Several Constructions of Cyclic Subspace Codes Utilizing (k + 1)-Dimensional Sidon Spaces

Abstract

In recent years, network coding has attracted widespread attention due to its important role in digital communication and network security. Among these, cyclic subspace codes have very important applications in error correction and rectification in random network coding, attracting many scholars to conduct research on them. As an important tool in constructing cyclic subspace codes, the present study focuses on the construction of Sidon spaces by leveraging the roots of irreducible polynomials and primitive elements over finite fields. In this paper, we construct some Sidon spaces with new parameters. Specifically, we let $k,m_1,m_2$ be three positive integers and define ${\rho }_1=⌈{\displaystyle \frac{m_1}{2k}}⌉-1,{\rho }_2=⌈{\displaystyle \frac{m_2}{2m_1}}⌉-1$. On the basis of these newly constructed Sidon spaces, we obtain new cyclic subspace codes with size ${\rho }_1{\rho }_2{q}^k{\displaystyle \frac{q^n-1}{q-1}}$ and minimum distance $2k$.

1. Introduction

With the rapid development of information technology and the popularity of the Internet, network communication has become an indispensable infrastructure in modern society. However, as the scale of the network expands and its structure becomes more complex, network coding and network communication face many challenges, such as increasingly prominent issues related to the reliability, effectiveness, and security of information transmission. To address these challenges, researchers continue to explore new coding technologies and methods to improve the performance of communication systems.

During the research process, it was discovered that monomial codes [1], cyclic codes, LDPC codes, and particularly cyclic subspace codes play significant roles in network coding. Among these codes, cyclic subspace codes exhibit a series of superior characteristics. Owing to their unique cyclic subspace structure, cyclic subspace codes cleverly utilize cyclic shifts and linear algebra operations in the encoding and decoding processes, thereby greatly simplifying the operational procedures and achieving efficient encoding and decoding algorithms [2,3,4]. Furthermore, with their special codeword structure design, cyclic subspace codes demonstrate excellent error detection and correction capabilities in complex communication environments with noise and interference. Notably, by meticulously constructing codewords with a larger minimum Hamming distance, cyclic subspace codes can maintain a low bit error rate in communication channels with high noise levels, ensuring the reliability of communication. However, there are still some deficiencies in cyclic subspace codes. Therefore, this paper is dedicated to constructing cyclic subspace codes with better performance using Sidon spaces.

Sidon spaces, as a key instrument in constructing such codes, optimize the encoding process by providing well-structured sets of subspaces [5]. Together, they provide a solid theoretical foundation for efficient data transmission, distributed storage, and future communication technologies. Next, we will introduce some theoretical knowledge related to cyclic subspace codes and Sidon spaces.

Let q be a prime power and let ${\mathbb{F}}_q$ be the finite field of q elements which is the field of scalars. The vector space of dimension n over ${\mathbb{F}}_q$ is expressed as the extension field ${\mathbb{F}}_{q^n}$.

The projective space of order n over ${\mathbb{F}}_q$, denoted as ${\mathcal{P}}_q(n)$, is the family of all subspaces of the vector space ${\mathbb{F}}_{q^n}$ [6]. To quantify the separation between arbitrary elements U and V in ${\mathcal{P}}_q(n)$, we introduce the subspace distance $d(U,V)$ given by the expression

$$d(U,V)=dim(U+V)-dim(U\cap V)=dim(U)+dim(V)-2dim(U\cap V).$$

where $dim(·)$ denotes the dimension of a vector space over ${\mathbb{F}}_q$. With this foundation, we can define an $(n,M,d)$ code $\mathcal{C}$ in the projective space, which is a subset of ${\mathcal{P}}_q(n)$ containing M elements, with the distance between any two codewords ( subspaces) being at least d [6]. For any element $U\in {\mathcal{G}}_q(n,k)$, the cyclic shift $\alpha U$ of U by an element $\alpha \in {\mathbb{F}}_{q^n}^{*}$ is the set $\{\alpha u\mid u\in U\}$. Notably, $\alpha U$ retains the same dimension as U. The set of all such shifts, orb$(U)$, comprises $\{\alpha U\mid \alpha \in {\mathbb{F}}_{q^n}^{*}\}$ and its size is ${\displaystyle \frac{q^n-1}{q^t-1}}$, where t divides n. A full-length orbit code arises when orb$(U)$ attains its maximum size of ${\displaystyle \frac{q^n-1}{q-1}}$. Such codes exhibit a minimum distance no greater than $2k-2$, and optimality is achieved when this bound is met [7]. The notation ${\mathcal{G}}_q(n,k)\subset {\mathcal{P}}_q(n)$ signifies the collection of all k-dimensional subspaces of ${\mathbb{F}}_{q^n}$.

Given positive integers n and k, a k-dimensional subspace code is a non-empty subset of ${\mathcal{G}}_q(n,k)$. Specifically, if $\mathcal{C}\subseteq {\mathcal{G}}_q(n,k)$ forms a subspace code and includes the orbit of every $U\in \mathcal{C}$, then $\mathcal{C}$ is classified as a cyclic subspace code.

The concept of subspace codes was initially introduced in paper [8], as documented in [9,10,11]. Among them, cyclic subspace codes stand out due to their superior properties, enabling efficient encoding and decoding processes, as discussed in [5]. The construction of cyclic subspace codes typically follows two distinct paths. The first approach leverages the roots of linearized polynomials, a method initially proposed by Ben-Sasson in [12]. Utilizing the roots of these specialized polynomials, this technique has been successfully employed to construct cyclic subspace codes in $G_q(n,k)$ with a minimum distance of $2k-2$, as detailed in [12,13,14]. Alternatively, the second approach exploits Sidon spaces to construct cyclic subspace codes, a topic explored in [15,16,17,18]. Given the extensive research conducted by numerous prominent scholars on cyclic subspace codes, we will delve into some of the pioneering construction methods in the subsequent paragraphs, as in Table 1.

(1) In [19], Roth et al. constructed a great number of large cyclic subspace codes in ${\mathcal{G}}_q(n,k)$ with a minimum distance of $2k-2$ by utilizing the Sidon space $U_i=\{u+u^q{\gamma }_i\mid u\in {\mathbb{F}}_{q^k}\}$, and subsequently obtained cyclic subspace codes of size ${\displaystyle \frac{\tau (q^n-1)}{q-1}}$. Furthermore, they confirmed that constructing a Sidon space in ${\mathcal{G}}_q(n,k)$ is equivalent to constructing a cyclic subspace code of size ${\displaystyle \frac{q^n-1}{q-1}}$ in the same space.

(2) In [20], Feng et al. constructed a great number of large cyclic subspace codes in ${\mathcal{G}}_q(n,k)$ with a minimum distance of $2k-2$ by utilizing the Sidon spaces $U_{i,j}=\{u+(u^{q^l}-u){\gamma }_{i,j}\mid u\in {\mathbb{F}}_{q^k}\}\mathrm{and}{U}_{q^k-1,j}=\{u+u^{q^l}{\gamma }^j\mid u\in {\mathbb{F}}_{q^k}\},$ and subsequently obtained cyclic subspace codes of size ${\displaystyle \frac{e{q}^k(q^n-1)}{q-1}}$.

