Lower Bound on the Optimal Access Bandwidth of ( $K+2,K,2$ )-MDS Array Code with Degraded Read Friendly

Ting-Yi Wu1, Yunghsiang S. Han21, Zhengrui Li31, Bo Bai1, Gong Zhang1, Liang Chen1, Xiang Wu1

1Huawei Technologies Co., Ltd. 2Dongguan University of Technology 3University of Science and Technology of China

I Preliminary

Notation:

•

${\mathcal{F}}_{q}$ denotes the field of size $q$ .
•

$\emptyset$ denotes empty set.
•

$[N]\triangleq\{1,2,3,\ldots,N\}$ and $[0]\triangleq\emptyset$ .
•

${\mathbb{I}}\triangleq\begin{bmatrix}1&0\\ 0&1\end{bmatrix}$ .
•

${\mathbb{O}}\triangleq\begin{bmatrix}0&0\\ 0&0\end{bmatrix}$ .
•

${\mathcal{R}}\triangleq\left\{\begin{bmatrix}\beta_{1}&0\\ \beta_{2}&0\end{bmatrix}:\beta_{i}\in{\mathcal{F}}_{q}\mbox{ for all }i\in[2]\mbox{ and }\beta_{1}\beta_{2}\neq 0\right\}$ .
•

${\mathcal{L}}\triangleq\left\{\begin{bmatrix}0&\beta_{1}\\ 0&\beta_{2}\end{bmatrix}:\beta_{i}\in{\mathcal{F}}_{q}\mbox{ for all }i\in[2]\mbox{ and }\beta_{1}\beta_{2}\neq 0\right\}$ .
•

${\mathcal{M}}\triangleq\left\{{\mathbb{B}}=\begin{bmatrix}\beta_{1}&\beta_{2}\\ \beta_{3}&\beta_{4}\end{bmatrix}\not\in\{{\mathcal{L}}\cup{\mathcal{R}}\cup{\mathbb{O}}\}:\beta_{i}\in{\mathcal{F}}_{q}\mbox{ for all }i\in[4]\right\}$
•

${\mathcal{M}}_{\mathrm{inv}}\triangleq\left\{{\mathbb{B}}=\begin{bmatrix}\beta_{1}&\beta_{2}\\ \beta_{3}&\beta_{4}\end{bmatrix}\in{\mathcal{M}}\mbox{, and }{\mathbb{B}}\mbox{ is invertible}:\beta_{i}\in{\mathcal{F}}_{q}\mbox{ for all }i\in[4]\right\}$

We let the parity-check matrix of the ( $K+2,K$ )-MDS array be

{\mathbb{H}}=\begin{bmatrix}{\mathbb{I}}&{\mathbb{I}}&\cdots&{\mathbb{I}}\\ {\mathbb{A}}_{1}&{\mathbb{A}}_{2}&\cdots&{\mathbb{A}}_{N}\end{bmatrix},

(1)

where $N=K+2$ and these upper identities make the degraded read friendly. To satisfy MDS property, $\begin{bmatrix}{\mathbb{I}}&{\mathbb{I}}\\ {\mathbb{A}}_{i}&{\mathbb{A}}_{j}\end{bmatrix}$ must be invertible for any $i\neq j$ , which is equivalent to that $({\mathbb{A}}_{i}+{\mathbb{A}}_{j})$ must be invertible [1] for any $i\neq j$ , For any node $i\in[N]$ , it stores two symbols as a vector $\alpha^{(i)}\triangleq\begin{bmatrix}\alpha^{(i)}_{1}\\ \alpha^{(i)}_{2}\end{bmatrix}$ .

To repair the $i$ th node, we have to find two check equations from the row space of ${\mathbb{H}}$ in order to retrieve $\alpha^{(i)}$ . Let ${\mathbb{M}}$ be a repair matrix of size $2\times 4$ , ${\mathbb{M}}{\mathbb{H}}$ denotes two check equations from the row space of ${\mathbb{H}}$ such that

{\mathbb{M}}{\mathbb{H}}\triangleq\begin{bmatrix}{\mathbb{H}}_{1}&{\mathbb{H}}_{2}&\cdots&{\mathbb{H}}_{N}\end{bmatrix},

(2)

and

{\mathbb{H}}_{1}\alpha^{(1)}+{\mathbb{H}}_{2}\alpha^{(2)}+\cdots+{\mathbb{H}}_{N}\alpha^{(N)}=\begin{bmatrix}0\\ 0\end{bmatrix},

(3)

where ${\mathbb{H}}_{i}={\mathbb{M}}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}$ for all $i\in[N]$ . If ${\mathbb{M}}$ can be used to recover $\alpha^{(i)}$ , we have to make sure that ${\mathbb{H}}_{i}$ is invertible and recover $\alpha^{(i)}$ as

\alpha^{(i)}={\mathbb{H}}_{i}^{-1}\left[\sum_{j\in[N]\setminus\{i\}}{\mathbb{H}}_{j}\alpha^{(j)}\right].

(4)

To be able to repair each node, we have to design the corresponding repair matrix for each $\alpha^{(i)}$ . Since some repair matrices can be used to recover multiple nodes, i.e., ${\mathbb{M}}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}$ and ${\mathbb{M}}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{j}\end{bmatrix}$ are both invertible for some $i\neq j$ , we are not restricted to find $N$ different repair matrices for recovering all $N$ nodes. Let $R$ be the number of repair matrices, $1\leq R\leq N$ , and $\{{\mathcal{I}}^{(1)},{\mathcal{I}}^{(2)},\ldots,{\mathcal{I}}^{(R)}\}$ be a partition of $[N]$ , a repair process of size $R$ is formed by $R$ repair matrices, ${\mathbb{M}}^{(1)},{\mathbb{M}}^{(2)},\ldots,{\mathbb{M}}^{(R)}$ , where ${\mathbb{M}}^{(r)}$ can be used to recover $\alpha^{(i)}$ is $i\in{\mathcal{I}}^{(r)}$ . The formal definition of a repiar process of size $R$ is given below.

Definition 1.

A repair process $P_{R}$ of $R$ repair matrices is defined as

P_{R}\triangleq\left\{\left({\mathbb{M}}^{(r)},{\mathcal{I}}^{(r)}\right):r\in[R]\right\},

(5)

such that ${\mathcal{I}}^{(1)}$ , ${\mathcal{I}}^{(2)}$ , $\ldots$ , ${\mathcal{I}}^{(R)}$ form a partition of $[N]$ and ${\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}$ is invertible for $i\in{\mathcal{I}}^{(r)}$ .

We then explain the process of recovering $\alpha^{(i)}$ by a $P_{R}$ , where $i\in{\mathcal{I}}^{(r)}$ . Let

{\mathbb{M}}^{(r)}{\mathbb{H}}\triangleq\begin{bmatrix}{\mathbb{H}}^{(r)}_{1}&{\mathbb{H}}^{(r)}_{2}&\cdots&{\mathbb{H}}^{(r)}_{N}\end{bmatrix},

(6)

where ${\mathbb{H}}^{(r)}_{i}={\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}$ . By Definition 1, ${\mathbb{H}}^{(r)}_{i}$ is invertible, therefore

\alpha^{(i)}=\left[{\mathbb{H}}^{(r)}_{i}\right]^{-1}\left[\sum_{j\in[N]\setminus\{i\}}{\mathbb{H}}^{(r)}_{j}\alpha^{(j)}\right].

(7)

Definition 2.

