C_{0}, \ C_{\ell}^{*} = C_{-\ell}, \ C_{2\ell}^{*} = C_{-2\ell}, \ C_{\mu^{k}}^{*} = C_{-\mu^{k}}, \ C_{2\mu^{k}}^{*} = C_{-2\mu^{k}}$, thus it remains to prove $C_{5\mu^{k}}^{*} = C_{5\mu^{k}}$. Let $t = \frac{f}{2}$. Since $f = \mathrm{ord}_{\ell}(q)$, it is trivial to see that $q^{t}\equiv -1 \pmod \ell$, and therefore we have that $-5\mu^{k}\equiv 5\mu^{k}q^{t} \pmod {5\ell}$. It follows immediately that $C_{5\mu^{k}}^{*} = C_{5\mu^{k}}$, for $0\leq k\leq e-1$.

(2) As in the first case, the conclusions that $C_{0}^{*} = C_{0}, \ C_{\ell}^{*} = C_{-\ell}, \ C_{2\ell}^{*} = C_{-2\ell}, \ C_{\mu^{k}}^{*} = C_{-\mu^{k}}, \ C_{2\mu^{k}}^{*} = C_{-2\mu^{k}}$ are clear, and now we prove that $C_{5\mu^{k^{'}}}^{*} = C_{-5\mu^{k^{'}}}$. To see this, we claim that for any $0\leq k_{1}^{'}, k_{2}^{'}\leq \frac{e}{2}-1$, $C_{5\mu^{k_{1}^{'}}}\neq C_{-5\mu^{k_{2}^{'}}}$, and $\{C_{5\mu^{k}}\} = \{C_{5\mu^{k^{'}}}\}\bigcup \{C_{-5\mu^{k^{'}}}\}$. Assume that $C_{5\mu^{k_{1}^{'}}} = C_{-5\mu^{k_{2}^{'}}}$ for some $0\leq k_{1}^{'}, k_{2}^{'}\leq \frac{e}{2}-1$, then we have that $5\mu^{k_{1}^{'}}\equiv -5\mu^{k_{2}^{'}}q^{j} \pmod {5\ell}$ for some $0\leq j\leq f-1$, which indicates that $-\mu^{k_{1}^{'}-k_{2}^{'}}\equiv q^{j} \pmod \ell$. Notice that $f$ is odd, therefore we have that $-\mu^{f(k_{1}^{'}-k_{2}^{'})}\equiv q^{jf} \equiv 1 \pmod \ell$ and $\mu^{f(k_{1}^{'}-k_{2}^{'})}\equiv -1 \pmod \ell$. It follows that $\mu^{2f(k_{1}^{'}-k_{2}^{'})}\equiv 1 \pmod \ell$, hence $\phi(\ell)\mid 2f(k_{1}^{'}-k_{2}^{'})$ and $\frac{e}{2}\mid k_{1}^{'}-k_{2}^{'}$. Since by the condition we have $0\leq k_{1}^{'}, k_{2}^{'}\leq \frac{e}{2}-1$, we deduce that $k_{1}^{'} = k_{2}^{'}$. Then the equation $5\mu^{k_{1}^{'}}\equiv$$-5\mu^{k_{2}^{'}}q^{j} \pmod {5\ell}$ can be reduced to $-1\equiv q^{j} \pmod \ell$. However, notice that $\mathrm{ord}_{\ell}(q) = f$ is odd, such a positive integer $j$ cannot exist, which is a contradiction. According to this, we have that for any $0\leq k_{1}^{'}, k_{2}^{'}\leq \frac{e}{2}-1$, $C_{5\mu^{k_{1}^{'}}}\neq C_{-5\mu^{k_{2}^{'}}}$. By comparing the number of elements, it is trivial to verify that $\{C_{5\mu^{k}}\} = \{C_{5\mu^{k^{'}}}\}\bigcup \{C_{-5\mu^{k^{'}}}\}$ holds. Then by the definition of reciprocal coset, one immediately get that $C_{5\mu^{k^{'}}}^{*} = C_{-5\mu^{k^{'}}}$.

With the same method we can prove the results for the rest of cases. The proofs will be omitted.

Lemma 6.3. Assume that $q\equiv 4 \pmod 5$. For the $q$-cyclotomic cosets, one of the following holds.

(1) If $f = 2t$ is even, then

(i) when $t$ is even, we have that

where $\{C_{\mu^{k^{'}}}\} = \{C_{\mu^{k}}\}\bigcup \{C_{-\mu^{k}}\}$, $\{C_{2\mu^{k^{'}}}\} = \{C_{2\mu^{k}}\}\bigcup \{C_{-2\mu^{k}}\}$, for $0\leq k\leq e-1$, $0\leq k^{'}\leq 2e-1$.

(ii) If $t$ is odd, we have that

where $0\leq k\leq e-1$, $0\leq k^{'}\leq 2e-1$.

(2) when $f$ is odd, then

where $\{C_{\mu^{k}}\} = \{C_{\mu^{k^{'}}}\}\bigcup \{C_{-\mu^{k^{'}}}\}$, $\{C_{2\mu^{k}}\} = \{C_{2\mu^{k^{'}}}\}\bigcup \{C_{-2\mu^{k^{'}}}\}$, $\{C_{5\mu^{k}}\} = \{C_{5\mu^{k^{'}}}\}\bigcup \{C_{-5\mu^{k^{'}}}\}$, for $0\leq k\leq e-1, 0\leq k^{'}\leq \frac{e}{2}-1$.