(3) In [21], Li et al. constructed a great number of large cyclic codes in ${\mathcal{G}}_q(n,k)$ whose minimum distance was $2k-2$ through utilizing the Sidon space $U_i=\{u+({\tau }_i{u}^q+u){\xi }_i{\gamma }_{l_i}\mid u\in {\mathbb{F}}_{q^k}\}.$ Furthermore, they successfully proved that if there exist m distinct Sidon spaces that satisfy some conditions, then the sum of these m Sidon spaces once again forms a Sidon space.

(4) In [16], Liu et al. constructed a great number of large cyclic codes in ${\mathcal{G}}_q(n,k)$ with a minimum distance of $2k-2$ by utilizing the Sidon space $U_{i,j,r}=\{u+u{\gamma }_{i,j}+u^{q^t}{\xi }_{i,r}\mid u\in {\mathbb{F}}_{q^k}\},$ and subsequently obtained cyclic subspace codes with size ${\displaystyle \frac{\theta \rho (q^k-1)(q^n-1)}{q-1}}$. Moreover, by making use of $U_{q^k-1,j,r}=\{u^{q^t}+u{\gamma }^j+u{\xi }^r\mid u\in {\mathbb{F}}_{q^k}\},$ they received additional cyclic subspace codes with size ${\displaystyle \frac{\theta \rho (q^n-1)}{q-1}}$. Finally, they received a new cyclic subspace code whose size was ${\displaystyle \frac{\theta \rho q^k(q^n-1)}{q-1}}$.

Table 1. The comparison of Sidon spaces and cyclic subspace codes.

Reference	Sidon Spaces	dim(C)	d(C)	Code Sizes
[18]	$U_{a_{e_0},i,t,e_1}=\{u{P}_{a_{e_0}}(\gamma )+$ $b{\omega }^i{u}^{q^{s_t}}{\gamma }^{e_1+1}\mid u\in {\mathbb{F}}_{q^k}\}$	k	$2k-2$	${\displaystyle \frac{q^k(q^{\left(e+1\right)k}-1)(q^n-1)}{q^k-1}}$
[21]	$U_i=\{a+u\gamma +\left(u^q+{\delta }_{i}u\right){\gamma }^{l_i}$ $\mid a\in {\mathbb{F}}_q,u\in {\mathbb{F}}_{q^k}\}$	$k+1$	$2k$	${\displaystyle \frac{e{q}^k\left(q^n-1\right)}{q-1}}$
[22]	$U=\{a+u\gamma +u^q{\gamma }^2$ $\mid a\in {\mathbb{F}}_q,u\in {\mathbb{F}}_{q^k}\}$	$k+1$	$2k$	$2{\displaystyle \frac{q^n-1}{q-1}}$
[23]	$W_{f_i,{\eta }_i}=\{x+{\eta }_i{f}_i(x):x\in {\mathbb{F}}_{q^k}\}$	k	$2k-2$	$r{\displaystyle \frac{q^n-1}{q-1}}$
[24]	$U_{i_1,\dots ,i_r,b,j}^l=\{u+\left(u^q-{\xi }^{b}u\right){\xi }^{i_l}{\gamma }^{l_j}$ $+{\sum }_{m=1,m\ne l}^{r}u{\xi }^{i_m}{\gamma }^{m_j}\mid u\in {\mathbb{F}}_{q^k}\}$	k	$2k-2$	$r\left({\displaystyle \frac{\left(\left(q^k-1\right)\left(q-1\right)+1\right)}{q-1}}\right)$ $\times \left({\displaystyle \frac{{\left(q^k-1\right)}^{r-1}\left(q^n-1\right)}{q-1}}\right)$
[25]	$U_{{\alpha }_h}^j(b,\gamma )=\{u^{q^t}+\left(u^{q^l}+a_h{u}^{q^t}\right)$ ${\omega }^b{\gamma }^{h_j}+u^{q^t}{f}_{{\alpha }_h}^j(\gamma )\mid u\in {\mathbb{F}}_{q^k}\}$	k	$2k-2$	$s{q}^k{\left(q^k-1\right)}^{s-1}\left(q^n-1\right)+{\displaystyle \frac{q^n-1}{q^k-1}}$
Theorem 3	$U_i=\{x+y^q+y{\gamma }^{l_i}+b_{i}x{\xi }^{t_i}$ $\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\}$	$k+1$	$2k$	${\rho }_1{\rho }_2\left(q^k-1\right){\displaystyle \frac{q^n-1}{q-1}}$
Theorem 4	$U_{q^k-1}=\{y^q+y{\gamma }^{l_i}+x{\xi }^{t_i}$ $\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\}$	$k+1$	$2k$	${\rho }_1{\rho }_2{\displaystyle \frac{q^n-1}{q-1}}$

Table 1. The comparison of Sidon spaces and cyclic subspace codes.

Reference	Sidon Spaces	dim(C)	d(C)	Code Sizes
[18]	$U_{a_{e_0},i,t,e_1}=\{u{P}_{a_{e_0}}(\gamma )+$ $b{\omega }^i{u}^{q^{s_t}}{\gamma }^{e_1+1}\mid u\in {\mathbb{F}}_{q^k}\}$	k	$2k-2$	${\displaystyle \frac{q^k(q^{\left(e+1\right)k}-1)(q^n-1)}{q^k-1}}$
[21]	$U_i=\{a+u\gamma +\left(u^q+{\delta }_{i}u\right){\gamma }^{l_i}$ $\mid a\in {\mathbb{F}}_q,u\in {\mathbb{F}}_{q^k}\}$	$k+1$	$2k$	${\displaystyle \frac{e{q}^k\left(q^n-1\right)}{q-1}}$
[22]	$U=\{a+u\gamma +u^q{\gamma }^2$ $\mid a\in {\mathbb{F}}_q,u\in {\mathbb{F}}_{q^k}\}$	$k+1$	$2k$	$2{\displaystyle \frac{q^n-1}{q-1}}$
[23]	$W_{f_i,{\eta }_i}=\{x+{\eta }_i{f}_i(x):x\in {\mathbb{F}}_{q^k}\}$	k	$2k-2$	$r{\displaystyle \frac{q^n-1}{q-1}}$
[24]	$U_{i_1,\dots ,i_r,b,j}^l=\{u+\left(u^q-{\xi }^{b}u\right){\xi }^{i_l}{\gamma }^{l_j}$ $+{\sum }_{m=1,m\ne l}^{r}u{\xi }^{i_m}{\gamma }^{m_j}\mid u\in {\mathbb{F}}_{q^k}\}$	k	$2k-2$	$r\left({\displaystyle \frac{\left(\left(q^k-1\right)\left(q-1\right)+1\right)}{q-1}}\right)$ $\times \left({\displaystyle \frac{{\left(q^k-1\right)}^{r-1}\left(q^n-1\right)}{q-1}}\right)$
[25]	$U_{{\alpha }_h}^j(b,\gamma )=\{u^{q^t}+\left(u^{q^l}+a_h{u}^{q^t}\right)$ ${\omega }^b{\gamma }^{h_j}+u^{q^t}{f}_{{\alpha }_h}^j(\gamma )\mid u\in {\mathbb{F}}_{q^k}\}$	k	$2k-2$	$s{q}^k{\left(q^k-1\right)}^{s-1}\left(q^n-1\right)+{\displaystyle \frac{q^n-1}{q^k-1}}$
Theorem 3	$U_i=\{x+y^q+y{\gamma }^{l_i}+b_{i}x{\xi }^{t_i}$ $\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\}$	$k+1$	$2k$	${\rho }_1{\rho }_2\left(q^k-1\right){\displaystyle \frac{q^n-1}{q-1}}$
Theorem 4	$U_{q^k-1}=\{y^q+y{\gamma }^{l_i}+x{\xi }^{t_i}$ $\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\}$	$k+1$	$2k$	${\rho }_1{\rho }_2{\displaystyle \frac{q^n-1}{q-1}}$

In this paper, we endeavor to construct new Sidon spaces by utilizing the roots of irreducible polynomials and primitive elements defined over finite fields. Furthermore, we introduce a new class of cyclic subspace codes, whose sizes are multiples of ${\displaystyle \frac{q^n-1}{q-1}}$, and possess a minimum distance of $2k$, thereby contributing to the advancement of the field.