The repair bandwidth for repairing node $i$ , where $i\in{\mathcal{I}}^{(r)}$ , by using $P_{R}$ is denoted by

B_{i}(P_{R})=\sum_{j\in[N]\setminus\{i\}}B_{i,j}(P_{R}),

(8)

and

B_{i,j}(P_{R})=\begin{cases}0,&\mbox{if }i=j\mbox{ or }{\mathbb{H}}^{(r)}_{j}={\mathbb{O}};\\ 1,&\mbox{if }{\mathbb{H}}^{(r)}_{j}\in{\mathcal{L}}\mbox{ or }{\mathbb{H}}^{(r)}_{j}\in{\mathcal{R}};\\ 2,&\mbox{otherwise.}\end{cases}

(9)

Note that $B_{i,j}(P_{R})$ denotes the number of symbols needed to be download from $j$ to repair node $i$ by using repair process $P_{R}$ .

Lemma 1.

Given a $P_{R}$ , $B_{i,j}(P_{R})=B_{j,i}(P_{R})=2$ if $i\neq j$ and $i,j\in{\mathcal{I}}^{(r)}$ for $r\in[R]$ .

Proof:

By Definition 1, ${\mathbb{H}}^{(r)}_{i}$ and ${\mathbb{H}}^{(r)}_{j}$ are both invertible since $i,j\in{\mathcal{I}}^{(r)}$ . Therefore, $B_{i,j}(P_{R})=B_{j,i}(P_{R})=2$ by (9). ∎

Lemma 2.

Given a $P_{R}$ , $B_{i}(P_{R})=B_{j}(P_{R})$ when both $i,j\in{\mathcal{I}}^{(r)}$ for some $r$ .

Proof:

By Definition 2, $B_{i}(P_{R})=\sum_{k\in[N]\setminus\{i\}}B_{i,k}(P_{R})$ and $B_{i,k}(P_{R})=B_{j,k}(P_{R})$ for all $k\in[N]\setminus\{i,j\}$ . Since $B_{i,j}(P_{R})=B_{j,i}(P_{R})$ , we have

$\displaystyle B_{i}(P_{R})$	$\displaystyle=$	$\displaystyle\sum_{k\in[N]\setminus\{i\}}B_{i,k}(P_{R})$	(10)
	$\displaystyle=$	$\displaystyle\sum_{k\in[N]\setminus\{i,j\}}B_{i,k}(P_{R})+B_{i,j}(P_{R})$	(11)
	$\displaystyle=$	$\displaystyle\sum_{k\in[N]\setminus\{i,j\}}B_{j,k}(P_{R})+B_{j,i}(P_{R})$	(12)
	$\displaystyle=$	$\displaystyle\sum_{k\in[N]\setminus\{j\}}B_{j,k}(P_{R})=B_{j}(P_{R}).$	(13)

∎

From Lemma 2, we have $B_{i}(P_{R})=B_{j}(P_{R})$ for all $i,j\in{\mathcal{I}}^{(r)}$ and $i\neq j$ . Hence, we define $B^{(r)}(P_{R})$ as below.

Definition 3.

Let $B^{(r)}(P_{R})$ be the repair bandwidth for repairing the node in ${\mathcal{I}}^{(r)}$ , i.e., $B^{(r)}(P_{R})=B_{i}(P_{R})$ for all $i\in{\mathcal{I}}^{(r)}$ .

Lemma 3.

The total repair bandwidth of $P_{R}$ is $B(P_{R})\triangleq\sum_{i\in[N]}B_{i}(P_{R})=\sum_{r\in[R]}|{\mathcal{I}}^{(r)}|B^{(r)}(P_{R})$ , where $|{\mathcal{I}}^{(r)}|$ is the number of distinct elements in set ${\mathcal{I}}^{(r)}$ .

Proof:

By Definition 3,

$\displaystyle\sum_{i\in[N]}B_{i}(P_{R})$	$\displaystyle=$	$\displaystyle\sum_{r\in[R]}\sum_{i\in{\mathcal{I}}^{(r)}}B_{i}(P_{R})$	(14)
	$\displaystyle=$	$\displaystyle\sum_{r\in[R]}\sum_{i\in{\mathcal{I}}^{(r)}}B^{(r)}(P_{R})$	(15)
	$\displaystyle=$	$\displaystyle\sum_{r\in[R]}\|{\mathcal{I}}^{(r)}\|B^{(r)}(P_{R}).$	(16)

∎

Example.

•

${\mathbb{H}}=\begin{bmatrix}1&0&1&0&1&0\\ 0&1&0&1&0&1\\ 2&0&0&4&0&8\\ 0&3&5&1&9&1\end{bmatrix}$ , where $N=3$ .
•

$P_{2}=\left({\mathbb{M}}^{(r)},{\mathcal{I}}^{(r)}\right)_{r\in[2]}$ , where

${\mathbb{M}}^{(1)}=\begin{bmatrix}1&0&0&0\\ 0&1&0&1\end{bmatrix}$ and ${\mathcal{I}}^{(1)}=\{1\}$ ,

${\mathbb{M}}^{(2)}=\begin{bmatrix}0&1&0&0\\ 0&0&0&1\end{bmatrix}$ and ${\mathcal{I}}^{(2)}=\{2,3\}$ .
•

${\mathbb{M}}^{(1)}{\mathbb{H}}=\begin{bmatrix}{\mathbb{H}}^{(1)}_{1}&{\mathbb{H}}^{(1)}_{2}&{\mathbb{H}}^{(1)}_{3}\end{bmatrix}=\begin{bmatrix}1&0&1&0&1&0\\ 0&2&5&0&9&0\end{bmatrix}\triangleq\begin{bmatrix}{\mathbb{B}}&{\mathbb{R}}_{1}&{\mathbb{R}}_{2}\end{bmatrix}$ , where ${\mathbb{B}}\in{\mathcal{M}}_{\mathrm{inv}}$ and ${\mathbb{R}}_{i}\in{\mathcal{R}}$ for all $i\in[2]$ .

${\mathbb{M}}^{(2)}{\mathbb{H}}=\begin{bmatrix}{\mathbb{H}}^{(2)}_{1}&{\mathbb{H}}^{(2)}_{2}&{\mathbb{H}}^{(2)}_{3}\end{bmatrix}=\begin{bmatrix}0&1&0&1&0&1\\ 0&3&5&1&9&1\end{bmatrix}\triangleq\begin{bmatrix}{\mathbb{L}}&{\mathbb{B}}_{1}&{\mathbb{B}}_{2}\end{bmatrix}$ , where ${\mathbb{B}}_{i}\in{\mathcal{M}}_{\mathrm{inv}}$ for all $i\in[2]$ and ${\mathbb{L}}\in{\mathcal{R}}$ .

