Generalized block inserting for constructing new constant dimension codes

Abstract

Constant dimension codes (CDCs) have drawn extensive attention due to their applications in random network coding. A fundamental problem for CDCs is to explore the maximum possible cardinality A_q(n,d,k) of a set of k-dimensional subspaces in $\mathbb {F}^{n}_{q}$ such that the subspace distance statisfies dis(U,V ) = 2k − 2 dim(U ∩ V ) ≥ d for all pairs of distinct subspaces U and V in this set. In this paper, by means of an appropriate combination of the matrix blocks from rank metric codes and small CDCs, we present three constructions of CDCs based on the generalized block inserting construction by Niu et al. in 2021. According to our constructions, we obtain 28 new lower bounds for CDCs which are better than the previously known lower bounds.

1 Introduction

Let $\mathbb {F}^{n}_{q}$ be an n-dimensional vector space over the finite field $\mathbb {F}_{q}$ with q elements. Given a nonnegative integer k ≤ n, let $\mathcal {G}_{q}(n,k)$ be the set of all k-dimensional subspaces in $\mathbb {F}^{n}_{q}$. The cardinality of $\mathcal {G}_{q}(n,k)$ is equal to the q-ary Gaussian coefficient$\left [\begin {array} {c} n \\ k \end {array} \right ]_{q} =\prod \limits _{i=0}^{k-1} \frac {q^{n-i}-1}{q^{k-i}-1}$. For any two subspaces $U,V \in \mathcal {G}_{q}(n,k)$, the subspace distance between U and V is defined as dis(U,V ) = dim(U) + dim(V ) − 2 dim(U ∩ V ) = 2k − 2 dim(U ∩ V ), where dim(⋅) denotes the dimension of a vector space over $\mathbb {F}_{q}$. An (n,d,k)_q constant dimension code (CDC) $\mathcal {C}$ is a nonempty set in $\mathcal {G}_{q}(n,k)$ such that dis(U,V ) ≥ d for all pairs of distinct subspaces $U,V\in \mathcal {C}$. In particular, we call $\mathcal {C}$ an (n,M,d,k)_q CDC if $\mathcal {C}$ has M codewords. The maximum cardinality among all (n,d,k)_q CDCs is denoted by A_q(n,d,k).

Due to the work of Kötter and Kschishang [18], where they presented an application of subspace codes for correcting errors and erasures in random network coding, constant dimension codes have been studied extensively. The main question of constant dimension subspace coding asks for the maximum cardinality A_q(n,d,k). However, it is hard to obtain the exact value of A_q(n,d,k). A lot of constant dimension codes were constructed in the literature, readers can refer to [5, 6, 10,11,12,13,14,15,16,17, 22, 26,27,28]. The homepage [15] lists the latest lower and upper bounds of A_q(n,d,k) for 4 ≤ n ≤ 19 and q ∈{2,3,4,5,7,8,9}. Rank metric codes, maximum rank distance codes in particular, are utilized to construct CDCs. The rank metric on the matrix space $\mathbb {F}^{m\times n}_{q}$ is defined by the rank of matrices. For any two matrices A, $B\in \mathbb {F}^{m\times n}_{q}$, the rank distance between A and B is defined as d_r(A,B) = rank(A − B). The minimum rank distance of a code ${\mathcal{M}}\subset \mathbb {F}^{m\times n}_{q}$ is defined as $d_{r}({\mathcal{M}})= \min \limits {\{d_{r}(A,B) : A\neq B, ~A,B\in {\mathcal{M}}\}}.$ If the minimum rank distance of a nonempty subset ${\mathcal{M}}$ in $\mathbb {F}^{m\times n}_{q}$ is at least d, then ${\mathcal{M}}$ is called an (m,n,d)_q rank metric code (RMC). Delsarte [4] and Gabidulin [8] independently showed that the cardinality of an (m,n,d)_q RMC is upper bounded by ${\Delta }(m,n,d)_{q}= q^{\max \limits {\{m,n\}\times (\min \limits {\{m,n\}-d+1}})}$. We call ${\mathcal{M}}$ a maximum rank distance (MRD) code if ${\mathcal{M}}$ is an RMC achieving this bound.

A method for constructing CDCs is the lifting construction [18]: for any fixed (m,n,d)_q MRD code ${\mathcal{M}}$, the corresponding CDC is the set consisting of all subspaces spanned by the m rows of matrix (I|M), where I is the m × m identity matrix and $M\in {\mathcal{M}}$. Nevertheless, this method usually does not generate codes with large cardinalities. In [5], by lifting Ferrers diagram rank metric codes, Etzion and Silberstein proposed the multilevel construction which generalizes the lifted MRD codes construction. Gluesing-Luerssen and Troha presented the linkage construction by linking two CDCs in [9] and Chen et al. in [1] further presented the so-called parrallel linkage construction by modifying the linkage construction. Later, He [11] constructed CDCs from two parallel linkage constructions which generalize the result of [1]. In [23], Liu et al. generalized the parallel construction and the multilevel construction by introducing rank-restricted rank metric codes. For other recent constructions of CDCs by the multilevel construction, we can refer to [7, 14, 19, 21, 25]. Cossidenta et al. [2] and Heinlei [16] respectively generalized the parallel linkage construction in different ways. In [20], Lao et al. proposed two block inserting constructions which insert flexibly CDCs constructed by matrix blocks from small CDCs and RMCs into the parallel linkage constructions in [11], see also [21]. Recently, Niu et al. in [24] generalized the block inserting construction by padding more matrices.

In this paper, inspired by the ideas of [20] and [24], we propose three constructions of CDCs with restricted parameters based on the generalized block inserting construction through an appropriate combination of matrix blocks from rank metric codes and small CDCs. Our results improve the previously best known lower bounds for CDCs. We organize the remaining part of this paper as follows. In Section 2, we introduce some essential definitions and results which are useful for our main results. In Section 3, we propose our three improved constructions of constant dimension codes and obtain 28 new lower bounds for CDCs. We conclude this paper in Section 4.

2 Preliminaries

In this section, we will recall some basic definitions and known results which are necessary to prove our results. More details are available in [2, 4, 5, 8].

2.1 Identifying vector and Delsarte Theorem

Let U be a k-dimensional subspace in $\mathbb {F}^{n}_{q}$, a generator matrix of U is a k × n matrix whose k rows form a basis of U. It is a well-known fact that there exists a unique generator matrix of U in reduced row echelon form (RREF) and denote it by E(U), refer to [5].

Definition 2.1

Let $U\in \mathcal {G}_{q}(n,k)$ and E(U) be the generator matrix in reduced row echelon form of U. The identifying vector of U is denoted by i(U), which is a binary vector of length n and weight k such that i(U) has exactly k ones in the positions of the pivot columns of E(U).

Example 2.2

Suppose U is a 4-dimensional subspace in $\mathbb {F}^{6}_{2}$ with the generator matrix

$$E(U)=\left( \begin{array}{cccccc} 1&0&1&0&0&0\\ 0&1&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1 \end{array}\right).$$

Then the identifying vector of U is i(U) = (110011).

The following Lemma 2.3 is crucial, we can obtain a lower bound of the subspace distance related to the Hamming distance.

Lemma 2.3 (Lemma 2 in 5)

Let U, $V\in \mathcal {G}_{q}(n,k)$. Assume i(U) and i(V ) are the identifying vectors of U and V, respectively. Then dis(U,V ) ≥ d_H(i(U),i(V )), where d_H denotes the Hamming distance.

If a rank metric code is a linear subspace over $\mathbb {F}_{q}$ in the matrix space $\mathbb {F}^{m\times n}_{q}$, then it is said to be $\mathbb {F}_{q}$-linear and we abbreviate it by linear in this paper. Linear MRD codes exist for all possible parameters (cf. [4, 8]). Moreover, if $d^{\prime }> d$, then we can assume that there exists a linear (m,n,d)_q MRD code containing a linear $(m,n, d^{\prime })_{q}$ MRD code as a subcode. If any codeword M in an (m,n,d)_q RMC satisfies rank(M) ≤ r, then we call it a rank-restricted rank metric code (RRMC) and denote it by (m,n,d;r)_q. The theory of Delsarte allows to determine the rank distribution of a linear MRD code by its parameters, refer to Theorem 5.6 in [4] or Corollary 26 in [3]. Thus a lower bound for the cardinality of RRMC can be given.

Theorem 2.4 (Delsarte Theorem 4)

Let m, n, d and r be positive integers such that m ≥ n and d ≤ r ≤ n. Assume M is an (m,n,d)_q MRD code. Then the number of codewords with rank r in M is given by

$$D(m,n, d,r)_{q} = \left[\begin{array} {c} n \\ r \end{array} \right]_{q} \sum\limits_{i=0}^{r-d} (-1)^{i} q^{\tbinom{i}{2}} \left[\begin{array} {c} r \\ i \end{array} \right]_{q}\big(q^{m(r-i-d+1)}-1 \big).$$

We can construct an (m,n,d;r)_q RRMC from a subset of an (m,n,d)_q linear MRD code satisfying the rank restriction (cf. [15, 16]). Hence, the maximum cardinality of an (m,n,d;r)_q RRMC is lower bounded by $1+{\sum }_{i=d}^{r} D(m,n,d;i)_{q}$, which is abbreviated by Δ(m,n,d;r)_q.