Lemma 6.4. Assume that $q\equiv 2$ or $3 \pmod 5$. For the $q$-cyclotomic cosets, one of the following holds.

(1) If $4\mid f$. Let $f = 4t$, then

(i) when $t$ is even, we have that

where $\{C_{\mu^{k^{'}}}\} = \{C_{\mu^{k^{''}}}\}\bigcup \{C_{-\mu^{k^{''}}}\}$, for $0\leq k\leq e-1$, $0\leq k^{''}\leq 2e-1$ and $0\leq k^{'}\leq 4e-1$.

(ii) If $t$ is odd, we have that

where $0\leq k\leq e-1$, $0\leq k^{'}\leq 4e-1$.

(2) If $2\mid f$ but $4\nmid f$, then

where $\{C_{\mu^{k^{'}}}\} = \{C_{\mu^{k}}\}\bigcup \{C_{-\mu^{k}}\}$, for $0\leq k\leq e-1$, $0\leq k^{'}\leq 2e-1$.

(3) If $f$ is odd, then

where $\{C_{\mu^{k}}\} = \{C_{\mu^{k^{'}}}\}\bigcup \{C_{-\mu^{k^{'}}}\}$, $\{C_{5\mu^{k}}\} = \{C_{5\mu^{k^{'}}}\}\bigcup \{C_{-5\mu^{k^{'}}}\}$, for $0\leq k^{'}\leq \frac{e}{2}-1$, $0\leq k\leq e-1$.

From the above lemmas, we give all the self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$ and their enumeration in the following theorems.

Theorem 6.6. Let $q\equiv 1 \pmod 5$, then one of the following holds.

(1) If $f = \mathrm{ord}_{\ell}(q)$ is even, there exist $(2^{s}+1)^{2+2e}$ self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$, which are given by

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \tau_{k}, \rho_{k}\leq 2^{s}$, for any $0\leq k\leq e-1$.

(2) If $f = \mathrm{ord}_{\ell}(q)$ is odd, there exist $(2^{s}+1)^{2+\dfrac{5e}{2}}$ self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$, which are given by

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \tau_{k}, \rho_{k}, \iota_{k^{'}}\leq 2^{s}$, for any $0\leq k\leq e-1$ and any $0\leq k^{'}\leq \dfrac{e}{2}-1$.

Proof.

(1) By Lemma 6.2, any self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$ has the form of

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \tau_{k}, \rho_{k}\leq 2^{s}$, for any $0\leq k\leq e-1$. Since each of $\varepsilon_{1}, \varepsilon_{2}$ and $\tau_{k}, \rho_{k}$, $0\leq k\leq e-1$, has $2^{s}+1$ possible values, we have in total $(2^{s}+1)^{2+2e}$ self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$.

(2) By Lemma 6.2, any self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$ has the form of

where $0\leq \varepsilon_{1}, \varepsilon_{2}, \tau_{k}, \rho_{k}, \iota_{k^{'}}\leq 2^{s}$, for any $0\leq k\leq e-1$ and any $0\leq k^{'}\leq \dfrac{e}{2}-1$. Each of $\varepsilon_{1}, \varepsilon_{2}$, $\tau_{k}, \rho_{k}, 0\leq k\leq e-1$, and $\iota_{k^{'}}$, $0\leq k^{'}\leq \dfrac{e}{2}-1$, has $2^{s}+1$ possible values, we have in total $(2^{s}+1)^{2+\dfrac{5e}{2}}$ self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$.

The proofs of theorems for the rest of cases are similar, and we will give them without proofs.

Theorem 6.7. Let $q\equiv 4 \pmod 5$, then one of the following holds.

(1) If $f = 2t$ is even, then

(i) when $t$ is even, there exist $(2^{s}+1)^{2e}$ self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$, which are given by

where $0\leq \tau_{k}, \rho_{k}\leq 2^{s}$, for any $0\leq k\leq e-1$.

(ii) when $t$ is odd, there exists only one self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$, which is given by

(2) If $f$ is odd, thenthere exist $(2^{s}+1)^{3e/2}$ self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$, which are given by

Theorem 6.8. Let $q\equiv 2$ or $3 \pmod 5$, then one of the following holds.

(1) If $4\mid f$. Let $f = 4t$, then

(i) when $t$ is even, there exist $(2^{s}+1)^{2e}$ self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$, which are given by

where $0\leq \tau_{k^{''}}\leq 2^{s}$, for any $0\leq k^{''}\leq 2e-1$.

(ii) when $t$ is odd, there exists only one self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$, which is given by

(2) If $2\mid f$ but $4\nmid f$, then there exist $(2^{s}+1)^{e}$ self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$, which are given by

where $0\leq \tau_{k}\leq 2^{s}$, for any $0\leq k\leq e-1$.

(3) If $f$ is odd, then there exist $(2^{s}+1)^{e}$ self-dual cyclic codes of length $5\cdot 2^{s}\ell$ over $\mathbb{F}_{2^{m}}$, which are given by

where $0\leq \tau_{k^{'}}, \iota_{k^{'}}\leq 2^{s}$, for any $0\leq k^{'}\leq \dfrac{e}{2}-1$.

Acknowledgments

The first author was supported by the Yuyou Team Support Program of North China University of Technology (No. 107051360019XN137/007) and Yujie Talent Project of North China University of Technology(No. 107051360022XN735).

Conflict of interest

The authors declare no conflict of interest.