The structure of this paper is outlined as follows. Section 1 presents the conceptual framework for constructing cyclic subspace codes. Section 2 establishes the preliminary theoretical foundations essential for the subsequent constructions. In Section 3, we construct Sidon spaces $U_i=\left\{x+y^q+y{\gamma }^{l_i}+b_{i}x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\right\}$ and $V_i=\{y^q+y{\gamma }^{l_i}+x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\}$ that have new parameters and better encoding. Based on the Sidon spaces constructed in Section 3, we present a cyclic subspace code of size ${\rho }_1{\rho }_2\left(q^k-1\right){\displaystyle \frac{q^n-1}{q-1}}$ and ${\rho }_1{\rho }_2{q}^k{\displaystyle \frac{q^n-1}{q-1}}$, along with its proof in Section 4. Finally, Section 5 concludes the paper by providing a comprehensive summary of our key findings and contributions, highlighting the significance and implications of our work within the broader research landscape.

2. Preliminaries

In this section, we will introduce some crucial notations and significant findings that will be utilized throughout the subsequent paper.

Definition 1

([19]). A subspace $V\in {\mathcal{G}}_q\left(n,k\right)$ is called a Sidon space if for all nonzero $a,b,c$, and $d\in V$, if $ab=cd$, then $\left\{a{\mathbb{F}}_q,b{\mathbb{F}}_q\right\}=\left\{c{\mathbb{F}}_q,d{\mathbb{F}}_q\right\}$.

Lemma 1

([19]). Let $U\in {\mathcal{G}}_q\left(n,k\right)$ and $\mathcal{C}=\left\{\alpha U:\alpha \in {\mathbb{F}}_{q^n}^{*}\right\}$. U is a Sidon space if and only if $\mathcal{C}$ is an optimal full-length orbit code.

Lemma 2

([19]). Let U be a subspace, $U\in {\mathcal{G}}_q\left(n,k\right)$. Only when U is a Sidon space, then the cyclic subspace code orb(U) has size ${\displaystyle \frac{q^n-1}{q-1}}$ with minimum distance $2k-2$.

Lemma 3

([19]). Let U and V be two distinct elements in ${\mathcal{G}}_q\left(n,k\right)$. Then, the next two conditions are equivalent.

(1) For any $\alpha \in {\mathbb{F}}_{q^n}^{*}$, $dim\left(U\bigcap \alpha V\right)\le 1$.

(2) For arbitrary nonzero elements a, $c\in U$ and nonzero elements b, $d\in V$, when $ab=cd\ne 0$, we will have the result that $\left\{a{\mathbb{F}}_q\right\}=\left\{c{\mathbb{F}}_q\right\}$ and $\left\{b{\mathbb{F}}_q\right\}=\left\{d{\mathbb{F}}_q\right\}$.

Lemma 3 has great importance in Section 3 and Section 4, which will be used widely.

Lemma 4

([16]). Let q and k be two positive integers, along with four nonzero elements u, v, s, and t of ${\mathbb{F}}_{q^k}$ which satisfy $uv=st$ and $u^{q}v=s^{q}t$. Then, we have the result that ${\displaystyle \frac{u}{s}}={\displaystyle \frac{t}{v}}\in {\mathbb{F}}_q^{*}$.

Through using Lemma 4, we can simplify the proof of subsequent theorems in the rest of the paper.

To achieve our results more efficiently in this paper, we have leveraged the inspiration and assistance provided by the above lemmas and definitions.

3. Constructions of Sidon Spaces

In this section, we first review some basic definitions. Subsequently, with the help of these facts, we construct several Sidon spaces with dimension $k+1$.

Given three positive integers k, $m_1$, and $m_2$, which satisfy the condition that $k|m_1|m_2$; and then given a prime power q and given a primitive element $\omega$ in ${\mathbb{F}}_{q^k}$; and, in addition, $\gamma \in {\mathbb{F}}_{q^{m_1}}^{*}$ being a root of an irreducible polynomial over ${\mathbb{F}}_{q^k}$ whose degree is ${\displaystyle \frac{m_1}{k}}$, and $\xi \in {\mathbb{F}}_{q^{m_2}}^{*}$ being a root of an irreducible polynomial over ${\mathbb{F}}_{q^{m_1}}$ whose degree is ${\displaystyle \frac{m_2}{m_1}}$, for any positive integer $k>1$ and $m_1=r_{1}k$ with $r_1>2$, we let ${\rho }_1=\lceil m_1/2k\rceil -1$. For any positive integer $m_1>1$ and $m_2=r_2{m}_1$ with $r_2>2$, we let ${\rho }_2=\lceil m_2/2m_1\rceil -1$, $r_2|r_1$. Moreover, we let $1\le l\le {\rho }_1,1\le t\le {\rho }_2$.

Theorem 1.

We keep the notations above. Moreover, we let $b_i\in {\mathbb{F}}_{q^k}^{*}$, where $1\le i\le {\rho }_1{\rho }_2\left(q^k-1\right)$. We define $U_i=\left\{x+y^q+y{\gamma }^{l_i}+b_{i}x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\right\}$. Hence, we can obtain that $U_i\in {\mathcal{G}}_q\left(n,k+1\right)$ is a $\left(k+1\right)$-dimensional Sidon space.

Proof. 

Firstly, we let

$$u_1=x_1+y_1^q+y_1{\gamma }^{l_i}+b_i{x}_1{\xi }^{t_i},u_2=x_2+y_2^q+y_2{\gamma }^{l_i}+b_i{x}_2{\xi }^{t_i},$$

$$u_3=x_3+y_3^q+y_3{\gamma }^{l_i}+b_i{x}_3{\xi }^{t_i},u_4=x_4+y_4^q+y_4{\gamma }^i+b_i{x}_4{\xi }^{t_i}.$$

If $u_1{u}_2=u_3{u}_4$, then we should check

$$\left\{u_1{\mathbb{F}}_q,u_2{\mathbb{F}}_q\right\}=\left\{u_3{\mathbb{F}}_q,u_4{\mathbb{F}}_q\right\}.$$

(1)

It is obvious that $\left\{1,{\gamma }^{l_i},{\gamma }^{2l_i},{\xi }^{t_i},{\xi }^{2t_i},{\gamma }^{l_i}{\xi }^{t_i}\right\}$ is an independent linear set. By comparing the coefficients of the equations in $u_1{u}_2=u_3{u}_4$, we obtain

$$\left\{\begin{array}{cc}\hfill x_1{x}_2& =x_3{x}_4\hfill \\ \hfill y_1{y}_2& =y_3{y}_4\hfill \\ \hfill x_1{y}_2^q+x_2{y}_1^q& =x_3{y}_4^q+x_4{y}_3^q\hfill \end{array}\right.$$

(2)

Next, with the help of $\left(2\right)$, we will prove this theorem from the subsequent four cases.