2.2 Parallel linkage construction and subcodes construction

In this subsection, we first review some essential notation and known results about the parallel linkage construction. We finally introduce the subcode construction in [2] since it is important for constructing the desired rank metric codes in our constructions.

For any set ${\mathcal{M}} \subset \mathbb {F}^{k\times n}_{q}$, if 1) for any $M\subset {\mathcal{M}}$, rank(M) = k, and 2) for any two distinct matrices M₁ and M₂ in ${\mathcal{M}}$, rs(M₁)≠rs(M₂), where rs(M₁) denotes the subspace spanned by the k rows of M₁, then ${\mathcal{M}}$ is called an SC-representation set[9]. It is obvious that $\{rs(M) : M\in {\mathcal{M}}\}$ corresponds to a constant dimension code. The following Lemma 2.5 is the parallel linkage construction which generalizes Theorem 4 in [1]. In this paper, (M₁|M₂) denotes a matrix concatenated from M₁ and M₂.

Lemma 2.5 (Theorem 2 in 11)

Let m, n, k and d be positive integers such that m ≥ k and n ≥ k. Let $\mathcal {A}_{1}$ and $\mathcal {A}_{2}$ be two SC-representation sets of (m,2d,k)_q and (n,2d,k)_q CDCs, respectively. Let $\mathcal {Q}_{1}$ be a (k,n,d)_q RMC and $\mathcal {Q}_{2}$ be a (k,m,d;k − d)_q RRMC. Define $\mathcal {C} = \mathcal {C}_{1}\cup \mathcal {C}_{2}$, where

$$\begin{array}{@{}rcl@{}} \mathcal{C}_{1} & =& \{ rs(A_{1}|Q_{1}) : A_{1} \in \mathcal{A}_{1}, Q_{1} \in \mathcal{Q}_{1}\},\\ \mathcal{C}_{2} &=& \{ rs(Q_{2}|A_{2}) : A_{2} \in \mathcal{A}_{2}, Q_{2} \in \mathcal{Q}_{2}\}. \end{array}$$

Then $\mathcal {C}$ is an (m + n,2d,k)_q CDC. Particularly, A_q(m + n,2d,k) ≥ A_q(m,2d,k) ⋅Δ(k,n,d)_q + A_q(n,2d,k) ⋅Δ(k,m,d;k − d)_q.

The subcode construction occured in [2], which can be described as the following Lemma 2.6. This construction can be used for the block inserting construction in Section 3. We give a complete proof here because [2] didn’t prove the uniqueness.

Lemma 2.6 (Subcode construction in 2)

Let ${\mathcal{M}}$ be an (m,n,d)_q linear MRD containing an $(m, n, d^{\prime })_{q}$ linear MRD code $\mathcal {C}$ as a subcode, where d and $d^{\prime }$ are positive integers such that $d^{\prime }>d$. Then there exist s subcodes of ${\mathcal{M}}$ fulfilling the following conditions: (1) ${\mathcal{M}}_{i}$ is an $(m, n, d^{\prime })_{q}$ MRD, 1 ≤ i ≤ s; (2) for $M\in {\mathcal{M}}_{i}$, $M^{\prime }\in {\mathcal{M}}_{j} (1\leq i<j\leq s)$, $M\neq M^{\prime }$ and rank$(M-M^{\prime })\geq d$. Here, $s=\frac {\Delta (m, n, d)_{q}}{\Delta (m, n, d^{\prime })_{q}}$. Moreover, $\mathcal {C}$ is the unique linear MRD code in these s subcodes.

Proof

Let $M_{j}\in {\mathcal{M}}$ and denote ${\mathcal{M}}_{j}= M_{j}+\mathcal {C}$. Then ${\mathcal{M}}_{j}$ is a subcode of ${\mathcal{M}}$. For any two distinct codewords C₁, $C_{2}\in \mathcal {C}$, we have M_j + C₁≠M_j + C₂, which implies that ${\mathcal{M}}_{j}$ is an $(m,n,d^{\prime })_{q}$ MRD subcode of ${\mathcal{M}}$. If M_i + C₁ = M_j + C₂, where M_i, $M_{j}\in {\mathcal{M}}$ and C₁, $C_{2}\in \mathcal {C}$, then $M_{i}-M_{j}\in \mathcal {C}$ since $\mathcal {C}$ is a linear MRD code. Thus ${\mathcal{M}}_{j}$ can be viewed as a coset of ${\mathcal{M}}$ and there exist $s=\frac {\Delta (m, n, d)_{q}}{\Delta (m, n, d^{\prime })_{q}}$ distinct $(m,n,d^{\prime })_{q}$ MRD codes. Assume $M_{j}+\mathcal {C}$ is a linear MRD code, then $a(M_{j}+C)\in M_{j}+\mathcal {C}$ for any $a\in \mathbb {F}_{q}$ and any $C\in \mathcal {C}$, which implies $M_{j}\in \mathcal {C}$ and thus $M_{j}+\mathcal {C}=\mathcal {C}$. So, we can conclude that $\mathcal {C}$ is the unique linear MRD code in these s subcodes.

3 Our constructions of constant dimension codes

In this section, we first introduce the generalized block inserting construction. Later, we describe our three improved constructions based on the generalized block inserting construction.

Suppose $A\in \mathbb {F}^{k\times m}_{q}$ is a matrix in RREF with rank(A) = k. Define the embedding map $\sigma _{A} :\mathbb {F}^{l\times (m-k)}_{q}\rightarrow \mathbb {F}^{l\times m}_{q}$ by inserting k zero columns ${\underbrace {(0, 0, \cdots , 0)}_{l}}^{T}$ into F in the positions of the pivot columns of A, where $F \in \mathbb {F}^{l\times (m-k)}_{q}$. For example,

$$A=\left( \begin{array}{cccccc} 1&1&1&0&0&1\\ 0&0&1&1&0&1\\ 0&0&0&0&1&1 \end{array} \right) \text{and} ~F=\left( \begin{array}{ccc} 1&0&0\\ 1&0&1\\ 0&1&1\\ 1&0&1 \end{array} \right),$$

where A has pivot columns 1,3,5. Then

$$\sigma_{A}(F)=\left( \begin{array}{cccccc} 0&1&0&0&0&0\\ 0&1&0&0&0&1\\ 0&0&0&1&0&1\\ 0&1&0&0&0&1 \end{array} \right).$$

In [20], Lao et al. proposed the block inserting construction which adds subspaces with generator matrix concatenated by small matrix blocks from CDCs and RMCs into the CDC in Lemma 2.5 by restricting the rank of matrices. Using the above embedding map σ_A(F), Niu et al. generalized the block inserting construction in the following Lemma 3.1.

Lemma 3.1 (Theorem 3.2 in 24)

Let m, n, k, d, a₁, a₂, b₁, b₂, t₁ and t₂ be positive integers with m ≥ k, n ≥ k, a₁ + a₂ = k, b₁ + b₂ ≥ d, a₁ ≤ t₁ ≤ m − d, a₂ ≤ t₂ ≤ n − d, and a_i ≥ d, 1 ≤ b_i ≤ d for i = 1,2. Let $\mathcal {U}_{i}=\{U_{i}\in \mathbb {F}^{a_{i}\times t_{i}}_{q} : U_{i} ~\text {in RREF}, \text {rank}(U_{i})=a_{i} \}$ be an SC-representation set of (t_i,2d,a_i)_q CDC for i = 1, 2. Let

$$\mathcal{M}_{3}\subset \{(F_{3}|M_{3}): F_{3}\in \mathbb{F}^{a_{1}\times (t_{2}-a_{2})}_{q}, M_{3}\in \mathbb{F}^{a_{1}\times(n-t_{2})}_{q} \}$$

and

$$\mathcal{M}_{4}\subset\{ (F_{4}|M_{4}): F_{4}\in \mathbb{F}^{a_{2}\times (t_{1}-a_{1})}_{q}, M_{4}\in \mathbb{F}^{a_{2}\times(m-t_{1})}_{q} \}$$

be two RRMCs with respective parameters (a₁,n − a₂,d;a₁ − d)_q and (a₂,m − a₁,d;a₂ − d)_q. Given integer s, for 1 ≤ r ≤ s, let ${\mathcal{M}}^{r}_{1}$ be an (a₁,m − t₁,d)_q MRD code and ${\mathcal{M}}^{r}_{2}$ be an (a₂,n − t₂,d)_q MRD code. For i = 1, 2, let $M\in {\mathcal{M}}^{r}_{i}$ and $M^{\prime }\in {\mathcal{M}}^{r^{\prime }}_{i}$ for $1\leq r<r^{\prime }\leq s$, assume that $M\neq M^{\prime }$ and rank$(M-M^{\prime })\geq b_{i}$. Define $\mathcal {C}_{3}= \cup ^{s}_{r=1}\mathcal {X}_{r}$,