•

Repairing node 1:

	$\displaystyle{\mathbb{H}}^{(1)}_{1}\alpha^{(1)}+{\mathbb{H}}^{(1)}_{2}\alpha^{(2)}+{\mathbb{H}}^{(1)}_{3}\alpha^{(3)}=\begin{bmatrix}0\\ 0\end{bmatrix}$	$\displaystyle\Rightarrow$	$\displaystyle\alpha^{(1)}=\begin{bmatrix}{\mathbb{H}}^{(1)}_{1}\end{bmatrix}^{-1}\left({\mathbb{H}}^{(1)}_{2}\alpha^{(2)}+{\mathbb{H}}^{(1)}_{3}\alpha^{(3)}\right)$		(18)
		$\displaystyle\Rightarrow$	$\displaystyle\alpha^{(1)}=\begin{bmatrix}1&0\\ 0&2\end{bmatrix}^{-1}\left(\begin{bmatrix}1&0\\ 5&0\end{bmatrix}\alpha^{(2)}+\begin{bmatrix}1&0\\ 9&0\end{bmatrix}\alpha^{(3)}\right).$		(19)

•

Repairing node 2:

	$\displaystyle{\mathbb{H}}^{(2)}_{1}\alpha^{(1)}+{\mathbb{H}}^{(2)}_{2}\alpha^{(2)}+{\mathbb{H}}^{(2)}_{3}\alpha^{(3)}=\begin{bmatrix}0\\ 0\end{bmatrix}$	$\displaystyle\Rightarrow$	$\displaystyle\alpha^{(2)}=\begin{bmatrix}{\mathbb{H}}^{(2)}_{2}\end{bmatrix}^{-1}\left({\mathbb{H}}^{(2)}_{1}\alpha^{(1)}+{\mathbb{H}}^{(2)}_{3}\alpha^{(3)}\right)$		(20)
		$\displaystyle\Rightarrow$	$\displaystyle\alpha^{(2)}=\begin{bmatrix}0&1\\ 5&1\end{bmatrix}^{-1}\left(\begin{bmatrix}0&1\\ 0&3\end{bmatrix}\alpha^{(2)}+\begin{bmatrix}0&1\\ 9&1\end{bmatrix}\alpha^{(3)}\right).$		(21)

•

Repairing node 3:

	$\displaystyle{\mathbb{H}}^{(2)}_{1}\alpha^{(1)}+{\mathbb{H}}^{(2)}_{2}\alpha^{(2)}+{\mathbb{H}}^{(2)}_{3}\alpha^{(3)}=\begin{bmatrix}0\\ 0\end{bmatrix}$	$\displaystyle\Rightarrow$	$\displaystyle\alpha^{(3)}=\begin{bmatrix}{\mathbb{H}}^{(2)}_{3}\end{bmatrix}^{-1}\left({\mathbb{H}}^{(2)}_{1}\alpha^{(1)}+{\mathbb{H}}^{(2)}_{2}\alpha^{(2)}\right)$		(22)
		$\displaystyle\Rightarrow$	$\displaystyle\alpha^{(2)}=\begin{bmatrix}0&1\\ 9&1\end{bmatrix}^{-1}\left(\begin{bmatrix}0&1\\ 0&3\end{bmatrix}\alpha^{(1)}+\begin{bmatrix}0&1\\ 5&1\end{bmatrix}\alpha^{(2)}\right).$		(23)

•

$B_{i,j}(P_{R})=\begin{bmatrix}0&1&1\\ 1&0&2\\ 1&2&0\end{bmatrix}$ , $B_{1}(P_{R})=2$ , $B_{2}(P_{R})=3$ , and $B_{3}(P_{R})=3$ .

II Some properties

Lemma 4.

Given a $P_{R}$ and $r\in[R]$ , if there is a $B_{i,j}(P_{R})=0$ for some $i\in{\mathcal{I}}^{(r)}$ and $j\in[N]\setminus{\mathcal{I}}^{(r)}$ , then $B^{(r)}(P_{R})=B_{i}(P_{R})=2K$ .

Proof:

Since $i\neq j$ , $B_{i,j}(P_{R})=0$ implies ${\mathbb{H}}^{(r)}_{j}={\mathbb{O}}$ due to $\eqref{eq:Bij}$ . Let ${\mathbb{M}}^{(r)}=\begin{bmatrix}{\mathbb{M}}^{(r)}_{1}&{\mathbb{M}}^{(r)}_{2}\end{bmatrix}$ be a repair matrix of $P_{R}$ , we have an invertible matrix ${\mathbb{H}}^{(r)}_{i}$ such that

{\mathbb{H}}^{(r)}_{i}={\mathbb{H}}^{(r)}_{i}+{\mathbb{H}}^{(r)}_{j}={\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}+{\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{j}\end{bmatrix}={\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}+{\mathbb{I}}\\ {\mathbb{A}}_{i}+{\mathbb{A}}_{j}\end{bmatrix}\\ =\begin{bmatrix}{\mathbb{M}}^{(r)}_{1}&{\mathbb{M}}^{(r)}_{2}\end{bmatrix}\begin{bmatrix}{\mathbb{O}}\\ {\mathbb{A}}_{i}+{\mathbb{A}}_{j}\end{bmatrix}={\mathbb{M}}^{(r)}_{1}{\mathbb{O}}+{\mathbb{M}}^{(r)}_{2}({\mathbb{A}}_{i}+{\mathbb{A}}_{j})={\mathbb{M}}^{(r)}_{2}({\mathbb{A}}_{i}+{\mathbb{A}}_{j}).

(24)

Since ${\mathbb{H}}^{(r)}_{i}$ and $({\mathbb{A}}_{i}+{\mathbb{A}}_{j})$ are invertible, we conclude that ${\mathbb{M}}^{(r)}_{2}$ is also invertible. From (24), we have ${\mathbb{H}}^{(r)}_{k}={\mathbb{M}}^{(r)}_{2}({\mathbb{A}}_{k}+{\mathbb{A}}_{j})$ for all $k\in[N]$ . Since both ${\mathbb{M}}^{(r)}_{2}$ and $({\mathbb{A}}_{k}+{\mathbb{A}}_{j})$ are invertible if $k\neq j$ , ${\mathbb{H}}^{(r)}_{k}$ is invertible if $k\neq j$ . Therefore, from (9), we have $B^{(r)}(P_{R})=B_{i}(P_{R})=\sum_{k\in[N]\setminus\{i\}}B_{i,k}(P_{R})=\sum_{k\in[N]\setminus\{i,j\}}2=2(N-2)=2K$ . ∎

Example.

{\mathbb{M}}^{(r)}{\mathbb{H}}=\begin{bmatrix}1&0&1&0&1&0&0&0\\ 0&1&2&3&4&5&0&0\end{bmatrix}\triangleq\begin{bmatrix}{\mathbb{B}}_{1}&{\mathbb{B}}_{2}&{\mathbb{B}}_{3}&{\mathbb{O}}\end{bmatrix}

is okay, but

{\mathbb{M}}^{(r)}{\mathbb{H}}=\begin{bmatrix}1&0&1&0&1&0&0&0\\ 0&1&2&3&4&0&0&0\end{bmatrix}\triangleq\begin{bmatrix}{\mathbb{B}}_{4}&{\mathbb{B}}_{5}&{\mathbb{R}}&{\mathbb{O}}\end{bmatrix}

is impossible, where ${\mathbb{R}}\in{\mathcal{R}}$ and ${\mathbb{B}}_{i}\in{\mathcal{M}}_{\mathrm{inv}}$ for all $i\in[5]$ .

Lemma 5.

Given a $P_{R}$ , then ${\mathbb{M}}^{(r)}$ is of rank $2$ for all $r\in[R]$ .

Note.

This lemma can be applied to $N-K\geq 2$ .

Proof:

Since $\mathrm{rank}({\mathbb{A}}{\mathbb{B}})\leq\min\{\mathrm{rank}({\mathbb{A}}),\mathrm{rank}({\mathbb{B}})\}$ and ${\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}$ is invertible if $i\in{\mathcal{I}}^{(r)}$ ,

$\displaystyle\mathrm{rank}\left({\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}\right)=2$	$\displaystyle\leq$	$\displaystyle\min\left\{\mathrm{rank}({\mathbb{M}}^{(r)}),\mathrm{rank}\left(\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}\right)\right\}$	(25)
	$\displaystyle\leq$	$\displaystyle\mathrm{rank}({\mathbb{M}}^{(r)})$	(26)
	$\displaystyle\leq$	$\displaystyle 2.$	(27)

Therefore, $\mathrm{rank}({\mathbb{M}}^{(r)})=2$ for all $r\in[R]$ . ∎

Lemma 6.