• Case$\left(1\right)$: $x_1,x_2\in {\mathbb{F}}_q^{*}$ and $y_1,y_2\in {\mathbb{F}}_{q^k}^{*}$.
  From $x_1{x}_2=x_3{x}_4$ and $y_1{y}_2=y_3{y}_4$ in $\left(2\right)$, we can have that ${\displaystyle \frac{x_1}{x_3}}={\displaystyle \frac{x_4}{x_2}}={\lambda }_1\in {\mathbb{F}}_q^{*}$, ${\displaystyle \frac{y_1}{y_3}}={\displaystyle \frac{y_4}{y_2}}={\lambda }_2\in {\mathbb{F}}_{q^k}^{*}$. Hence, the equation $x_1{y}_2^q+x_2{y}_1^q=x_3{y}_4^q+x_4{y}_3^q$ can be simplified to $\left({\lambda }_1-{\lambda }_2^q\right)x_3{y}_2^q=\left({\lambda }_1-{\lambda }_2^q\right)x_2{y}_3^q$.
  $\left(I\right)$. ${\lambda }_1\ne {\lambda }_2^q$.
  By simplifying the equations above, we obtain ${\displaystyle \frac{x_2}{x_3}}={\displaystyle \frac{y_2^q}{y_3^q}}$. Then, we let $t_1={\displaystyle \frac{x_2}{x_3}}\in {\mathbb{F}}_q^{*},t_2={\displaystyle \frac{y_2^q}{y_3^q}}\in {\mathbb{F}}_{q^k}^{*}$. Hence, we obtain $t_1=t_2^q$, as well as $t_1=t_2$. Thus, we obtain ${\displaystyle \frac{u_2}{u_3}}={\displaystyle \frac{u_4}{u_1}}$.
  $\left(II\right)$. ${\lambda }_1={\lambda }_2^q$.
  When ${\lambda }_1={\lambda }_2^q$, as well as ${\lambda }_1={\lambda }_2$, we have ${\displaystyle \frac{x_1}{x_3}}={\displaystyle \frac{x_4}{x_2}}={\displaystyle \frac{y_1}{y_3}}={\displaystyle \frac{y_4}{y_2}}\in {\mathbb{F}}_q^{*}$, and then we obtain ${\displaystyle \frac{u_1}{u_3}}={\displaystyle \frac{u_4}{u_2}}$.
• Case$\left(2\right)$: $x_1,x_2\in {\mathbb{F}}_q^{*}$ and $y_1{y}_2=0$.
  Next, we will discuss this case from the following three aspects.
  $\left(I\right)$. The case that $y_1$, $y_2$, $y_3$, and $y_4$ have only one or three zeroes is impossible.
  $\left(II\right)$. If $y_1$, $y_2$, $y_3$, and $y_4$ have two zeroes, assuming that $y_1=y_3=0$ and $y_2,y_4\ne 0$, then we have ${\displaystyle \frac{x_4}{x_2}}={\displaystyle \frac{x_1}{x_3}}={\displaystyle \frac{y_4}{y_2}}\in {\mathbb{F}}_q^{*}$. Hence, we have ${\displaystyle \frac{u_1}{u_3}}={\displaystyle \frac{u_4}{u_2}}\in {\mathbb{F}}_q^{*}$.
  $\left(III\right)$. If $y_1$, $y_2$, $y_3$, and $y_4$ have four zeroes, as well as $y_1=y_2=y_3=y_4=0$, we have ${\displaystyle \frac{u_1}{u_3}}={\displaystyle \frac{u_4}{u_2}}\in {\mathbb{F}}_q^{*}$.
• Case$\left(3\right)$: $x_1{x}_2=0$ and $y_1,y_2\in {\mathbb{F}}_{q^k}^{*}$.
  Next, we will discuss this case from the following three aspects.
  $\left(I\right)$. The case that $x_1$, $x_2$, $x_3$, and $x_4$ have only one or three zeroes is impossible.
  $\left(II\right)$. If $x_1$, $x_2$, $x_3$, and $x_4$ have two zeroes, assuming that $x_1=x_3=0$ and $x_2,x_4\ne 0$, then we have ${\displaystyle \frac{x_4}{x_2}}={\displaystyle \frac{y_1}{y_3}}={\displaystyle \frac{y_4}{y_2}}\in {\mathbb{F}}_q^{*}$. Hence, we have ${\displaystyle \frac{u_1}{u_3}}={\displaystyle \frac{u_4}{u_2}}\in {\mathbb{F}}_q^{*}$.
  $\left(III\right)$. If $x_1$, $x_2$, $x_3$, and $x_4$ have four zeroes, as well as $x_1=x_2=x_3=x_4=0$, we have ${\displaystyle \frac{u_1}{u_3}}={\displaystyle \frac{u_4}{u_2}}\in {\mathbb{F}}_q^{*}$.
• Case$\left(4\right)$: $x_1{x}_2=0$ and $y_1{y}_2=0$.
  If we have four zeroes, as well as $x_1=x_3=y_1=y_3=0$, then we let ${\displaystyle \frac{x_4}{x_2}}={\displaystyle \frac{y_4}{y_2}}\in {\mathbb{F}}_q^{*}$. Hence, we have ${\displaystyle \frac{u_1}{u_3}}={\displaystyle \frac{u_4}{u_2}}\in {\mathbb{F}}_q^{*}$.

From Definition 1, we can obtain that $U_i$ is a $\left(k+1\right)$-dimensional Sidon space.

This ends the proof. □

Corollary 1.

Similarly, we can obtain that $W_i=\left\{b_{i}x+y^q+y{\gamma }^{l_i}+x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\right\}$ is a $\left(k+1\right)$-dimensional Sidon space.

Proof. 

We let

$$w_1=b_i{x}_1+y_1^q+y_1{\gamma }^{l_i}+x_1{\xi }^{t_i},w_2=b_i{x}_2+y_2^q+y_2{\gamma }^{l_i}+x_2{\xi }^{t_i},$$

$$w_3=b_i{x}_3+y_3^q+y_3{\gamma }^{l_i}+x_3{\xi }^{t_i},w_4=b_i{x}_4+y_4^q+y_4{\gamma }^{l_i}+x_4{\xi }^{t_i}.$$

If $w_1{w}_2=w_3{w}_4$, then we should check

$$\left\{w_1{\mathbb{F}}_q,w_2{\mathbb{F}}_q\right\}=\left\{w_3{\mathbb{F}}_q,w_4{\mathbb{F}}_q\right\}.$$

(3)

It is obvious that $\left\{1,{\gamma }^{l_i},{\gamma }^{2l_i},{\xi }^{t_i},{\xi }^{2t_i},{\gamma }^{l_i}{\xi }^{t_i}\right\}$ is an independent linear set. By comparing the coefficients of the equations in $w_1{w}_2=w_3{w}_4$, we obtain

$$\left\{\begin{array}{cc}\hfill x_1{x}_2& =x_3{x}_4\hfill \\ \hfill y_1{y}_2& =y_3{y}_4\hfill \\ \hfill x_1{y}_2+x_2{y}_1& =x_3{y}_4+x_4{y}_3\hfill \end{array}\right.$$

(4)

The remaining proofs are similar to Theorem 1, and we omit them here. □

Theorem 2.