$$\mathcal{X}_{r}=\Biggl\{ rs \left( \begin{array}{cccc} U_{1}&M_{1}&\sigma_{U_{2}}(F_{3})&M_{3}\\ \sigma_{U_{1}}(F_{4})&M_{4}&U_{2}&M_{2} \end{array}\right)\Biggr\},$$

where $U_{1}\in \mathcal {U}_{1}$, $U_{2}\in \mathcal {U}_{2}$, $M_{i}\in {\mathcal{M}}^{r}_{i}$ for i = 1, 2, and $(F_{i}|M_{i})\in {\mathcal{M}}_{i}$ for i = 3, 4. Then $\mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}$ is an (m + n,2d,k)_q CDC, where $\mathcal {C}_{1}$ and $\mathcal {C}_{2}$ are defined in Lemma 2.5. As a consequence,

$$\begin{array}{@{}rcl@{}} A_{q}(m+n, 2d, k) &\geq & A_{q}(m,2d,k)\cdot {\Delta} (k, n, d)_{q}+ A_{q}(n,2d, k)\cdot {\Delta} (k, m, d; k-d)_{q}\\ &&+ s\cdot A_{q}(t_{1}, 2d, a_{1})\cdot A_{q}(t_{2}, 2d, a_{2}) \cdot {\Delta}(a_{1}, m-t_{1}, d)_{q}\cdot {\Delta}(a_{2}, n-t_{2}, d)_{q}\\ &&\cdot {\Delta} (a_{1}, n-a_{2},d;a_{1}-d)_{q}\cdot {\Delta}(a_{2}, m-a_{1}, d; a_{2}-d)_{q}. \end{array}$$

Next, we construct new CDCs with larger cardinalities based on Lemma 3.1. If the parameters a₂ and t₂ satisfy n − t₂ ≥ a₂, then more subspaces can be inserted into the CDC in Lemma 3.1. Thus, using the similar block inserting construction via exchanging the positions of the matrix blocks, we propose the following Construction A. In this paper, O_m×n denotes the zero matrix with size m × n.

Theorem 3.2 (Construction A)

With the same notation used in Lemma 3.1. Additionally, n − t₂ ≥ a₂ and k − t₁ ≥ 2d. Let c₁ and c₂ be positive integers with c₁ + c₂ ≥ d and 1 ≤ c_i ≤ d for i = 1, 2. Let $\mathcal {E}=\{E\in \mathbb {F}^{a_{1}\times t_{1}}_{q} : E ~\text {in RREF}, \text {rank}(E)=a_{1} \}$ $\big (\text {resp.}~ {\mathcal{H}}=\{ H\in \mathbb {F}^{a_{2}\times (n-t_{2})}_{q} : H ~\text {in RREF}, \text {rank}(H)=a_{2}\} \big )$ be an SC-representation set of (t₁,2d,a₁)_q (resp. (n − t₂,2d,a₂)_q) CDC. Let

$$\mathcal{M}_{1,2}\subset \{ (M_{1,2}|F_{1,2}): F_{1,2}\in \mathbb{F}^{a_{1}\times (n-t_{2}-a_{2})}_{q}, M_{1,2}\in \mathbb{F}^{a_{1}\times t_{2}}_{q} \}$$

and

$$\mathcal{M}_{2,1}\subset\{ (F_{2,1}|M_{2,1}): F_{2,1}\in \mathbb{F}^{a_{2}\times (t_{1}-a_{1})}_{q}, M_{2,1}\in \mathbb{F}^{a_{2}\times (m-t_{1})}_{q} \}$$

be two RRMCs with respective parameters (a₁,n − a₂,d;a₁ − d)_q and (a₂,m − a₁,d;a₂ − d)_q. Given integer t, for 1 ≤ r ≤ t, let ${\mathcal{M}}^{r}_{1,1}$ be an (a₁,m − t₁,d)_q MRD code and ${\mathcal{M}}^{r}_{2,2}$ be an (a₂,t₂,d;k − t₁ − d)_q RRMC. For i = 1, 2, let $M\in {\mathcal{M}}^{r}_{i,i}$ and $M^{\prime }\in {\mathcal{M}}^{r^{\prime }}_{i,i}$ for $1\leq r<r^{\prime }\leq t$, assume that $M\neq M^{\prime }$ and rank$(M-M^{\prime })\geq c_{i}$. Let $\mathcal {C}_{1}$, $\mathcal {C}_{2}$ and $\mathcal {C}_{3}$ be the codes defined in Lemma 3.1. Define $\mathcal {C}_{4}= \cup ^{l}_{r=t}{\mathcal{B}}_{r}$,

$$\mathcal{B}_{r}=\Biggl\{ rs \left( \begin{array}{cccc} E&M_{1,1}&M_{1,2}&\sigma_{H}(F_{1,2})\\ \sigma_{E}(F_{2,1})&M_{2,1}&M_{2,2}&H \end{array}\right)\Biggr\},$$

where $E\in \mathcal {E}$, $H\in {\mathcal{H}}$, $M_{i,i}\in {\mathcal{M}}^{r}_{i,i}$ for i = 1, 2, $(M_{1,2}|F_{1,2})\in {\mathcal{M}}_{1,2}$ and $(F_{2,1}|M_{2,1})\in {\mathcal{M}}_{2,1}$. Then $\mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}\cup \mathcal {C}_{4}$ is an (m + n,2d,k)_q CDC.

Proof

Obviously, the subpsaces in $\mathcal {C}_{4}$ are k-dimensional. We first show that $\mathcal {C}_{4}$ is an (m + n,2d,k)_q CDC. Let

$$W_{1}=rs(P_{1})=\Biggl\{ rs \left( \begin{array}{cccc} E&M_{1,1}&M_{1,2}&\sigma_{H}(F_{1,2})\\ \sigma_{E}(F_{2,1})&M_{2,1}&M_{2,2}&H \end{array}\right)\Biggr\}\in \mathcal{B}_{r},$$

$$W^{\prime}_{1}=rs(P^{\prime}_{1})=\Biggl\{ rs \left( \begin{array}{cccc} E^{\prime}&M^{\prime}_{1,1}&M^{\prime}_{1,2}&\sigma_{H^{\prime}}(F^{\prime}_{1,2})\\ \sigma_{E^{\prime}}(F^{\prime}_{2,1})&M^{\prime}_{2,1}&M^{\prime}_{2,2}&H^{\prime} \end{array}\right)\Biggr\}\in \mathcal{B}_{r^{\prime}}$$

be two distinct k-dimensionl subspaces. Then the dimension of $W_{1}\cap W^{\prime }_{1}$ is

$$\dim(W_{1}\cap W_{1}^{\prime})= \dim\left( \{ \left( \alpha_{1}, \alpha_{2}\right): \exists \left( \beta_{1}, \beta_{2}\right)~\text{s.t}.~ \left( \alpha_{1}, \alpha_{2}\right)P_{1}=\left( \beta_{1},\beta_{2}\right)P^{\prime}_{1} \} \right),$$

where α₁, $\beta _{1}\in \mathbb {F}^{a_{1}}_{q}$, and α₂, $\beta _{2}\in \mathbb {F}^{a_{2}}_{q}$. We analyze the dimension of $W_{1}\cap W^{\prime }_{1}$ from three cases.

Case 1 If $E\neq E^{\prime }$, then $\dim (W_{1}\cap W^{\prime }_{1})\leq \dim (E\cap E^{\prime })+a_{2}\leq a_{1}-d+a_{2}=k-d$.

Case 2 If $H\neq H^{\prime }$, then $\dim (W_{1}\cap W^{\prime }_{1})\leq \dim (H\cap H^{\prime })+a_{1}\leq a_{2}-d+a_{1}=k-d$.