Given a $P_{R}$ , if ${\mathbb{H}}^{(r_{1})}_{i}$ and ${\mathbb{H}}^{(r_{2})}_{i}$ are both in ${\mathcal{L}}$ (or ${\mathcal{R}}$ ), ${\mathbb{H}}^{(r_{1})}_{j}$ and ${\mathbb{H}}^{(r_{2})}_{j}$ are both in ${\mathcal{L}}$ (or ${\mathcal{R}}$ ) for some $r_{1},r_{2}\in[R]$ , $i,j\in[N]\setminus\{{\mathcal{I}}^{(r_{1})}\cup{\mathcal{I}}^{(r_{2})}\}$ , and $i\neq j$ , then $\begin{bmatrix}{\mathbb{M}}^{(r_{1})}\\ {\mathbb{M}}^{(r_{2})}\end{bmatrix}$ is of rank $2$ .

Note.

This lemma can be applied to $N-K\geq 2$ .

Proof:

Since $\begin{bmatrix}{\mathbb{I}}&{\mathbb{I}}\\ {\mathbb{A}}_{i}&{\mathbb{A}}_{j}\end{bmatrix}$ is invertible and $\mathrm{rank}({\mathbb{A}})=\mathrm{rank}({\mathbb{A}}{\mathbb{B}})$ if ${\mathbb{B}}$ is of full rank,

\mathrm{rank}\left(\begin{bmatrix}{\mathbb{M}}^{(r_{1})}\\ {\mathbb{M}}^{(r_{2})}\end{bmatrix}\right)=\mathrm{rank}\left(\begin{bmatrix}{\mathbb{M}}^{(r_{1})}\\ {\mathbb{M}}^{(r_{2})}\end{bmatrix}\begin{bmatrix}{\mathbb{I}}&{\mathbb{I}}\\ {\mathbb{A}}_{i}&{\mathbb{A}}_{j}\end{bmatrix}\right)=\mathrm{rank}\left(\begin{bmatrix}{\mathbb{H}}^{(r_{1})}_{i}&{\mathbb{H}}^{(r_{2})}_{j}\\ {\mathbb{H}}^{(r_{2})}_{i}&{\mathbb{H}}^{(r_{2})}_{j}\end{bmatrix}\right)\leq 2.

(28)

Combining (28) with Lemma 5, we conclude that $\begin{bmatrix}{\mathbb{M}}^{(r_{1})}\\ {\mathbb{M}}^{(r_{2})}\end{bmatrix}$ is of rank $2$ for all $r_{1},r_{2}\in[R]$ . ∎

Lemma 7.

If ${\mathbb{M}}^{(r_{1})}={\mathbb{T}}{\mathbb{M}}^{(r_{2})}$ for some $r_{1},r_{2}\in[R]$ , $r_{1}\neq r_{2}$ , and invertible matrix ${\mathbb{T}}$ , then both ${\mathbb{M}}^{(r_{1})}$ and ${\mathbb{M}}^{(r_{2})}$ can be used to repair nodes in ${\mathcal{I}}^{(r_{1})}\cup{\mathcal{I}}^{(r_{2})}$ and $B^{(r_{1})}(P_{R})=B^{(r_{2})}(P_{R})$ .

Proof:

Since ${\mathbb{T}}$ is invertible, ${\mathbb{H}}^{(r_{2})}_{i}$ is invertible if and only if ${\mathbb{H}}^{(r_{1})}_{i}={\mathbb{T}}{\mathbb{H}}^{(r_{2})}_{i}$ is invertible. With the facts that ${\mathbb{H}}^{(r_{1})}_{i}$ is invertible when $i\in{\mathcal{I}}^{(r_{1})}$ and ${\mathbb{H}}^{(r_{2})}_{j}$ is invertible when $j\in{\mathcal{I}}^{(r_{2})}$ , we can conclude that both ${\mathbb{H}}^{(r_{1})}_{k}$ and ${\mathbb{H}}^{(r_{2})}_{k}$ are invertible when $k\in{\mathcal{I}}^{(r_{1})}\cup{\mathcal{I}}^{(r_{2})}$ . Therefore, both ${\mathbb{H}}^{(r_{1})}$ and ${\mathbb{H}}^{(r_{2})}$ can be used to repair nodes in ${\mathcal{I}}^{(r_{1})}\cup{\mathcal{I}}^{(r_{2})}$ .

From (9), $B_{i,j}(P_{R})=1$ , for some $i\in{\mathcal{I}}^{(r_{2})}$ , denotes that ${\mathbb{H}}^{(r_{2})}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}$ , so ${\mathbb{H}}^{(r_{1})}_{j}={\mathbb{T}}{\mathbb{H}}^{(r_{2})}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}$ . Also, $B_{i,j}(P_{R})=1$ , for some $i\in{\mathcal{I}}^{(r_{1})}$ , denotes that ${\mathbb{H}}^{(r_{1})}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}$ , so ${\mathbb{H}}^{(r_{2})}_{j}={\mathbb{T}}^{-1}{\mathbb{M}}^{(r_{1})}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}$ .

Furthermore, $B_{i,j}(P_{R})=0$ , for some $i\in{\mathcal{I}}^{(r_{2})}$ and $i\neq j$ , denotes that ${\mathbb{H}}^{(r_{2})}_{j}={\mathbb{O}}$ , so ${\mathbb{H}}^{(r_{1})}_{j}={\mathbb{T}}{\mathbb{H}}^{(r_{2})}_{j}={\mathbb{O}}$ . Also, $B_{i,j}(P_{R})=0$ , for some $i\in{\mathcal{I}}^{(r_{1})}$ and $i\neq j$ , denotes that ${\mathbb{H}}^{(r_{1})}_{j}={\mathbb{O}}$ , so ${\mathbb{H}}^{(r_{2})}_{j}={\mathbb{T}}^{-1}{\mathbb{M}}^{(r_{1})}_{j}={\mathbb{O}}$ .

We then prove that if $B_{i,j}(P_{R})=2$ for some $i\in{\mathcal{I}}^{(r_{1})}$ and $i\neq j$ , then ${\mathbb{H}}^{(r_{2})}_{j}\not\in\{{\mathbb{O}}\cup{\mathcal{R}}\cup{\mathcal{L}}\}$ . Assuming ${\mathbb{H}}^{(r_{2})}_{j}\in\{{\mathbb{O}}\cup{\mathcal{R}}\cup{\mathcal{L}}\}$ , so ${\mathbb{H}}^{(r_{1})}_{j}={\mathbb{T}}^{-1}{\mathbb{H}}^{(r_{2})}_{j}\in\{{\mathbb{O}}\cup{\mathcal{R}}\cup{\mathcal{L}}\}$ . Hence we can conclude that $B_{i,j}(P_{R})\neq 2$ from (9). By contradiction, $B_{i,j}(P_{R})=2$ , for some $i\in{\mathcal{I}}^{(r_{1})}$ and $i\neq j$ , implies ${\mathbb{H}}^{(r_{2})}_{j}\not\in\{{\mathbb{O}}\cup{\mathcal{R}}\cup{\mathcal{L}}\}$ . Following the similar way, we can also prove that if $B_{i,j}(P_{R})=2$ for some $i\in{\mathcal{I}}^{(r_{2})}$ and $i\neq j$ , then ${\mathbb{H}}^{(r_{1})}_{j}\not\in\{{\mathbb{O}}\cup{\mathcal{R}}\cup{\mathcal{L}}\}$ .