We keep the notations above. Moreover, we let $1\le i\le {\rho }_1{\rho }_2$. We define $V_i=\left\{y^q+y{\gamma }^{l_i}+x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\right\}$. Hence, we can obtain that $V_i\in {\mathcal{G}}_q\left(n,k+1\right)$ is a $\left(k+1\right)$-dimensional Sidon space.

Proof. 

When $x=0$, the case is trivial and clearly holds. We only consider the case where $x\ne 0$ below.

Firstly, we let

$$v_1=y_1^q+y_1{\gamma }^{l_i}+x_1{\xi }^{t_i},v_2=y_2^q+y_2{\gamma }^{l_i}+x_2{\xi }^{t_i},$$

$$v_3=y_3^q+y_3{\gamma }^{l_i}+x_3{\xi }^{t_i},v_4=y_4^q+y_4{\gamma }^{l_i}+x_4{\xi }^{t_i}.$$

If $v_1{v}_2=v_3{v}_4$, then we should check

$$\left\{v_1{\mathbb{F}}_q,v_2{\mathbb{F}}_q\right\}=\left\{v_3{\mathbb{F}}_q,v_4{\mathbb{F}}_q\right\}.$$

(5)

It is obvious that $\left\{1,{\gamma }^{l_i},{\gamma }^{2l_i},{\xi }^{t_i},{\xi }^{2t_i},{\gamma }^{l_i}{\xi }^{t_i}\right\}$ is an independent linear set. By comparing the coefficients of the equations in $v_1{v}_2=v_3{v}_4$, we obtain

$$\left\{\begin{array}{cc}\hfill x_1{x}_2& =x_3{x}_4\hfill \\ \hfill y_1{y}_2& =y_3{y}_4\hfill \\ \hfill x_1{y}_2& =x_3{y}_4\hfill \end{array}\right.$$

(6)

From the equations above, we can have that ${\displaystyle \frac{x_1}{x_3}}={\displaystyle \frac{x_4}{x_2}}={\displaystyle \frac{y_1}{y_3}}={\displaystyle \frac{y_4}{y_2}}\in {\mathbb{F}}_q^{*}$, and then we get ${\displaystyle \frac{v_1}{v_3}}={\displaystyle \frac{v_4}{v_2}}\in {\mathbb{F}}_q^{*}$.

From Definition 1, we can obtain that $V_i$ is a $\left(k+1\right)$-dimensional Sidon space.

This ends the proof. □

4. Constructions of Cyclic Subspace Codes

In this section, we first review some basic definitions. Subsequently, with the help of these facts and the newly constructed Sidon spaces in Section 3, we construct several cyclic subspace codes.

Theorem 3.

We use the notations from Theorem 1. Moreover, we let $b_i\in {\mathbb{F}}_{q^k}^{*}$, where $1\le i\le {\rho }_1{\rho }_2\left(q^k-1\right)$, $b_{i_1}\ne b_{i_2}$. We define $U_i=\left\{x+y^q+y{\gamma }^{l_i}+b_{i}x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\right\}$. We let ${\mathcal{C}}_{u_i}=\left\{\alpha U_i\mid \alpha \in {\mathbb{F}}_{q^n}^{*}\right\}$, and then the subspace code ${\mathbb{C}}_1={\bigcup }_{i=1}^{{\rho }_1{\rho }_2\left(q^k-1\right)}{\mathcal{C}}_{u_i}$ is a cyclic constant dimension subspace code whose size is ${\rho }_1{\rho }_2\left(q^k-1\right){\displaystyle \frac{q^n-1}{q-1}}$ and minimum distance is $2k$.

Proof. 

We know that each $U_i$ is a Sidon space according to Theorem 1 and every ${\mathcal{C}}_i$ is a cyclic subspace code of size ${\displaystyle \frac{q^n-1}{q-1}}$ and minimum distance $2k$ by Lemma 2. In order to verify that ${\mathbb{C}}_1$ has a minimum distance of $2k$, we should prove that $dim\left(U_{i_1}\bigcap \lambda U_{i_2}\right)\le 1$. From Lemma 3, we know that it is equivalent to verify

$$\left\{u_1{\mathbb{F}}_q\right\}=\left\{u_2{\mathbb{F}}_q\right\},\left\{u_3{\mathbb{F}}_q\right\}=\left\{u_4{\mathbb{F}}_q\right\},$$

(7)

where $u_1,u_2\in U_{i_1},u_3,u_2\in U_{i_2}$.

Firstly, we let

$$u_1=x_1+y_1^q+y_1{\gamma }^{l_{i_1}}+b_{i_1}{x}_1{\xi }^{t_{i_1}},u_2=x_2+y_2^q+y_2{\gamma }^{l_{i_1}}+b_{i_1}{x}_2{\xi }^{t_{i_1}},$$

$$u_3=x_3+y_3^q+y_3{\gamma }^{l_{i_2}}+b_{i_2}{x}_3{\xi }^{t_{i_2}},u_4=x_4+y_4^q+y_4{\gamma }^{l_{i_2}}+b_{i_2}{x}_4{\xi }^{t_{i_2}}.$$

be four nonzero elements such that $u_1{u}_3=u_2{u}_4$.

$\left(I\right)$. $l_{i_1}=l_{i_2},t_{i_1}=t_{i_2}$.

It is obvious that $\left\{1,{\gamma }^{l_i},{\gamma }^{2l_i},{\xi }^{t_i},{\xi }^{2t_i},{\gamma }^{l_i}{\xi }^{t_i}\right\}$ is an independent linear set. By comparing the coefficients of the equations in $u_1{u}_3=u_2{u}_4$, we obtain

$$\left\{\begin{array}{cc}\hfill x_1{x}_3& =x_2{x}_4\hfill \\ \hfill y_1{y}_3& =y_2{y}_4\hfill \\ \hfill x_1{y}_3^q+x_3{y}_1^q& =x_2{y}_4^q+x_4{y}_2^q\hfill \\ \hfill b_{i_1}{x}_1{y}_3^q+b_{i_2}{x}_3{y}_1^q& =b_{i_1}{x}_2{y}_4^q+b_{i_2}{x}_4{y}_2^q\hfill \end{array}\right.$$

(8)

From $x_1{x}_3=x_2{x}_4$ and $y_1{y}_3=y_2{y}_4$ in $\left(8\right)$, we can have that ${\displaystyle \frac{x_1}{x_2}}={\displaystyle \frac{x_4}{x_3}}={\lambda }_1\in {\mathbb{F}}_q^{*}$, ${\displaystyle \frac{y_1}{y_2}}={\displaystyle \frac{y_4}{y_3}}={\lambda }_2\in {\mathbb{F}}_{q^k}^{*}$. Hence, the equations $b_{i_1}{x}_1{y}_3^q+b_{i_2}{x}_3{y}_1^q=b_{i_1}{x}_2{y}_4^q+b_{i_2}{x}_4{y}_2^q$ and $x_1{y}_3^q+x_3{y}_1^q=x_2{y}_4^q+x_4{y}_2^q$ can be simplified to $b_{i_1}\left({\lambda }_1-{\lambda }_2^q\right)x_2{y}_3^q=b_{i_2}\left({\lambda }_1-{\lambda }_2^q\right)x_3{y}_2^q$ and $\left({\lambda }_1-{\lambda }_2^q\right)x_2{y}_3^q=\left({\lambda }_1-{\lambda }_2^q\right)x_3{y}_2^q$.