Case 3 If $E=E^{\prime }$ and $H=H^{\prime }$, then

$$\left\{ \begin{aligned} \alpha_{1} E +\alpha_{2}\sigma_{E}(F_{2,1})&=\beta_{1} E+\beta_{2} \sigma_{E}(F^{\prime}_{2,1}),\\ \alpha_{1}\sigma_{H}(F_{1,2})+\alpha_{2} H&=\beta_{1}\sigma_{H}(F^{\prime}_{1,2})+\beta_{2} H. \end{aligned}\right.$$

By the above equation system, we get

$$\left\{\begin{aligned} \left( \alpha_{1}-\beta_{1} \right)E&=\beta_{2} \sigma_{E}(F^{\prime}_{2,1})-\alpha_{2} \sigma_{E}(F_{2,1}),\\ \left( \alpha_{2}-\beta_{2} \right) H&=\beta_{1}\sigma_{H}(F^{\prime}_{1,2})-\alpha_{1}\sigma_{H}(F_{1,2}). \end{aligned}\right.$$

According to the definitions of σ_E and σ_H, we can induce that α₁ = β₁ and α₂ = β₂. Thus,

$$\begin{array}{@{}rcl@{}} \dim(W_{1}\cap W^{\prime}_{1})={}& \dim\left\{ \left( \alpha_{1}, \alpha_{2} \right): \left( \alpha_{1},\alpha_{2} \right) P=0\right\}, \end{array}$$

where

$$\begin{array}{@{}rcl@{}} P={} &\left\{\left( \begin{array}{cccc} O_{a_{1}\times t_{1}}&M_{1,1}-M^{\prime}_{1,1}&M_{1,2}-M^{\prime}_{1,2}&\sigma_{H}(F_{1,2}-F^{\prime}_{1,2})\\ \sigma_{E}(F_{2,1}-F^{\prime}_{2,1})&M_{2,1}-M^{\prime}_{2,1}&M_{2,2}-M^{\prime}_{2,2}&O_{a_{2}\times (n-t_{2})} \end{array} \right)\right\}. \end{array}$$

We continue analyzing the dimension of $\dim (W_{1}\cap W^{\prime }_{1})$ from the following three subscases.

• (a) If $(M_{1,2}|F_{1,2})\neq (M^{\prime }_{1,2}|F^{\prime }_{1,2})$ or $(F_{2,1}|M_{2,1})\neq (F^{\prime }_{2,1}|M^{\prime }_{2,1})$, then $\dim (W_{1}\cap W^{\prime }_{1})=k-\text {rank}(P)\leq k-d$.

• (b) If $(M_{1,2}|F_{1,2})= (M^{\prime }_{1,2}|F^{\prime }_{1,2})$, $(F_{2,1}|M_{2,1})= (F^{\prime }_{2,1}|M^{\prime }_{2,1})$ and $r=r^{\prime }$, then $M_{1,1}\neq M^{\prime }_{1,1}$ or $M_{2,2}\neq M^{\prime }_{2,2}$. It follows that $\dim (W_{1}\cap W^{\prime }_{1})=k-\text {rank}(P)\leq k-d$.

• (c) If $(M_{1,2}|F_{1,2})= (M^{\prime }_{1,2}|F^{\prime }_{1,2})$, $(F_{2,1}|M_{2,1})= (F^{\prime }_{2,1}|M^{\prime }_{2,1})$ and $r\neq r^{\prime }$, then $\text {rank}(M_{1,1}-M^{\prime }_{1,1})\geq c_{1}$ and $\text {rank}(M_{2,2}-M^{\prime }_{2,2})\geq c_{2}$. It follows that

  $$\begin{array}{@{}rcl@{}} \dim(W_{1}\cap W_{1}^{\prime})&=& k-\text{rank}(P)\\ &=& k-(\text{rank}(M_{1,1}-M^{\prime}_{1,1})+\text{rank}(M_{2,2}-M^{\prime}_{2,2}))\\ &\leq& k-(c_{1}+c_{2})\\ &\leq& k-d. \end{array}$$

Thus, $\text {dis}(W_{1}, W_{1}^{\prime })\geq 2k-2~\!\dim (W_{1}\cap W_{1}^{\prime })\geq 2d.$

Finally, we prove that the distance between $W_{1}\in \mathcal {C}_{4}$ and $W_{2}\in \mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}$ is at least 2d. We can discuss from the following three cases.

• (1) If $W_{2}\in \mathcal {C}_{1}$, then there exist k ones in the first m positions of i(W₂). However, i(W₁) has no more than k − d ones in the first m positions since rank(E) + rank(F_2,1|M_2,1) ≤ a₁ + a₂ − d = k − d.

• (2) If $W_{2}\in \mathcal {C}_{2}$, then W₂ = rs(P₂) = rs(Q₂|A₂). The dimension of W₁ ∩ W₂ is

  $$\dim(\{({\alpha}_{1}, {\alpha}_{2}): \exists ~ (\beta_{1}, \beta_{2} )~ \text{s.t.}~ ({\alpha}_{1}, {\alpha}_{2})P_{1}=({\beta}_{1}, {\beta}_{2})P_{2}\}),$$

  where α_i, $\beta _{i}\in \mathbb {F}^{a_{i}}_{q}$ for i = 1,2. As rank(A₂) = k and

  $$\begin{array}{@{}rcl@{}} \text{rank}\left( \begin{array}{cc} M_{1,2}&\sigma_{H}(F_{1,2})\\ M_{2,2}& H \end{array} \right) &\leq & \text{rank}(H) + \text{rank}(M_{1,2}|\sigma_{H}(F_{1,2}))\\ &= &\text{rank}(H)+ \text{rank}(M_{1,2}|F_{1,2}) \\ &\leq &a_{2}+a_{1}-d\\ &= &k-d, \end{array}$$

  we have

  $$\dim(W_{1}\cap W_{2}) \leq \dim (\{({\alpha}_{1}, {\alpha}_{2}): \exists ~ (\beta_{1}, \beta_{2} )~ \text{s.t.}~ ({\alpha}_{1}, {\alpha}_{2})\left( \begin{array}{cc} M_{1,2}&\sigma_{H}(F_{1,2})\\ M_{2,2}&H \end{array} \right)=({\beta}_{1}, {\beta}_{2})A_{2}\})\leq k-d.$$

• (3) If $W_{2}\in \mathcal {C}_{3}$, then $W_{2}=rs(P_{3})= \left (\begin {array}{cccc} U_{1}&M_{1}&\sigma _{U_{2}}(F_{3})&M_{3}\\ \sigma _{U_{1}}(F_{4})&M_{4}&U_{2}&M_{2} \end {array}\right )$. Thus, the subspace distance between W₁ and W₂ is

  $$\begin{array}{@{}rcl@{}} \text{dis}(W_{1}, W_{2})&=& 2 \text{rank}\left( \begin{array}{cccc} U_{1} &M_{1}&\sigma_{U_{2}}(F_{3})&M_{3}\\ \sigma_{U_{1}}(F_{4})&M_{4}&U_{2}&M_{2}\\ E &M_{1,1}&M_{1,2}&\sigma_{H}(F_{1,2})\\ \sigma_{E}(F_{2,1})&M_{2,1}&M_{2,2}&H \end{array}\right)-2k\\ &=& 2~\!\text{rank}\left( \begin{array}{cccc} U_{1} &\sigma_{U_{2}}(F_{3})&M_{1}&M_{3}\\ \sigma_{U_{1}}(F_{4})&U_{2}&M_{4}&M_{2}\\ E&M_{1,2}&M_{1,1} &\sigma_{H}(F_{1,2})\\ \sigma_{E}(F_{2,1})&M_{2,2}&M_{2,1}&H \end{array}\right)-2k\\ &=& \text{dis}(W^{\prime}_{1}, W^{\prime}_{2}), \end{array}$$

  where

  $$W^{\prime}_{1}=rs\left( \begin{array}{cccc} E&M_{1,2}&M_{1,1} &\sigma_{H}(F_{1,2})\\ \sigma_{E}(F_{2,1})&M_{2,2}&M_{2,1}&H \end{array}\right)$$

  and

  $$W^{\prime}_{2}=rs\left( \begin{array}{cccc} U_{1} &\sigma_{U_{2}}(F_{3})&M_{1}&M_{3}\\ \sigma_{U_{1}}(F_{4})&U_{2}&M_{4}&M_{2} \end{array}\right).$$

It is obvious that $i(W_{2}^{\prime })$ has k ones in the first t₁ + t₂ positions. But $i(W_{1}^{\prime })$ has t₁ + rank(M_2,2) ones in the first t₁ + t₂ positions since a₁ + rank(F_2,1|M_2,2) ≤ a₁ + (t₁ − a₁ + rank(M_2,2)) = t₁ + rank(M_2,2). In consequence,

$$\begin{array}{@{}rcl@{}} \text{dis}(W_{1}, W_{2})&=&\text{dis}(W^{\prime}_{1}, W^{\prime}_{2})\\ &\geq & d_{H}(i(W^{\prime}_{1}), i(W^{\prime}_{2}))\\ &\geq& (k-t_{1}-\text{rank}(M_{2,2}))+(k-t_{1}-\text{rank}(M_{2,2}))\\ &=&2(k-t_{1}-\text{rank}(M_{2,2})) \\ &\geq& 2d \end{array}$$

by Lemma 2.3. In conclusion, $\cup ^{4}_{i=1}\mathcal {C}_{i}$ is an (m + n,2d,k)_q CDC. This completes the proof.

Remark 3.3

The condition k − t₁ ≥ 2d is vital for Construction A. According to the subcode construction in Lemma 2.6, this condition can guarantee the existence of $M_{2,2}^{r}$. Hence, comapred to Lemma 3.1, Construction A inserts more subspace into $\mathcal {C}_{1}\cup \mathcal {C}_{2}$ by exchanging the positions of matrix blocks and improves the lower bounds of CDCs for some parameters.