Combining above results and (9), we can conclude that $B_{i,k}(P_{R})=B_{j,k}(P_{R})$ for some $i\in{\mathcal{I}}^{(r_{1})}$ , $j\in{\mathcal{I}}^{(r_{2})}$ , and for all $k\in[N]\setminus\{i,j\}$ . Therefore, $B^{(r_{1})}(P_{R})=B^{(r_{2})}(P_{R})$ . ∎

Theorem 1.

For any $P_{R}$ such that ${\mathbb{H}}^{(r_{1})}_{i}$ and ${\mathbb{H}}^{(r_{2})}_{i}$ are both in ${\mathcal{L}}$ (or ${\mathcal{R}}$ ), ${\mathbb{H}}^{(r_{1})}_{j}$ and ${\mathbb{H}}^{(r_{2})}_{j}$ are both in ${\mathcal{L}}$ (or ${\mathcal{R}}$ ) for some $r_{1},r_{2}\in[R]$ , and $i,j\in[N]\setminus\{{\mathcal{I}}^{(r_{1})}\cup{\mathcal{I}}^{(r_{2})}\}$ , where $r_{1}\neq r_{2}$ and $i\neq j$ , then the repair process

P_{R-1}=\left\{\left({\mathbb{M}}^{(r)},{\mathcal{I}}^{(r)}\right):r\in[R]\setminus\{r_{1},r_{2}\}\right\}\cup\left\{\left({\mathbb{M}}^{(r_{1})},{\mathcal{I}}^{(r_{1})}\cup{\mathcal{I}}^{(r_{2})}\right)\right\}

(29)

has the same total repair bandwidth as $P_{R}$ has.

Proof:

By Lemmas 5 and 6, we can have an invertible transform matrix ${\mathbb{T}}$ such that ${\mathbb{M}}^{(r_{1})}={\mathbb{T}}{\mathbb{M}}^{(r_{2})}$ . Then according to Lemma 7, ${\mathbb{M}}^{(r_{1})}$ can repair nodes in ${\mathcal{I}}^{(r_{2})}$ and $B^{(r_{1})}(P_{R})=B^{(r_{2})}(P_{R})$ . Therefore,

$\displaystyle B(P_{R})$	$\displaystyle=$	$\displaystyle\sum_{r\in[R]}\|{\mathcal{I}}^{(r)}\|B^{(r)}(P_{R})$	(30)
	$\displaystyle=$	$\displaystyle\sum_{r\in[R]\setminus\{r_{1},r_{2}\}}\|{\mathcal{I}}^{(r)}\|B^{(r)}(P_{R})+\|{\mathcal{I}}^{(r_{1})}\|B^{(r_{1})}(P_{R})+\|{\mathcal{I}}^{(r_{2})}\|B^{(r_{2})}(P_{R})$	(31)
	$\displaystyle=$	$\displaystyle\sum_{r\in[R]\setminus\{r_{1},r_{2}\}}\|{\mathcal{I}}^{(r)}\|B^{(r)}(P_{R})+\|{\mathcal{I}}^{(r_{1})}\|B^{(r_{1})}(P_{R})+\|{\mathcal{I}}^{(r_{2})}\|B^{(r_{1})}(P_{R})$	(32)
	$\displaystyle=$	$\displaystyle\sum_{r\in[R]\setminus\{r_{1},r_{2}\}}\|{\mathcal{I}}^{(r)}\|B^{(r)}(P_{R})+\|{\mathcal{I}}^{(r_{1})}+{\mathcal{I}}^{(r_{1})}\|B^{(r_{1})}(P_{R})$	(33)
	$\displaystyle=$	$\displaystyle B(P_{R-1}).$	(34)

∎

Lemma 8.

Given a $P_{R}$ , if $B_{i}(P_{R})<2K$ for some $i\in{\mathcal{I}}^{(r)}$ , then

	$\displaystyle B_{i}(P_{R})$	$\displaystyle=$	$\displaystyle 2(N-1)-\left\|\left\{j:j\in[N]\setminus\{i\}\mbox{ and }{\mathbb{H}}^{(r)}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}\right\}\right\|$		(35)
		$\displaystyle=$	$\displaystyle 2(N-1)-\left\|\left\{j:j\in[N]\setminus\{i\}\mbox{ and }B_{i,j}(P_{R})=1\right\}\right\|,$		(36)

where ${\mathbb{H}}^{(r)}_{j}={\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{j}\end{bmatrix}$ .

Proof:

Due to Lemma 4 and $B_{i}(P_{R})<2K$ , we can not have $B_{i,j}(P_{R})=0$ for all $j\in[N]\setminus\{i\}$ . By Definition 2,

$\displaystyle B_{i}(P_{R})$	$\displaystyle=$	$\displaystyle\sum_{j\in[N]\setminus\{i\}}B_{i,j}(P_{R})$	(37)
	$\displaystyle=$	$\displaystyle 2\left\|\{j:j\in[N]\setminus\{i\}\mbox{ and }{\mathbb{H}}^{(r)}_{j}\in{\mathcal{M}}\}\right\|+\left\|\{j:j\in[N]\setminus\{i\}\mbox{ and }{\mathbb{H}}^{(r)}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}\}\right\|$	(38)
	$\displaystyle=$	$\displaystyle 2(N-1)-\left\|\{j:j\in[N]\setminus\{i\}\mbox{ and }{\mathbb{H}}^{(r)}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}\}\right\|$	(39)
	$\displaystyle=$	$\displaystyle 2(N-1)-\left\|\left\{j:j\in[N]\setminus\{i\}\mbox{ and }B_{i,j}(P_{R})=1\right\}\right\|$	(40)

∎

We then present the properties of repair matrices of any $P_{R}$ .

Lemma 9.

Given a $P_{R}$ , if $B_{i}(P_{R})<2K$ for some $i\in{\mathcal{I}}^{(r)}$ , then ${\mathbb{M}}^{(r)}_{2}$ is not invertible, where ${\mathbb{M}}^{(r)}=\begin{bmatrix}{\mathbb{M}}^{(r)}_{1}&{\mathbb{M}}^{(r)}_{2}\end{bmatrix}$ .

Proof:

Since $B_{i}(P_{R})<2K$ , we have

B_{i}(P_{R})=2(N-1)-\left|\left\{j:j\in[N]\setminus\{i\}\mbox{ and }{\mathbb{H}}^{(r)}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}\right\}\right|<2K,

(41)

which induces

\left|\left\{j:j\in[N]\setminus\{i\}\mbox{ and }{\mathbb{H}}^{(r)}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}\right\}\right|>2(N-1)-2K=2.