$\left(\mathrm{i}\right)$. ${\lambda }_1\ne {\lambda }_2^q$.

From the equations above, we obtain ${\displaystyle \frac{x_2}{x_3}}={\displaystyle \frac{y_2^q}{y_3^q}}$ and ${\displaystyle \frac{b_{i_1}}{b_{i_2}}}={\displaystyle \frac{x_3{y}_2^q}{x_2{y}_3^q}}$. Then, we can obtain ${\displaystyle \frac{b_{i_1}}{b_{i_2}}}=1$, which is contradictory to the fact that $b_{i_1}\ne b_{i_2}$. Hence, we will not discuss it.

$\left(\mathrm{ii}\right)$. ${\lambda }_1={\lambda }_2^q$.

When ${\lambda }_1={\lambda }_2^q$, as well as ${\lambda }_1={\lambda }_2$, we have ${\displaystyle \frac{x_1}{x_2}}={\displaystyle \frac{x_4}{x_3}}={\displaystyle \frac{y_1}{y_2}}={\displaystyle \frac{y_4}{y_3}}\in {\mathbb{F}}_q^{*}$, and then we obtain ${\displaystyle \frac{u_1}{u_2}}={\displaystyle \frac{u_4}{u_3}}\in {\mathbb{F}}_q^{*}$.

$\left(II\right)$. $l_{i_1}\ne l_{i_2},t_{i_1}\ne t_{i_2}$.

It is obvious that $\left\{1,{\gamma }^{l_{i_1}},{\gamma }^{l_{i_2}},{\gamma }^{l_{i_1}+l_{i_2}},{\xi }^{t_{i_1}},{\xi }^{t_{i_2}},{\xi }^{t_{i_1}+t_{i_2}},{\gamma }^{l_{i_1}}{\xi }^{t_{i_2}},{\gamma }^{l_{i_2}}{\xi }^{t_{i_1}}\right\}$ is an independent linear set. By comparing the coefficients of the equations in $u_1{u}_3=u_2{u}_4$, we obtain

$$\left\{\begin{array}{cc}\hfill x_1{x}_3& =x_2{x}_4\hfill \\ \hfill y_1{y}_3& =y_2{y}_4\hfill \\ \hfill x_1{y}_3& =x_2{y}_4\hfill \\ \hfill x_3{y}_1& =x_4{y}_2\hfill \end{array}\right.$$

(9)

From the equations above, we have ${\displaystyle \frac{x_1}{x_2}}={\displaystyle \frac{x_4}{x_3}}={\displaystyle \frac{y_1}{y_2}}={\displaystyle \frac{y_4}{y_3}}\in {\mathbb{F}}_q^{*}$, and then we obtain ${\displaystyle \frac{u_1}{u_2}}={\displaystyle \frac{u_4}{u_3}}\in {\mathbb{F}}_q^{*}$.

To conclude, we have successfully verified that ${\mathbb{C}}_1$ has a minimum distance of $2k$ by Lemma 2. In particular, we have also verified that the ${\rho }_1{\rho }_2\left(q^k-1\right){\displaystyle \frac{q^n-1}{q-1}}$ elements of ${\mathcal{C}}_i$ are different, so

$$\left|{\mathbb{C}}_1\right|={\rho }_1{\rho }_2\left(q^k-1\right){\displaystyle \frac{q^n-1}{q-1}}.$$

This ends the proof. □

Corollary 2.

Similarly, we define $W_i=\left\{b_{i}x+y^q+y{\gamma }^{l_i}+x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\right\}$ to be as in Corollary 1. We let ${\mathcal{C}}_{w_i}=\left\{\alpha U_i\mid \alpha \in {\mathbb{F}}_{q^n}^{*}\right\}$, and then the subspace code ${\mathbb{C}}_3={\bigcup }_{i=1}^{{\rho }_1{\rho }_2\left(q^k-1\right)}{\mathcal{C}}_{w_i}$ is a cyclic constant dimension subspace code whose size is ${\rho }_1{\rho }_2\left(q^k-1\right){\displaystyle \frac{q^n-1}{q-1}}$ and minimum distance is $2k$.

Proof. 

We know that each $U_i$ is a Sidon space according to Corollary 1 and every ${\mathcal{C}}_i$ is a cyclic subspace code of size ${\displaystyle \frac{q^n-1}{q-1}}$ and minimum distance $2k$ by Lemma 2. In order to verify that ${\mathbb{C}}_3$ has a minimum distance of $2k$, we should prove that $dim\left(U_{i_1}\bigcap \lambda U_{i_2}\right)\le 1$. From Lemma 3, we know that it is equivalent to verify

$$\left\{w_1{\mathbb{F}}_q\right\}=\left\{w_2{\mathbb{F}}_q\right\},\left\{w_3{\mathbb{F}}_q\right\}=\left\{w_4{\mathbb{F}}_q\right\},$$

(10)

where $w_1,w_2\in W_{i_1},w_3,w_2\in W_{i_2}$. Let

$$w_1=b_{i_1}{x}_1+y_1^q+y_1{\gamma }^{l_{i_1}}+x_1{\xi }^{t_{i_1}},w_2=b_{i_1}{x}_2+y_2^q+y_2{\gamma }^{l_{i_1}}+x_2{\xi }^{t_{i_1}},$$

$$w_3=b_{i_2}{x}_3+y_3^q+y_3{\gamma }^{l_{i_2}}+x_3{\xi }^{t_{i_2}},w_4=b_{i_2}{x}_4+y_4^q+y_4{\gamma }^{l_{i_2}}+x_4{\xi }^{t_{i_2}}.$$

be four nonzero elements such that $w_1{w}_3=w_2{w}_4$.

The remaining proofs are similar to Theorem 3, and we omit them here. □

Theorem 4.

We use the notations from Theorem 2. Moreover, we let $1\le i\le {\rho }_1{\rho }_2$. We define the Sidon space $V_i=\left\{y^q+y{\gamma }^{l_i}+x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\right\}$. We let ${\mathcal{C}}_{v_i}=\left\{\alpha V_i\mid \alpha \in {\mathbb{F}}_{q^n}^{*}\right\}$, and then the subspace code ${\mathbb{C}}_2={\bigcup }_{i=1}^{{\rho }_1{\rho }_2}{\mathcal{C}}_{v_i}$; then, the subspace code ${\mathbb{C}}_2$ is a cyclic constant dimension subspace code whose size is ${\rho }_1{\rho }_2{\displaystyle \frac{q^n-1}{q-1}}$ and minimum distance is $2k$.