From Theorem 3.2, we obtain some new lower bounds for CDCs in the following corollary. Denote the cardinality of $\mathcal {C}$ by $|\mathcal {C}|$. We define ${\Lambda }(m,n, d;r)_{q}={\sum }_{i=d}^{r}D(m, n, d,i)_{q}$ if r ≥ d and Λ(m,n,d;r)_q = 1 if r < d.

Corollary 3.4

Let m, n, d, k, a₁, a₂, b₁, b₂, c₁, c₂, t₁ and t₂ be positive integers with m ≥ k, n ≥ k, a₁ + a₂ = k, b₁ + b₂ ≥ d, c₁ + c₂ ≥ d, m − t₁ ≥ d, n − t₂ ≥ a₂, d ≤ a_i ≤ t_i, k − t₁ ≥ 2d, 1 ≤ b_i ≤ d and 1 ≤ c_i ≤ d for i = 1, 2. Let $\mathcal {C}_{1}$, $\mathcal {C}_{2}$, $\mathcal {C}_{3}$, and $\mathcal {C}_{4}$ be the codes defined in Thoerem 3.2. Then

$$\begin{array}{@{}rcl@{}} A_{q}(m+n, 2d, k) &\geq& A_{q}(m,2d,k)\cdot {\Delta} (k, n, d)_{q}+ A_{q}(n,2d, k)\cdot {\Delta} (k, m, d; k-d)_{q}\\ &&+ s\cdot A_{q}(t_{1}, 2d, a_{1})\cdot A_{q}(t_{2}, 2d, a_{2}) \cdot {\Delta}(a_{1}, m-t_{1}, d)_{q}\cdot {\Delta}(a_{2}, n-t_{2}, d)_{q}\\ && \cdot {\Delta} (a_{1}, n-a_{2},d;a_{1}-d)_{q}\cdot {\Delta}(a_{2}, m-a_{1}, d; a_{2}-d)_{q} + A_{q}(t_{1}, 2d, a_{1})\\ && \cdot A_{q}(n-t_{2}, 2d, a_{2}) \cdot {\Delta}(a_{1}, n-a_{2}, d; a_{1}-d)_{q} \cdot {\Delta} (a_{2}, m-a_{1}, d; a_{2}-d)_{q} \\ && \cdot {\Delta}(a_{1}, m-t_{1}, d)_{q}\cdot \Big({\Delta}(a_{2}, t_{2}, d;k-t_{1}-d)_{q}+(t-1)\cdot {\Lambda}(a_{2}, t_{2}, d; k-t_{1}-d)_{q}\Big), \end{array}$$

where $s=\min \limits \Big (\frac {\Delta (a_{1}, m-t_{1},b_{1})_{q}}{\Delta (a_{1}, m-t_{1},d)_{q}}, \frac {\Delta (a_{2}, n-t_{2},b_{2})_{q}}{\Delta (a_{2}, n-t_{2},d)_{q}}\Big )$ and $t=\min \limits \Big (\frac {\Delta (a_{1}, m-t_{1},c_{1})_{q}}{\Delta (a_{1}, m-t_{1},d)_{q}}, \frac {\Delta (a_{2}, t_{2},c_{2})_{q}}{\Delta (a_{2}, t_{2},d)_{q}}\Big )$.

Proof

By Lemma 2.6, we can construct $M_{1,1}^{r}$ and $M_{2,2}^{r}$ satisfying Theorem 3.2. Note that the subcode construction in Lemma 2.6 only provides a unique linear MRD code. Let M₂ be the unique linear (a₂,t₂,d)_q MRD code constructed by Lemma 2.6. Without loss of generality, let $M_{2,2}^{1}=\{M\in M_{2}: \text {rank}(M)\leq k-t_{1}-d\}$, and $M_{2,2}^{r}$ be constructed from a subset of a non-linear MRD code such that the rank of each matrix is at most k − t₁ − d for 2 ≤ r ≤ t. Then, $|\mathcal {C}_{4}|=|\mathcal {E}|\cdot |{\mathcal{H}}|\cdot |{\mathcal{M}}_{1,2}|\cdot |{\mathcal{M}}_{2,1}|\cdot |M_{1,1}^{1}|\cdot \big (|{\mathcal{M}}_{2,2}^{1}|+(t-1)\cdot |{\mathcal{M}}_{2,2}^{2}|\big )$. We can see that $|{\mathcal{M}}_{2,2}^{2}|= {\Lambda }(a_{2}, t_{2}, d; k-t_{1}-d)_{q}.$ Therefore, the desired conclusion follows.

Example 3.5

We adopt the notation in Corollary 3.4. Take m = n = k = 8, d = 2, a₁ = a₂ = 4, b₁ = c₁ = 1, b₂ = c₂ = 1, and t₁ = t₂ = 4. Then t = q⁴. Take $M_{1,1}^{r}$ and $M_{2,2}^{r}$ in the same way as the proof of Corollary 3.4 for 1 ≤ r ≤ t. Then $|{\mathcal{M}}_{1,1}^{1}|={\Delta }(4,4,2)_{q}$, $|{\mathcal{M}}_{2,2}^{1}|={\Delta }(4,4,2;2)_{q}$ and $|{\mathcal{M}}_{2,2}^{1}|={\Lambda }(4,4,2;2)_{q}$. Moreover, A_q(4,4,4) = 1. It follows that

$$|\mathcal{C}_{4}|={\Delta}(4,4,2)_{q}\cdot \big({\Delta}(4,4,2;2)_{q}+(q^{4}-1)\cdot {\Lambda}(4,4,2;2)_{q}\big)\cdot{\Delta}(4,4,2;2)_{q}\cdot {\Delta}(4,4,2;2)_{q}.$$

For q = 2, $|\mathcal {C}_{4}|= 9520558391296$. The cardinality of $\mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}$ is 61680045647822848 in [20] and [24], which is the previously best known lower bound of A₂(16,4,8). Hence, $A_{2}(16,4,8)\geq |\mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}|+|\mathcal {C}_{4}|=61680045647822848 + 9520558391296=61689566206214144$.

We continue presenting new construction of CDCs based on Lemma 3.1, which is slightly different from Theorem 3.2. If the parameters a₁, a₂, t₁ and t₂ satisfy m − t₁ ≥ a₁ and n − t₂ ≥ a₂, then more subspaces can be inserted into the CDC in Lemma 3.1. Hence, using the similar block inserting construction through an appropriate combination of the matrix blocks from small CDCs and RRMCs, we give the following Construction B and improve the lower bounds for CDCs.

Theorem 3.6 (Construction B)

With the same notation used in Lemma 3.1. Additionally, m − t₁ ≥ a₁ and n − t₂ ≥ a₂. Let $c^{\prime }_{1}$, $c^{\prime }_{2}$, e₁ and e₂ be positive integers with $c^{\prime }_{1}+c^{\prime }_{2}\geq d$, e₁ + e₂ ≤ k − d and $1\leq c^{\prime }_{i}\leq d$ for i = 1, 2. Let $\mathcal {V}_{1}=\{V_{1}\in \mathbb {F}^{a_{1}\times (m-t_{1})}_{q} : V_{1} ~\text {in RREF}, \text {rank}(V_{1})=a_{1} \}$ $\big (\text {resp.}~ \mathcal {V}_{2}=\{ V_{2}\in \mathbb {F}^{a_{2}\times (n-t_{2})}_{q} : V_{2} ~\text {in RREF}, \text {rank}(V_{2})=a_{2}\} \big )$ be an SC-representation set of (m − t₁,2d,a₁)_q (resp. (n − t₂,2d,a₂)_q) CDC. Let

$$\mathcal{N}_{3}\subset \{ (N_{3}|G_{3}): G_{3}\in \mathbb{F}^{a_{1}\times (n-t_{2}-a_{2})}_{q}, N_{3}\in \mathbb{F}^{a_{1}\times t_{2}}_{q} \}$$

and

$$\mathcal{N}_{4}\subset\{ (N_{4}|G_{4}): G_{4}\in \mathbb{F}^{a_{2}\times (m-t_{1}-a_{1})}_{q}, N_{4}\in \mathbb{F}^{a_{2}\times t_{1}}_{q} \}$$

be two RRMCs with respective parameters (a₁,n − a₂,d;a₁ − d)_q and (a₂,m − a₁,d;a₂ − d)_q. Given integer l, for 1 ≤ r ≤ l, let $\mathcal {N}^{r}_{1}$ be an (a₁,t₁,d;e₁)_q RRMC and $\mathcal {N}^{r}_{2}$ be an (a₂,t₂,d;e₂)_q RRMC. For i = 1, 2, let $N\in \mathcal {N}^{r}_{i}$ and $N^{\prime }\in \mathcal {N}^{r^{\prime }}_{i}$ for $1\leq r<r^{\prime }\leq l$, assume that $N\neq N^{\prime }$ and rank$(N-N^{\prime })\geq c^{\prime }_{i}$. Define $\mathcal {C}_{4}^{\prime }= \cup ^{l}_{r=1}{\mathcal{B}}^{\prime }_{r}$,

$$\mathcal{B}^{\prime}_{r}=\Biggl\{ rs \left( \begin{array}{cccc} N_{1}&V_{1}&N_{3}&\sigma_{V_{2}}(G_{3})\\ N_{4}&\sigma_{V_{1}}(G_{4})&N_{2}&V_{2} \end{array}\right)\Biggr\},$$

where $V_{1}\in \mathcal {V}_{1}$, $V_{2}\in \mathcal {V}_{2}$, $N_{i}\in \mathcal {N}^{r}_{i}$ for i = 1, 2, and $(N_{i}|G_{i})\in \mathcal {N}_{i}$ for i = 3, 4. Then $\mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}\cup \mathcal {C}_{4}^{\prime }$ is an (m + n,2d,k)_q CDC, where $\mathcal {C}_{1}$, $\mathcal {C}_{2}$ and $\mathcal {C}_{3}$ are defined in Lemma 3.1.