(42)

By pigeonhole principle, there must be $j_{1},j_{2}\in[N]\setminus\{i\}$ , such that $j_{1}\neq j_{2}$ and both ${\mathbb{H}}^{(r)}_{j_{1}},{\mathbb{H}}^{(r)}_{j_{2}}\in{\mathcal{L}}$ (or ${\mathcal{R}}$ ). WLOG, we assume ${\mathbb{H}}^{(r)}_{j_{1}},{\mathbb{H}}^{(r)}_{j_{2}}\in{\mathcal{L}}$ and then we have

$\displaystyle{\mathbb{H}}^{(r)}_{j_{1}}+{\mathbb{H}}^{(r)}_{j_{2}}$	$\displaystyle=$	$\displaystyle{\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{j_{1}}\end{bmatrix}+{\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{j_{2}}\end{bmatrix}$	(43)
	$\displaystyle=$	$\displaystyle\begin{bmatrix}{\mathbb{M}}^{(r)}_{1}&{\mathbb{M}}^{(r)}_{2}\end{bmatrix}\begin{bmatrix}{\mathbb{O}}\\ {\mathbb{A}}_{j_{1}}+{\mathbb{A}}_{j_{2}}\end{bmatrix}$	(44)
	$\displaystyle=$	$\displaystyle{\mathbb{M}}^{(r)}_{2}({\mathbb{A}}_{j_{1}}+{\mathbb{A}}_{j_{2}})\in{\mathcal{L}}.$	(45)

Since ${\mathbb{M}}^{(r)}_{2}({\mathbb{A}}_{j_{1}}+{\mathbb{A}}_{j_{2}})$ is not invertible and ${\mathbb{A}}_{j_{1}}+{\mathbb{A}}_{j_{2}}$ is invertible, we can conclude that ${\mathbb{M}}^{(r)}_{2}$ is not invertible. ∎

Lemma 10.

Given a $P_{R}=\left\{\left({\mathbb{M}}^{(r)},{\mathcal{I}}^{(r)}\right):r\in[R]\right\}$ and ${\hat{r}}\in[R]$ such that $B_{i}(P_{R})<2K$ for some ${i}\in{\mathcal{I}}^{({\hat{r}})}$ . By letting $\hat{{\mathbb{M}}}^{({\hat{r}})}=\left[{\mathbb{H}}^{({\hat{r}})}_{i}\right]^{-1}{\mathbb{M}}^{({\hat{r}})}$ ,

(1)

we can construct an repair process $\hat{P}_{R}$ as

\hat{P}_{R}=\left\{\left({\mathbb{M}}^{(r)},{\mathcal{I}}^{(r)}\right):r\in[R]\setminus\{{\hat{r}}\}\right\}\cup\left\{\left(\hat{{\mathbb{M}}}^{({\hat{r}})},{\mathcal{I}}^{({\hat{r}})}\right)\right\},

(46)

where $B_{i}(P_{R})=B_{i}(\hat{P}_{R})$ .

(2)

For $\hat{P}_{R}$ ,

\hat{{\mathbb{M}}}^{({\hat{r}})}=\begin{bmatrix}1&0&0&0\\ \beta^{({\hat{r}})}_{1}&\beta^{({\hat{r}})}_{2}&\beta^{({\hat{r}})}_{3}&\beta^{({\hat{r}})}_{4}\end{bmatrix},

(47)

if $\exists j_{1},j_{2}\in[N]$ and $j_{1}\neq j_{2}$ such that $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{1}},\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{2}}\in{\mathcal{L}}$ , where $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j}=\hat{{\mathbb{M}}}^{({\hat{r}})}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{j}\end{bmatrix}$ for all $j\in[N]$ and $\beta^{({\hat{r}})}_{k}\in{\mathcal{F}}_{q}$ for all $k\in[4]$ .

(3)

For $\hat{P}_{R}$ ,

\hat{{\mathbb{M}}}^{({\hat{r}})}=\begin{bmatrix}\beta^{({\hat{r}})}_{1}&\beta^{({\hat{r}})}_{2}&\beta^{({\hat{r}})}_{3}&\beta^{({\hat{r}})}_{4}\\ 0&1&0&0\end{bmatrix},

(48)

if $\exists j_{1},j_{2}\in[N]$ and $j_{1}\neq j_{2}$ such that $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{1}},\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{2}}\in{\mathcal{R}}$ .

Proof:

(1) is followed directly from Lemma 7.

Next we prove (2). Since $\hat{{\mathbb{M}}}^{({\hat{r}})}=\left[{\mathbb{H}}^{({\hat{r}})}_{i}\right]^{-1}{\mathbb{M}}^{({\hat{r}})}$ ,

\hat{{\mathbb{H}}}^{({\hat{r}})}_{i}\triangleq\hat{{\mathbb{M}}}^{({\hat{r}})}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}=\left[{\mathbb{H}}^{({\hat{r}})}_{i}\right]^{-1}{\mathbb{M}}^{({\hat{r}})}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}=\left[{\mathbb{H}}^{({\hat{r}})}_{i}\right]^{-1}{\mathbb{H}}^{({\hat{r}})}_{i}={\mathbb{I}}.

(49)

Since $B_{i}(\hat{P}_{R})<2K$ , $\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}$ is not invertible due to Lemma 9.

We first prove that $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j}=\begin{bmatrix}1&0\\ \beta&0\end{bmatrix}$ for some $\beta\in{\mathcal{F}}_{q}$ if $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j}\in{\mathcal{L}}$ . Let $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j}=\begin{bmatrix}\beta^{\prime}&0\\ \beta&0\end{bmatrix}\in{\mathcal{L}}$ for some $\beta,\beta^{\prime}\in{\mathcal{F}}_{q}$ , then we have

\hat{{\mathbb{H}}}^{({\hat{r}})}_{i}+\hat{{\mathbb{H}}}^{({\hat{r}})}_{j}={\mathbb{I}}+\begin{bmatrix}\beta^{\prime}&0\\ \beta&0\end{bmatrix}=\begin{bmatrix}1+\beta^{\prime}&0\\ \beta&1\end{bmatrix}=\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}({\mathbb{A}}_{i}+{\mathbb{A}}_{j}).

(50)

Since $\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}$ is not invertible, then $\begin{bmatrix}1+\beta^{\prime}&0\\ \beta&1\end{bmatrix}$ is not invertible as well, which implies that the determinant of $\begin{bmatrix}1+\beta^{\prime}&0\\ \beta&1\end{bmatrix}$ is $0$ . Therefore, $1\times(1+\beta^{\prime})=0$ implies $\beta^{\prime}=1$ .

We next prove that $\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}=\begin{bmatrix}0&0\\ \beta^{({\hat{r}})}_{3}&\beta^{({\hat{r}})}_{4}\end{bmatrix}$ for some $\beta^{({\hat{r}})}_{3},\beta^{({\hat{r}})}_{4}\in{\mathcal{F}}_{q}$ , if $\exists j_{1},j_{2}\in[N]$ and $j_{1}\neq j_{2}$ such that $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{1}},\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{2}}\in{\mathcal{L}}$ . Let $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{1}}=\begin{bmatrix}1&0\\ \beta_{1}&0\end{bmatrix}$ and $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{2}}=\begin{bmatrix}1&0\\ \beta_{2}&0\end{bmatrix}$ , we have

	$\displaystyle\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{1}}+\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{2}}$	$\displaystyle=$	$\displaystyle\begin{bmatrix}0&0\\ \beta_{1}+\beta_{2}&0\end{bmatrix}$		(51)
		$\displaystyle=$	$\displaystyle\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}({\mathbb{A}}_{j_{1}}+{\mathbb{A}}_{j_{2}}).$		(52)

Since $({\mathbb{A}}_{j_{1}}+{\mathbb{A}}_{j_{2}})$ is invertible, then

\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}=\begin{bmatrix}0&0\\ \beta_{1}+\beta_{2}&0\end{bmatrix}({\mathbb{A}}_{j_{1}}+{\mathbb{A}}_{j_{2}})^{-1}=\begin{bmatrix}0&0\\ \beta^{({\hat{r}})}_{3}&\beta^{({\hat{r}})}_{4}\end{bmatrix},