Proof. 

We know that each $V_i$ is a Sidon space according to Theorem 2 and every ${\mathcal{C}}_i$ is a cyclic subspace code of size ${\displaystyle \frac{q^n-1}{q-1}}$ and minimum distance $2k$ by Lemma 2. In order to verify that ${\mathbb{C}}_2$ has a minimum distance of $2k$, we should prove that $dim\left(V_{i_1}\bigcap \lambda V_{i_2}\right)\le 1$. From Lemma 3, we know that it is equivalent to verify

$$\left\{v_1{\mathbb{F}}_q\right\}=\left\{v_2{\mathbb{F}}_q\right\},\left\{v_3{\mathbb{F}}_q\right\}=\left\{v_4{\mathbb{F}}_q\right\},$$

(11)

where $v_1,v_2\in V_{i_1},v_3,v_4\in V_{i_2}$. Firstly, we let

$$v_1=y_1^q+y_1{\gamma }^{l_{i_1}}+x_1{\xi }^{t_{i_1}},v_2=y_2^q+y_2{\gamma }^{l_{i_1}}+x_2{\xi }^{t_{i_1}},$$

$$v_3=y_3^q+y_3{\gamma }^{l_{i_2}}+x_3{\xi }^{t_{i_2}},v_4=y_4^q+y_4{\gamma }^{l_{i_2}}+x_4{\xi }^{t_{i_2}}.$$

be four nonzero elements such that $v_1{v}_3=v_2{v}_4$.

It is obvious that $\left\{1,{\gamma }^{l_{i_1}},{\gamma }^{l_{i_2}},{\gamma }^{l_{i_1}+l_{i_2}},{\xi }^{t_{i_1}},{\xi }^{t_{i_2}},{\xi }^{t_{i_1}+t_{i_2}},{\gamma }^{l_{i_1}}{\xi }^{t_{i_2}},{\gamma }^{l_{i_2}}{\xi }^{t_{i_1}}\right\}$ is an independent linear set. By comparing the coefficients of the equations in $v_1{v}_3=v_2{v}_4$, we obtain

$$\left\{\begin{array}{cc}\hfill x_1{x}_3& =x_2{x}_4\hfill \\ \hfill y_1{y}_3& =y_2{y}_4\hfill \\ \hfill x_1{y}_3& =x_2{y}_4\hfill \\ \hfill x_3{y}_1& =x_4{y}_2\hfill \end{array}\right.$$

(12)

From the equations above, we have ${\displaystyle \frac{x_1}{x_2}}={\displaystyle \frac{x_4}{x_3}}={\displaystyle \frac{y_1}{y_2}}={\displaystyle \frac{y_4}{y_3}}\in {\mathbb{F}}_q^{*}$, and then we obtain ${\displaystyle \frac{v_1}{v_2}}={\displaystyle \frac{v_4}{v_3}}\in {\mathbb{F}}_q^{*}$.

To conclude, we have successfully verified that ${\mathbb{C}}_2$ has a minimum distance of $2k$ by Lemma 2. In particular, we have also verified that the ${\rho }_1{\rho }_2{\displaystyle \frac{q^n-1}{q-1}}$ elements of ${\mathcal{C}}_i$ are different, so

$$\left|{\mathbb{C}}_2\right|={\rho }_1{\rho }_2{\displaystyle \frac{q^n-1}{q-1}}.$$

This end the proof. □

Theorem 5.

We use the notations from Theorems 1 and 2. Then, we let ${\mathbb{C}}_1$ and ${\mathbb{C}}_2$ be the codes constructed in Theorems 3 and 4. Let $b_i\in {\mathbb{F}}_{q^k}^{*}$, where $1\le i\le {\rho }_1{\rho }_2{q}^k$, $b_{i_1}\ne b_{i_2}$. The subspace code $\mathcal{C}={\mathbb{C}}_1\bigcup {\mathbb{C}}_2$ is a cyclic constant dimension subspace code whose size is ${\rho }_1{\rho }_2{q}^k{\displaystyle \frac{q^n-1}{q-1}}$ and minimum distance is $2k$.

Proof. 

We know that each of $U_i,V_i$ is a Sidon space according to Theorems 3 and 4. In order to verify that $\mathcal{C}$ has a minimum distance of $2k$, we should prove that $dim\left(U_i\bigcap \lambda V_i\right)\le 1$. From Lemma 3, we know that is equivalent to verify

$$\left\{u_1{\mathbb{F}}_q\right\}=\left\{u_2{\mathbb{F}}_q\right\},\left\{v_3{\mathbb{F}}_q\right\}=\left\{v_4{\mathbb{F}}_q\right\},$$

(13)

where $u_1,u_2\in U_{i_1},v_3,v_4\in V_{i_2}$. Firstly, we let

$$\begin{array}{cc}\hfill u_1& =x_1+y_1^q+y_1{\gamma }^{l_{i_1}}+b_{i_1}{x}_1{\xi }^{t_{i_1}},u_2=x_2+y_2^q+y_2{\gamma }^{l_{i_1}}+b_{i_1}{x}_2{\xi }^{t_{i_1}},\hfill \\ \hfill v_3& =y_3^q+y_3{\gamma }^{l_{i_2}}+x_3{\xi }^{t_{i_2}},v_4=y_4^q+y_4{\gamma }^{l_{i_2}}+x_4{\xi }^{t_{i_2}}\hfill \end{array}$$

be four nonzero elements such that $u_1{v}_3=u_2{v}_4$.

$\left(I\right)$. $l_{i_1}=l_{i_2},t_{i_1}=t_{i_2}$.

It is obvious that $\left\{1,{\gamma }^{l_i},{\gamma }^{2l_i},{\xi }^{t_i},{\xi }^{2t_i},{\gamma }^{l_i}{\xi }^{t_i}\right\}$ is an independent linear set. By comparing the coefficients of the equations in $u_1{v}_3=u_2{v}_4$, we obtain

$$\left\{\begin{array}{cc}\hfill x_1{x}_3& =x_2{x}_4\hfill \\ \hfill y_1{y}_3& =y_2{y}_4\hfill \\ \hfill x_1{y}_3^q+x_3{y}_1^q& =x_2{y}_4^q+x_4{y}_2^q\hfill \\ \hfill b_{i_1}{x}_1{y}_3^q+b_{i_2}{x}_3{y}_1^q& =b_{i_1}{x}_2{y}_4^q+b_{i_2}{x}_4{y}_2^q\hfill \end{array}\right.$$

(14)

From $x_1{x}_3=x_2{x}_4$ and $y_1{y}_3=y_2{y}_4$ in $\left(14\right)$, we can have that ${\displaystyle \frac{x_1}{x_2}}={\displaystyle \frac{x_4}{x_3}}={\lambda }_1\in {\mathbb{F}}_q^{*}$, ${\displaystyle \frac{y_1}{y_2}}={\displaystyle \frac{y_4}{y_3}}={\lambda }_2\in {\mathbb{F}}_{q^k}^{*}$. Hence, the equations $x_1{y}_3^q+x_3{y}_1^q=x_2{y}_4^q+x_4{y}_2^q$ and $b_{i_1}{x}_1{y}_3^q+b_{i_2}{x}_3{y}_1^q=b_{i_1}{x}_2{y}_4^q+b_{i_2}{x}_4{y}_2^q$ can be simplified to $\left({\lambda }_1-{\lambda }_2^q\right)x_2{y}_3^q=\left({\lambda }_1-{\lambda }_2^q\right)x_3{y}_2^q$ and $b_{i_1}\left({\lambda }_1-{\lambda }_2^q\right)x_2{y}_3^q=b_{i_2}\left({\lambda }_1-{\lambda }_2^q\right)x_3{y}_2^q$.