Proof

Let W₄ be a subspace in $\mathcal {C}_{4}^{\prime }$, then

$$W_{4}=rs(P_{4})=rs \left( \begin{array}{cccc} N_{1}&V_{1}&N_{3}&\sigma_{V_{2}}(G_{3})\\ N_{4}&\sigma_{V_{1}}(G_{4})&N_{2}&V_{2} \end{array}\right).$$

We can prove that $\mathcal {C}_{4}^{\prime }$ is an (m + n,2d,k)_q CDC from a similar proof as Theorem 3.2, hence we omit the detail of it here. It remains to show that the distance between $W_{4}\in \mathcal {C}_{4}^{\prime }$ and $W_{2}\in \mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}$ is at least 2d. There are the following three cases.

• (1) If $W_{2}\in \mathcal {C}_{1}$, then there exist k ones in the first m positions of i(W₂). But i(W₄) has no more than k − d ones in the first m positions since $\text {rank}(V_{1})+ \text {rank}(N_{4}|\sigma _{V_{1}}(G_{4}))=a_{1}+\text {rank}(N_{4}|G_{4})\leq a_{1}+a_{2}-d= k-d$. Thus dis(W₄,W₂) ≥ d_H(i(W₄),i(W₂)) ≥ 2d.

• (2) If $W_{2}\in \mathcal {C}_{2}$, then W₂ = rs(P₂) = rs(Q₂|A₂). The dimension of W₄ ∩ W₂ is dim ({(α₁,α₂) : ∃ (β₁,β₂) s.t. (α₁,α₂)P₄ = (β₁,β₂)P₂}), where α_i, $\beta _{i}\in \mathbb {F}^{a_{i}}_{q}$ for i = 1,2. Since rank(A₂) = k and

  $$\begin{array}{@{}rcl@{}} \text{rank}\left( \begin{array}{cc} N_{3}&\sigma_{V_{2}}(G_{3})\\ N_{2}&V_{2} \end{array} \right)&\leq & \text{rank}(V_{2}) + \text{rank}(N_{3}|\sigma_{V_{2}}(G_{3}))\\ &=&\text{rank}(V_{2})+ \text{rank}(N_{3}|G_{3})\\ &\leq& a_{2}+a_{1}-d\\ &=&k-d, \end{array}$$

  we have

  $$\dim(W_{4}\cap W_{2}) \leq \dim (\{({\alpha}_{1}, {\alpha}_{2}): \exists ~ (\beta_{1}, \beta_{2} )~ \text{s.t.}~ ({\alpha}_{1}, {\alpha}_{2}) \left( \begin{array}{cc} N_{3}&\sigma_{V_{2}}(G_{3})\\ N_{2}&V_{2} \end{array}\right)=({\beta}_{1}, {\beta}_{2})A_{2}\})\leq k-d.$$

• (3) If $W_{2}\in \mathcal {C}_{3}$, then $W_{2}=rs(P_{3})= \left (\begin {array}{cccc} U_{1}&M_{1}&\sigma _{U_{2}}(F_{3})&M_{3}\\ \sigma _{U_{1}}(F_{4})&M_{4}&U_{2}&M_{2} \end {array}\right )$. Since rank(N₁) ≤ e₁, rank(N₂) ≤ e₂, and rank(U_i) = a_i for i = 1, 2,

  $$\begin{array}{@{}rcl@{}} \dim(W_{4}\cap W_{2}) &\leq& \dim (\{ \alpha_{1} : \exists~ \beta_{1} ~\text{s.t.}~ \alpha_{1} U_{1}= \beta_{1} N_{1}, ~\alpha_{1}, \beta_{1}\in \mathbb{F}^{a_{1}}_{q}\}) \\ && +\dim (\{ \alpha_{2} : \exists~ \beta_{2} ~\text{s.t.}~ \alpha_{2} U_{2}= \beta_{2} N_{2}, ~\alpha_{2}, \beta_{2}\in \mathbb{F}^{a_{2}}_{q} \})\\ &\leq & e_{1}+e_{2} \\ &\leq & k-d. \end{array}$$

By the above discussion, we can conclude that $\text{dis}(W_{4}, W_{2})= \dim (W_{4}) + \dim (W_{2})-2~ \dim (W_{4}\cap W_{2}) \geq 2k-2(k-d)=2d$. Hence, we complete the proof.

Remark 3.7

It is easy to see that Construction B is different from Theorem 5 in [20], which we pad more matrix blocks and use subcode construction in Lemma 2.6. Moreover, compared to Lemma 3.1, Construction B inserts more subspaces into $\mathcal {C}_{1}\cup \mathcal {C}_{2}$ for some parameters.

Combining with Lemma 3.1, we have the following result from Theorem 3.6 which improves some lower bounds for CDCs.

Corollary 3.8

Let m, n, d, k, a₁, a₂, b₁, b₂, $c^{\prime }_{1}$, $c^{\prime }_{2}$, e₁, e₂, t₁ and t₂ be positive integers with m ≥ k, n ≥ k, a₁ + a₂ = k, b₁ + b₂ ≥ d, $c^{\prime }_{1}+c^{\prime }_{2}\geq d$, e₁ + e₂ ≤ k − d, a₁ + t₁ ≤ m, a₂ + t₂ ≤ n, d ≤ a_i ≤ t_i, 1 ≤ b_i ≤ d and $1\leq c^{\prime }_{i}\leq d$ for i = 1, 2. Let $\mathcal {C}_{1}$, $\mathcal {C}_{2}$, $\mathcal {C}_{3}$, and $\mathcal {C}_{4}^{\prime }$ be the codes defined in Thoerem 3.6. Then

$$\begin{array}{@{}rcl@{}} A_{q}(m+n, 2d, k)&\geq& A_{q}(m,2d,k)\cdot {\Delta} (k, n, d)_{q}+ A_{q}(n,2d, k)\cdot {\Delta} (k, m, d; k-d)_{q}\\ &&+ s\cdot A_{q}(t_{1}, 2d, a_{1})\cdot A_{q}(t_{2}, 2d, a_{2}) \cdot {\Delta}(a_{1}, m-t_{1}, d)_{q}\cdot {\Delta}(a_{2}, n-t_{2}, d)_{q}\\ && \cdot {\Delta} (a_{1}, n-a_{2},d;a_{1}-d)_{q}\cdot {\Delta}(a_{2}, m-a_{1}, d; a_{2}-d)_{q}\\ && + A_{q}(m-t_{1}, 2d, a_{1})\cdot A_{q}(n-t_{2}, 2d, a_{2}) \cdot {\Delta}(a_{1}, n-a_{2}, d; a_{1}-d)_{q}\\ &&\cdot {\Delta} (a_{2}, m-a_{1}, d; a_{2}-d)_{q} \cdot \Big({\Delta}(a_{1}, t_{1}, d;e_{1})_{q}\cdot {\Delta}(a_{2}, t_{2}, d; e_{2})_{q}\\ &&+(l-1)\cdot {\Lambda}(a_{1}, t_{1}, d;e_{1})_{q}\cdot {\Lambda}(a_{2}, t_{2}, d; e_{2})_{q}\Big). \end{array}$$

Here $s=\min \limits \Big (\frac {\Delta (a_{1}, m-t_{1},b_{1})_{q}}{\Delta (a_{1}, m-t_{1},d)_{q}}, \frac {\Delta (a_{2}, n-t_{2},b_{2})_{q}}{\Delta (a_{2}, n-t_{2},d)_{q}}\Big )$ and

$$l=\min \Big(l^{\prime} ,{\Delta}(a_{1}, t_{1}, c^{\prime}_{1};e_{1})_{q}-{\Delta}(a_{1},t_{1},d;e_{1})_{q} , {\Delta}(a_{2},t_{2}, c^{\prime}_{2};e_{2})_{q}-{\Delta}(a_{2},t_{2},d;e_{2})_{q} \Big)$$

with $l^{\prime }=\min \limits \Big (\frac {\Delta (a_{1}, t_{1},c^{\prime }_{1})_{q}}{\Delta (a_{1}, t_{1},d)_{q}}, \frac {\Delta (a_{2}, t_{2},c^{\prime }_{2})_{q}}{\Delta (a_{2}, t_{2},d)_{q}}\Big )$.