(53)

for some $\beta^{({\hat{r}})}_{3},\beta^{({\hat{r}})}_{4}\in{\mathcal{F}}_{q}$ . Lastly, we prove that $\hat{{\mathbb{M}}}^{({\hat{r}})}_{1}=\begin{bmatrix}1&0\\ \beta^{({\hat{r}})}_{1}&\beta^{({\hat{r}})}_{2}\end{bmatrix}$ for some $\beta^{({\hat{r}})}_{1},\beta^{({\hat{r}})}_{2}\in{\mathcal{F}}_{q}$ if $\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}=\begin{bmatrix}0&0\\ \beta^{({\hat{r}})}_{3}&\beta^{({\hat{r}})}_{4}\end{bmatrix}$ for some $\beta^{({\hat{r}})}_{3},\beta^{({\hat{r}})}_{4}\in{\mathcal{F}}_{q}$ . Since

\hat{{\mathbb{H}}}^{({\hat{r}})}_{i}=\begin{bmatrix}\hat{{\mathbb{M}}}^{({\hat{r}})}_{1}&\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}\end{bmatrix}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{i}\end{bmatrix}=\hat{{\mathbb{M}}}^{({\hat{r}})}_{1}+\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}{\mathbb{A}}_{i}={\mathbb{I}}

(54)

and $\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}=\begin{bmatrix}0&0\\ \beta^{({\hat{r}})}_{3}&\beta^{({\hat{r}})}_{4}\end{bmatrix}$ , $\hat{{\mathbb{M}}}^{({\hat{r}})}_{1}=\hat{{\mathbb{M}}}^{({\hat{r}})}_{2}{\mathbb{A}}_{i}+{\mathbb{I}}=\begin{bmatrix}1&0\\ \beta^{({\hat{r}})}_{1}&\beta^{({\hat{r}})}_{2}\end{bmatrix}$ for some $\beta^{({\hat{r}})}_{1},\beta^{({\hat{r}})}_{2}\in{\mathcal{F}}_{q}$ .

Therefore, we can conclude that $\hat{{\mathbb{M}}}^{({\hat{r}})}=\begin{bmatrix}1&0&0&0\\ \beta^{({\hat{r}})}_{1}&\beta^{({\hat{r}})}_{2}&\beta^{({\hat{r}})}_{3}&\beta^{({\hat{r}})}_{4}\end{bmatrix}$ , if $\exists j_{1},j_{2}\in[N]$ and $j_{1}\neq j_{2}$ such that $\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{1}},\hat{{\mathbb{H}}}^{({\hat{r}})}_{j_{2}}\in{\mathcal{L}}$ .

A similar proof can be made for (3). ∎

Theorem 2.

Given a $P_{R}$ and let ${\mathcal{J}}_{i}\triangleq\{j:j\in[N]\mbox{ and }B_{i,j}(P_{R})=1\}$ , if $B_{i}(P_{R})<2K$ for some $i\in{\mathcal{I}}^{(r)}$ , then either ${\mathbb{H}}^{(r)}_{j}\in{\mathcal{L}}$ for all $j\in{\mathcal{J}}_{i}$ or ${\mathbb{H}}^{(r)}_{j}\in{\mathcal{R}}$ for all $j\in{\mathcal{J}}_{i}$ .

Proof:

From Lemma 8 and $B_{i}(P_{R})<2K$ , we can find at least $3$ different indexes from $[N]\setminus{\mathcal{I}}^{(r)}$ , $j_{1}$ , $j_{2}$ , and $j_{3}$ , such that ${\mathbb{H}}^{(r)}_{j_{1}},{\mathbb{H}}^{(r)}_{j_{2}},{\mathbb{H}}^{(r)}_{j_{3}}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}$ . Hence, $|{\mathcal{J}}_{i}|\geq 3$ . By pigeonhole principle, there must be two of $\left\{{\mathbb{H}}^{(r)}_{j_{1}},{\mathbb{H}}^{(r)}_{j_{2}},{\mathbb{H}}^{(r)}_{j_{3}}\right\}$ are either in ${\mathcal{L}}$ or ${\mathcal{R}}$ .

WLOG, we assume ${\mathbb{H}}^{(r)}_{j_{1}},{\mathbb{H}}^{(r)}_{j_{2}}\in{\mathcal{L}}$ . By Lemma 10, we can construct a repair matrix

\hat{{\mathbb{M}}}^{(r)}=\begin{bmatrix}1&0&0&0\\ \beta^{(r)}_{1}&\beta^{(r)}_{2}&\beta^{(r)}_{3}&\beta^{(r)}_{4}\end{bmatrix},

(55)

where $\hat{{\mathbb{M}}}^{(r)}=\left[{\mathbb{H}}^{(r)}_{i}\right]^{-1}{\mathbb{M}}^{(r)}$ and $\beta^{(r)}_{k}\in{\mathcal{F}}_{q}$ for all $k\in[4]$ . Suppose there is a $\hat{j}\in[N]\setminus\{j_{1},j_{2}\}$ such that ${\mathbb{H}}^{(r)}_{\hat{j}}\in{\mathcal{R}}$ , then

{\mathbb{H}}^{(r)}_{\hat{j}}={\mathbb{M}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{\hat{j}}\end{bmatrix}={\mathbb{H}}^{(r)}_{i}\hat{{\mathbb{M}}}^{(r)}\begin{bmatrix}{\mathbb{I}}\\ {\mathbb{A}}_{\hat{j}}\end{bmatrix}={\mathbb{H}}^{(r)}_{i}\begin{bmatrix}1&0\\ \beta_{1}&\beta_{2}\end{bmatrix}={\mathbb{H}}^{(r)}_{i}\hat{{\mathbb{H}}}^{(r)}_{\hat{j}}

(56)

for some $\beta_{1},\beta_{2}\in{\mathcal{F}}_{q}$ . Since $\hat{{\mathbb{H}}}^{(r)}_{\hat{j}}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}$ , we have $\beta_{2}=0$ and $\hat{{\mathbb{H}}}^{(r)}_{\hat{j}}\in{\mathcal{L}}$ . Therefore, ${\mathbb{H}}^{(r)}_{\hat{j}}={\mathbb{H}}^{(r)}_{i}\hat{{\mathbb{H}}}^{(r)}_{\hat{j}}\in{\mathcal{L}}$ , which contradicts to the assumption of ${\mathbb{H}}^{(r)}_{\hat{j}}\in{\mathcal{R}}$ . Therefore, we can prove that, for all $j\in{\mathcal{J}}_{i}$ , ${\mathbb{H}}^{(r)}_{j}\in{\mathcal{L}}$ .