$\left(\mathrm{i}\right)$. ${\lambda }_1\ne {\lambda }_2^q$.

From the equations above, we obtain ${\displaystyle \frac{x_2}{x_3}}={\displaystyle \frac{y_2^q}{y_3^q}}$ and ${\displaystyle \frac{b_{i_1}}{b_{i_2}}}={\displaystyle \frac{x_3{y}_2^q}{x_2{y}_3^q}}$. Then, we can obtain ${\displaystyle \frac{b_{i_1}}{b_{i_2}}}=1$, which is contradictory to the fact that $b_{i_1}\ne b_{i_2}$. Hence, we will not discuss it.

$\left(\mathrm{ii}\right)$. ${\lambda }_1={\lambda }_2^q$.

When ${\lambda }_1={\lambda }_2^q$, we have ${\displaystyle \frac{x_1}{x_2}}={\displaystyle \frac{x_4}{x_3}}={\displaystyle \frac{y_1}{y_2}}={\displaystyle \frac{y_4}{y_3}}\in {\mathbb{F}}_q^{*}$, and then we obtain ${\displaystyle \frac{u_1}{u_2}}={\displaystyle \frac{v_4}{v_3}}\in {\mathbb{F}}_q^{*}$.

$\left(II\right)$. $l_{i_1}\ne l_{i_2},t_{i_1}\ne t_{i_2}$.

It is obvious that $\left\{1,{\gamma }^{l_{i_1}},{\gamma }^{l_{i_2}},{\gamma }^{l_{i_1}+l_{i_2}},{\xi }^{t_{i_1}},{\xi }^{t_{i_2}},{\xi }^{t_{i_1}+t_{i_2}},{\gamma }^{l_{i_1}}{\xi }^{t_{i_2}},{\gamma }^{l_{i_2}}{\xi }^{t_{i_1}}\right\}$ is an independent linear set. By comparing the coefficients of the equations in $u_1{v}_3=u_2{v}_4$, we obtain

$$\left\{\begin{array}{cc}\hfill x_1{x}_3& =x_2{x}_4\hfill \\ \hfill y_1{y}_3& =y_2{y}_4\hfill \\ \hfill x_1{y}_3& =x_2{y}_4\hfill \\ \hfill x_3{y}_1& =x_4{y}_2\hfill \end{array}\right.$$

(15)

From the equations above, we have ${\displaystyle \frac{x_1}{x_2}}={\displaystyle \frac{x_4}{x_3}}={\displaystyle \frac{y_1}{y_2}}={\displaystyle \frac{y_4}{y_3}}\in {\mathbb{F}}_q^{*}$, and then we obtain ${\displaystyle \frac{u_1}{u_2}}={\displaystyle \frac{v_4}{v_3}}\in {\mathbb{F}}_q^{*}$.

To conclude, we have successfully verified that $\mathcal{C}$ has a minimum distance of $2k$ by Lemma 2. Thus,

$$\left|\mathcal{C}\right|={\rho }_1{\rho }_2{q}^k{\displaystyle \frac{q^n-1}{q-1}}.$$

This ends the proof. □

Corollary 3.

Similarly, we let ${\mathbb{C}}_2$ and ${\mathbb{C}}_3$ be the codes constructed in Theorem 4 and Corollary 2. The subspace code $\mathcal{C}={\mathbb{C}}_2\bigcup {\mathbb{C}}_3$ is a cyclic constant dimension subspace code whose size is ${\rho }_1{\rho }_2{q}^k{\displaystyle \frac{q^n-1}{q-1}}$ and minimum distance is $2k$.

Proof. 

We know that each of $W_i,V_i$ is a Sidon space according to Corollary 1 and Theorem 2. In order to verify that $\mathcal{C}$ has a minimum distance of $2k$, we should prove that $dim\left(W_i\bigcap \lambda V_i\right)\le 1$. From Lemma 3, we know that it is equivalent to verify

$$\left\{w_1{\mathbb{F}}_q\right\}=\left\{w_2{\mathbb{F}}_q\right\},\left\{v_3{\mathbb{F}}_q\right\}=\left\{v_4{\mathbb{F}}_q\right\},$$

(16)

where $w_1,w_2\in W_{i_1},v_3,v_4\in V_{i_2}$. Let

$$\begin{array}{cc}\hfill w_1& =b_{i_1}{x}_1+y_1^q+y_1{\gamma }^{l_{i_1}}+x_1{\xi }^{t_{i_1}},w_2=b_{i_1}{x}_2+y_2^q+y_2{\gamma }^{l_{i_1}}+x_2{\xi }^{t_{i_1}},\hfill \\ \hfill v_3& =y_3^q+y_3{\gamma }^{l_{i_2}}+x_3{\xi }^{t_{i_2}},v_4=y_4^q+y_4{\gamma }^{l_{i_2}}+x_4{\xi }^{t_{i_2}}\hfill \end{array}$$

be four nonzero elements such that $w_1{v}_3=w_2{v}_4$.

The remaining proofs are similar to Theorem 5, and we omit them here. □

Example 1.

We let $q=k=4,m_1=16,m_2=112$. Then, $r_1={\displaystyle \frac{m_1}{k}}=4>2,r_2={\displaystyle \frac{m_2}{m_1}}=7>2$. Thus, ${\rho }_1=\lceil m_1/2k\rceil -1=1,{\rho }_2=\lceil m_2/2m_1\rceil -1=3$. Hence, the cyclic subspace code constructed in Theorem 5 has a size ${\rho }_1{\rho }_2{q}^k{\displaystyle \frac{q^n-1}{q-1}}=4^5\left(4^{80}-1\right)$.

5. Conclusions

In this paper, we obtain several new $\left(k+1\right)$-dimensional Sidon spaces in the form of $U_i=\left\{x+y^q+y{\gamma }^{l_i}+b_{i}x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\right\}$ and $V_i=\left\{y^q+y{\gamma }^{l_i}+x{\xi }^{t_i}\mid x\in {\mathbb{F}}_q,y\in {\mathbb{F}}_{q^k}\right\}.$ Furthermore, by combining these Sidon spaces, we obtain cyclic subspace codes with sizes ${\rho }_1{\rho }_2(q^k-1){\displaystyle \frac{q^n-1}{q-1}}$, ${\rho }_1{\rho }_2{\displaystyle \frac{q^n-1}{q-1}}$, and ${\rho }_1{\rho }_2{q}^k{\displaystyle \frac{q^n-1}{q-1}}.$ Compared to the known constructions in [15,20,22], our work introduces different construction forms of Sidon spaces and cyclic subspace codes. Additionally, when compared to the constructions in [16,18,20], the cyclic subspace codes we construct have a minimum distance of $2k$, leading to superior error correction performance. In the future, we will continue to delve into the research on the construction forms of Sidon spaces and cyclic subspace codes.