Proof

By Lemma 2.6, we can construct the desired ${\mathcal{M}}^{r}_{1}$ and ${\mathcal{M}}^{r}_{2}$ MRD codes used to construct $\mathcal {C}_{3}$ for 1 ≤ r ≤ s. Similarly, for $1\leq r^{\prime }\leq l^{\prime }= {\min \limits } \Big (\frac {\Delta (a_{1}, t_{1},c^{\prime }_{1})_{q}}{\Delta (a_{1}, t_{1},d)_{q}}, \frac {\Delta (a_{2}, t_{2},c^{\prime }_{2})_{q}}{\Delta (a_{2}, t_{2},d)_{q}} \Big )$, let $\mathcal {D}_{i}$ be an $(a_{i}, t_{i}, c^{\prime }_{i})_{q}$ MRD code, then we can construct an (a_i,t_i,d)_q MRD code $\mathcal {E}^{r^{\prime }}_{i}\subset \mathcal {D}_{i}$ for i = 1, 2. By Lemma 2.6 again, we know that there exists only one linear MRD code in the constructed (a_i,t_i,= d)_q MRD codes for i = 1, 2. Without loss of generality, assume $\mathcal {E}^{1}_{1}$ (resp. $\mathcal {E}^{1}_{2}$) is the unique linear (a₁,t₁,d)_q (resp. (a₂,t₂,d)_q) MRD code. Let $\mathcal {N}^{1}_{i}=\{N_{i}: N_{i}\in \mathcal {E}^{r^{\prime }}_{i} , \text {rank}(N_{i})\leq e_{i}\}$ for i = 1, 2. Then, $|\mathcal {N}^{1}_{i}|={\Delta }(a_{i},t_{i}, d;e_{i})_{q}$ for i = 1, 2. If e₁ < d, let $\mathcal {N}^{r^{\prime }}_{1}$ contain only one matrix in $\mathcal {D}_{1}\setminus \mathcal {E}^{1}_{1}$ with rank at most e₁ for $2\leq r^{\prime }$, otherwise, let $\mathcal {N}^{r^{\prime }}_{1}=\{N_{1}: N_{1}\in \mathcal {E}^{r^{\prime }}_{1} , \text {rank}(N_{1})\leq e_{1}\}$ for $2\leq r^{\prime }$. We construct $\mathcal {N}^{r^{\prime }}_{2}$ for $2\leq r^{\prime }$ in the same way. Then $|\mathcal {N}^{r^{\prime }}_{1}|= {\Lambda }(a_{1}, t_{1}, d; e_{1})_{q}$ and $|\mathcal {N}^{r^{\prime }}_{2}|= {\Lambda }(a_{2}, t_{2}, d; e_{2})_{q}$ for $2\leq r^{\prime }$. From the construction of $\mathcal {N}^{r^{\prime }}_{i}$, we get l (a₁,t₁,d;e₁)_q RRMCs and l (a₂,t₂,d;e₂)_q RRMCs, where

$$l=\min \Big(l^{\prime} ,{\Delta}(a_{1}, t_{1}, c^{\prime}_{1};e_{1})_{q}-{\Delta}(a_{1},t_{1},d;e_{1})_{q} , {\Delta}(a_{2},t_{2}, c^{\prime}_{2};e_{2})_{q}-{\Delta}(a_{2},t_{2},d;e_{2})_{q} \Big).$$

Then from Theorem 3.6, we have $A_{q}(m+n, 2d,k)\geq |\mathcal {C}_{1}|+|\mathcal {C}_{2}|+|\mathcal {C}_{3}|+|\mathcal {C}_{4}^{\prime }|$, where $|\mathcal {C}_{4}^{\prime }|=\big (|\mathcal {V}_{1}|\cdot |\mathcal {V}_{2}|\cdot |\mathcal {N}_{3}|\cdot |\mathcal {N}_{4}|\big )\cdot \big (|\mathcal {N}^{1}_{1}|\cdot |\mathcal {N}^{1}_{2}|+(l-1)\cdot (|\mathcal {N}^{2}_{1}|\cdot |\mathcal {N}^{2}_{2}|)\big )$. Combining this result with the lower bound of $|\mathcal {C}_{1}|+|\mathcal {C}_{2}|+|\mathcal {C}_{3}|$ in Lemma 3.1, the desired conclusion follows.

Example 3.9

We adopt the notation in Corollary 3.8. Take m = n = k = 8, d = 3, a₁ = a₂ = 4, $b_{1}=c^{\prime }_{1}=2$, $b_{2}=c^{\prime }_{2}=1$, e₁ = 3, e₂ = 2 and t₁ = t₂ = 4. Then l = q⁴. Take $\mathcal {N}^{r^{\prime }}_{1}$ and $\mathcal {N}^{r^{\prime }}_{2}$ in the same way as the proof of Corollary 3.8 for $1\leq r^{\prime }\leq l$. Then $|\mathcal {N}^{r^{\prime }}_{2}|=1$ for $2\leq r^{\prime }\leq l$ since e₂ < d. Moreover, A_q(4,6,4) = 1. Thus, from Corollary 3.8,

$$|\mathcal{C}_{4}^{\prime}|={\Delta}(4,4,3;1)_{q}\cdot{\Delta}(4,4,3;1)_{q}\cdot\big({\Delta}(4,4,3;3)_{q}\cdot {\Delta}(4,4,3;2)_{q} + (q^{4}-1)\cdot {\Lambda}(4,4,3;3)_{q}\big).$$

For q = 2, $|\mathcal {C}_{4}^{\prime }|= 3601$. Then our construction leads to A₂(16,6,8) ≥ 282927684888529.

If the parameters a₂ and d satisfy a₂ < 2d, then Δ(a₂,m − a₁,d;a₂ − d) = 1 by Lemma 2.4. Hence, we can assume that $\mathcal {N}_{4}$ contains only zero matrix. Based on this assumption, more subspaces can be added into the CDC in Theorem 3.6. Niu et al. further showed that the CDC in Lemma 3.1 can combine with other special subspaces. Hence, for constructing CDCs with larger cardinalities, we quote their result in the following Lemma 3.10 and give the following Construction C.

Lemma 3.10 (Theorem 3.3 in 24)

With the same notation used in Lemma 3.1. In addition, a₂ < 2d. Let $\mathcal {D}=\{(D_{1}|D_{2})\in \mathbb {F}^{k\times 2(k-d)}_{q}:D_{1},D_{2}\in \mathbb {F}^{k\times (k-d)}_{q} \}$ be an SC-representation set of a (2(k − d),2d,k)_q CDC. Define $\mathcal {C}_{5}=\{ rs(O_{k\times (m-k+d)}|D_{1}|O_{k\times (n-k+d)}|D_{2}):(D_{1}|D_{2})\in \mathcal {D}\}$. Assume x = t₁ − m + k − d ≥ 0 and y = t₂ − n + k − d ≥ 0. Let $\mathcal {C}_{i}$ be the code defined in Lemma 3.1 for i = 1, 2, 3. If k − x − y ≥ d, then $\mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}\cup \mathcal {C}_{5}$ is an (m + n,2d,k)_q CDC.

Theorem 3.11 (Construction C)

With the same notation used in Lemma 3.1 and Theorem 3.6. In addition, a₂ < 2d. Let $\mathcal {C}_{4}^{\prime }$ be the code defined in Theorem 3.6 and $\mathcal {N}_{4}=\{O_{{a_{2}}\times (m-a_{1})}\}$. Let $\mathcal {C}_{5}$ be the same as the one in Lemma 3.10. Assume x = t₁ − m + k − d ≥ 0 and y = t₂ − n + k − d ≥ 0. If k − x − y ≥ d and e₁ + e₂ − x − y ≥ d, then $\mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}\cup \mathcal {C}_{4}^{\prime }\cup \mathcal {C}_{5}$ is an (m + n,2d,k)_q CDC, where $\mathcal {C}_{i}$ is defined in Lemma 3.1 for 1 ≤ i ≤ 3. Consequently, $A_{q}(m+n, 2d,k)\geq |\mathcal {C}_{1}|+|\mathcal {C}_{2}|+|\mathcal {C}_{3}|+|\mathcal {C}_{4}^{\prime }|+A_{q}(2(k-d), 2d, k)$.

Proof

From Lemma 3.10, we know that $\mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}\cup \mathcal {C}_{5}$ is an (m + n,2d,k)_q CDC. Thus, it remains to show that $\mathcal {C}_{4}^{\prime }\cup \mathcal {C}_{5}$ is also an (m + n,2d,k)_q CDC, i.e., dis$(\mathcal {C}_{4}^{\prime },\mathcal {C}_{5})\geq 2d$. Let

$$W_{4}= rs\left( \begin{array}{cccc} N_{1}& v_{1} &N_{3}& \sigma_{V_{2}}(G_{3})\\ O_{a_{2}\times t_{1}}&O_{a_{2}\times (m-t_{1})}&N_{2}&V_{2} \end{array}\right)\in \mathcal{C}_{4}^{\prime}$$

and $W_{5}=rs(O_{k\times (m-k+d)}|D_{1}|O_{k\times (n-k+d)}|D_{2})\in \mathcal {C}_{5}$ be two k-dimensional subspaces with respective identifying vectors i(W₄) and i(W₅) = (v^m,vⁿ). where $v^{m}\in \mathbb {F}^{m}_{2}$ and $v^{n}\in \mathbb {F}^{n}_{2}$. By Lemma 2.3,

$$\begin{array}{@{}rcl@{}} \text{dis}(W_{4}, W_{5})&\geq& d_{H}(i(W_{4}), i(W_{5})) \\ &\geq& e_{1}-x+w_{H}(v^{m})-(a_{1}-e_{1}+x)+e_{2}-y+w_{H}(v^{n})-(a_{2}-e_{2}+y)\\ &=&2(e_{1}+e_{2})-2(x+y)-a_{1}-a_{2}+w_{H}(v^{m})+w_{H}(v^{n})\\ &=& 2(e_{1}+e_{2}-x-y)\\ &\geq& 2d, \end{array}$$

where w_H(v^m) and w_H(vⁿ) denote the Hamming weights of v^m and vⁿ, respectively. It is easy to check the lower bound of A_q(m + n,2d,k). Thus, we complete the proof.

Example 3.12

Take the same values for all parameters used in Example 3.9. Then the maximum cardinality of $\mathcal {C}_{5}$ is A₂(10,6,8) = A₂(10,6,2) = 341 [2]. Combining this result with the lower bound in Example 3.9, we have $A_{2}(16,6,8)\geq |\mathcal {C}_{1}\cup \mathcal {C}_{2}\cup \mathcal {C}_{3}\cup \mathcal {C}^{\prime }_{4}|+|\mathcal {C}_{5}|\geq 282927684888529 + 341=282927684888870$, which is better than 282927684887704 in [20].

In the end of this section, we give some new lower bounds for CDCs from Corollary 3.4, Corollary 3.8 and Theorem 3.11 in what follows, where q ∈{2,3,4,5,7,8,9}.

Corollary 3.13

With m = n = 8, d = 2, k = 8, a₁ = a₂ = 4, b₁ = b₂ = 1, c₁ = c₂ = 1 and t₁ = t₂ = 4, we have

$$\begin{array}{@{}rcl@{}} A_{q}(16,4,8) &\geq& q^{56}+ q^{52} + q^{51}+ 2 q^{50}+ q^{49}+ q^{48} - q^{46} - 3 q^{45}- 3 q^{44}- 3 q^{43} + q^{42} + 5 q^{41}\\ &&+ 8 q^{40} +9 q^{39}+ 10 q^{38} + 14 q^{37} + 19 q^{36} + 21 q^{35} + 10q^{34}- 14 q^{33} - 49 q^{32} \\ && - 74 q^{31} - 75 q^{30} - 42 q^{29} + 11 q^{28}+ 56 q^{27} + 78 q^{26} + 66 q^{25} + 37 q^{24} + 3 q^{23}\\ && - 21 q^{22} -28 q^{21}- 26 q^{20} - 16 q^{19}- 5 q^{18} + q^{17} + 4 q^{16}+ 3 q^{15} + q^{14}. \end{array}$$

With m = 6, n = 12, d = 2, k = 6, a₁ = 2, a₂ = 4, b₁ = b₂ = 1 c₁ = c₂ = 1, t₁ = 2 and t₂ = 8, we have

$$\begin{array}{@{}rcl@{}} A_{q}(18,4,6)&\geq & q^{60}+A_{q}(12,4,6)\big(q^{26}+ q^{25}+ 2 q^{24}+ q^{23}+ q^{22}- q^{21}- 3 q^{20}- 4 q^{19}- 3 q^{18} \\ && - 2 q^{17}+ 4 q^{15}+ 5 q^{14}+ 5 q^{13}+ 3 q^{12}+ q^{11}- q^{10}- 3 q^{9}- 3 q^{8}- 2 q^{7}-q^{6}\big)\\ && +A_{q}(8,4,4)\big(q^{28}+ q^{27}+ 2 q^{26} + q^{25}- q^{23} - 2 q^{22}-q^{21} \big)+ q^{28} + 2 q^{27}\\ && + 5 q^{26}+ 6 q^{25} + 7 q^{24} + 4 q^{23}- 4 q^{21}- 7 q^{20} - 7 q^{19} - 8 q^{18}- 7 q^{17}- 7 q^{16}\\ && - 4 q^{15} + 4 q^{13} + 7 q^{12} + 6 q^{11}+ 5 q^{10}+ 2 q^{9}- q^{7}- 2 q^{6}+ q^{5}. \end{array}$$

With m = 6, n = 13, d = 2, k = 6, a₁ = 2, a₂ = 4, t₁ = 2, t₂ = 8, b₁ = b₂ = 1, and c₁ = c₂ = 1, we have

$$\begin{array}{@{}rcl@{}} A_{q}(19,4,6)&\geq & q^{65}+A_{q}(13,4,6)\big(q^{26}+ q^{25}+ 2 q^{24}+ q^{23}+ q^{22}- q^{21}- 3 q^{20}- 4 q^{19}- 3 q^{18}\\ &&- 2 q^{17}+ 4 q^{15}+ 5 q^{14}+ 5 q^{13}+ 3 q^{12}+ q^{11}- q^{10} - 3 q^{9}- 3 q^{8}- 2 q^{7}-q^{6}\big)\\ &&+ A_{q}(8,4,4)\big(q^{31}+ q^{30} + 2 q^{29}+ q^{28}- q^{26} - 2 q^{25}-q^{24} \big)+ q^{28}+ 2 q^{27}\\ && + 5 q^{26}+ 6 q^{25}+ 7 q^{24}+ 4 q^{23}- 4 q^{21}- 7 q^{20} - 7 q^{19}- 8 q^{18}- 7 q^{17}- 7 q^{16}\\ && - 4 q^{15} + 4 q^{13} + 7 q^{12}+ 6 q^{11} + 5 q^{10}+ 2 q^{9}- q^{7}- 2 q^{6} -q^{5}. \end{array}$$

With m = n = 8, d = 3, k = 8, a₁ = a₂ = 4, t₁ = t₂ = 4, $b_{1}=c^{\prime }_{1}=2$, $b_{2}=c^{\prime }_{2}=1$, e₁ = 3 and e₂ = 2, we have

$$\begin{array}{@{}rcl@{}} A_{q}(16,6,8) &\geq&q^{48}+ q^{39}+ q^{38}+ 2 q^{37}+ 3 q^{36}+ 3 q^{35} + 3 q^{34} + 2 q^{33} - 4 q^{31} - 6 q^{30}- 10 q^{29} \\ && - 10 q^{28}- 11 q^{27}- 7 q^{26} - 3 q^{25}+ 6 q^{24}+ 12 q^{23}+ 19 q^{22}+ 23 q^{21}+ 25 q^{20} \\ &&+ 22 q^{19} + 16 q^{18}+ 9 q^{17} - 7 q^{15} - 13 q^{14}- 15 q^{13}-17 q^{12}- 13 q^{11}- 11 q^{10} \\ &&- 7q^{9}- 4 q^{8}- 3 q^{7} - 2 q^{6} - q^{5}- q^{4}+1. \end{array}$$

From Corollary 3.13, we can obtain 28 new lower bounds for CDCs. In Table 1, we list the improved lower bounds for CDCs from our constructions for q = 2 and the previously best known results with the corresponding references. It should be noted that the authors in [13] obtained lower bounds of A₂(18,4,6) and A₂(19,4,6), but Lao et al. [20] pointed out that the result in Theorem 4 in [13] is incorrect.

Table 1 New lower bounds of A_q(n,d,k) for q = 2

4 Conclusion

A lot of new lower bounds for CDCs have been given from a variety of constructions. However, it is obvious that there exists still a big gap between the best lower bounds and upper bounds for small parameters n ≤ 19 and q ≤ 9 in [2]. In this paper, we present three constructions of constant dimension codes via an appropriate combination of the matrix blocks from rank metric codes and small constant dimension codes. Both Construction A and Construction B construct CDCs through exchanging the positions of matrix blocks. The Construction A provides some new CDCs of distance 4, while the Construction B provides some new CDCs of distance 6. The Construction C further improves the lower bounds for certain CDCs of distance 6. Our constructions provide 28 improved lower bounds for CDCs.