A similar proof can be made for ${\mathbb{H}}^{(r)}_{j}\in{\mathcal{R}}$ for all $j\in{\mathcal{J}}_{i}$ when we assume ${\mathbb{H}}^{(r)}_{j_{1}},{\mathbb{H}}^{(r)}_{j_{2}}\in{\mathcal{R}}$ . ∎

III Lower bound on $B(P_{R})$

Let ${\mathcal{P}}_{R}$ be the set of the all effective¹¹1From Theorem 1, we know that $P_{R}$ can be reduced to $P_{R-1}$ if there exists different $r_{1},r_{2}\in[R]$ and different $i,j\in[N]$ such that ${\mathbb{H}}^{(r_{1})}_{i},{\mathbb{H}}^{(r_{2})}_{i}\in{\mathcal{L}}$ (or ${\mathcal{R}}$ ) and ${\mathbb{H}}^{(r_{1})}_{j},{\mathbb{H}}^{(r_{2})}_{j}\in{\mathcal{L}}$ (or ${\mathcal{R}}$ ). For those repair processes of size $R$ satisfy the above constraint are said to be not effective and excluded from ${\mathcal{P}}_{R}$ to avoid duplicate computation. repair processes of size $R$ , our goal is to minimizes the total repair bandwidth over ${\mathcal{P}}_{R}$ for a given $R$ , i.e.,

B^{*}(R)\triangleq\min_{P_{R}\in{\mathcal{P}}_{R}}B(P_{R}).

(57)

Let

{\mathcal{J}}_{r}(P_{R})\triangleq\left\{j:j\in[N]\mbox{ and }{\mathbb{H}}^{(r)}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}\right\},

(58)

as stated in Lemmas 3 and 8, the total repair bandwidth of $P_{R}$ is

$\displaystyle B(P_{R})$	$\displaystyle=$	$\displaystyle\sum_{r\in[R]}\|{\mathcal{I}}^{(r)}\|\times\Big{[}2(N-1)-\|{\mathcal{J}}_{r}(P_{R})\|\Big{]}$	(59)
	$\displaystyle=$	$\displaystyle\sum_{r\in[R]}\|{\mathcal{I}}^{(r)}\|\times\Big{[}2(N-1)\Big{]}-\sum_{r\in[R]}\|{\mathcal{I}}^{(r)}\|\times\|{\mathcal{J}}_{r}(P_{R})\|$	(60)
	$\displaystyle=$	$\displaystyle N\times\Big{[}2(N-1)\Big{]}-\sum_{r\in[R]}\|{\mathcal{I}}^{(r)}\|\times\|{\mathcal{J}}_{r}(P_{R})\|.$	(61)

Therefore,

B^{*}(R)=\min_{P_{R}\in{\mathcal{P}}_{R}}B(P_{R})=N\times\Big{[}2(N-1)\Big{]}-\max_{P_{R}\in{\mathcal{P}}_{R}}\sum_{r\in[R]}|{\mathcal{I}}^{(r)}|\times|{\mathcal{J}}_{r}(P_{R})|,

(62)

which means that minimizing $B(P_{R})$ over $P_{R}\in{\mathcal{P}}_{R}$ is equivalent to maximizing $\sum_{r\in[R]}|{\mathcal{I}}^{(r)}|\times|{\mathcal{J}}_{r}(P_{R})|$ over $P_{R}\in{\mathcal{P}}_{R}$ .

For any repair process $P_{R}$ , we define ${\mathbb{H}}_{P_{R}}$ as

{\mathbb{H}}_{P_{R}}\triangleq\begin{bmatrix}{\mathbb{M}}^{(1)}\\ {\mathbb{M}}^{(2)}\\ \vdots\\ {\mathbb{M}}^{(R)}\end{bmatrix}{\mathbb{H}}=\begin{bmatrix}{\mathbb{H}}^{(1)}_{1}&{\mathbb{H}}^{(1)}_{2}&\cdots&{\mathbb{H}}^{(1)}_{N}\\ {\mathbb{H}}^{(2)}_{1}&{\mathbb{H}}^{(2)}_{2}&\cdots&{\mathbb{H}}^{(2)}_{N}\\ \vdots&\vdots&\ddots&\vdots\\ {\mathbb{H}}^{(R)}_{1}&{\mathbb{H}}^{(R)}_{2}&\cdots&{\mathbb{H}}^{(R)}_{N}\end{bmatrix}.

(63)

From Theorem 2, we know that either ${\mathbb{H}}^{(r)}_{i}\in{\mathcal{L}}$ for all $i\in{\mathcal{J}}_{r}(P_{R})$ or ${\mathbb{H}}^{(r)}_{i}\in{\mathcal{R}}$ for all $i\in{\mathcal{J}}_{r}(P_{R})$ . Hence we can categorize all repair matrices into $2$ types. Since we can swap any ${\mathbb{M}}^{(i)}$ and ${\mathbb{M}}^{(j)}$ , $i,j\in[R]$ , without effecting the total repair bandwidth, WLOG, we assume that there is a $0\leq R_{\mathrm{L}}\leq R$ such that

{\mathbb{H}}^{(r)}_{k}\in{\mathcal{L}}\mbox{ if }k\in{\mathcal{J}}_{r}(P_{R})\mbox{ and }r\in[R_{\mathrm{L}}]

(64)

and

{\mathbb{H}}^{(r)}_{k}\in{\mathcal{R}}\mbox{ if }k\in{\mathcal{J}}_{r}(P_{R})\mbox{ and }r\in[R]\setminus[R_{\mathrm{L}}].

(65)

Since we can also swap any ${\mathbb{A}}_{i}$ and ${\mathbb{A}}_{j}$ in ${\mathbb{H}}$ without effecting the total repair bandwidth of $P_{R}$ , WLOG, we can assume that any $i\in{\mathcal{I}}^{(r)}$ is less than any $j\in{\mathcal{I}}^{(r+1)}$ for all $r\in[R-1]$ . Let $\ell_{r}\triangleq|{\mathcal{I}}^{(r)}|$ for all $r\in[R]$ , then $\{{\mathcal{I}}^{(r)}\}_{r\in[R]}$ can be uniquely decided by $\vec{\ell}\triangleq(\ell_{1},\ell_{2},\ldots,\ell_{R})$ as

$\displaystyle{\mathcal{I}}^{(1)}$	$\displaystyle=$	$\displaystyle\left\{1,2,\ldots,\ell_{1}\right\}=[\ell^{*}_{1}];$	(66)
$\displaystyle{\mathcal{I}}^{(2)}$	$\displaystyle=$	$\displaystyle\left\{\ell^{}_{1}+1,\ell^{}_{1}+2,\ldots,\ell^{}_{2}\right\}=[\ell^{}_{2}]\setminus[\ell^{*}_{1}];$	(67)
	$\displaystyle\vdots$		(68)
$\displaystyle{\mathcal{I}}^{(r)}$	$\displaystyle=$	$\displaystyle\left\{\ell^{}_{r-1}+1,\ell^{}_{r-1}+2,\ldots,\ell^{}_{r}\right\}=[\ell^{}_{r}]\setminus[\ell^{*}_{r-1}];$	(69)
	$\displaystyle\vdots$		(70)
$\displaystyle{\mathcal{I}}^{(R)}$	$\displaystyle=$	$\displaystyle\left\{\ell^{}_{R-1}+1,\ell^{}_{R-1}+2,\ldots,N\right\}=[N]\setminus[\ell^{*}_{R-1}],$	(71)

where $\ell^{*}_{r}\triangleq\sum_{k\in[r]}\ell_{k}$ .

By summarizing Theorems 1 and 2, we can have the following constraints on ${\mathbb{H}}_{P_{R}}$ .

•

Constraint 1-L: There exists no distinct $r_{1},r_{2}\in[R_{\mathrm{L}}]$ and distinct $i,j\in[N]$ such that ${\mathbb{H}}^{(r_{1})}_{i},{\mathbb{H}}^{(r_{2})}_{i},{\mathbb{H}}^{(r_{1})}_{j},{\mathbb{H}}^{(r_{2})}_{j}\in{\mathcal{L}}$ ;
•

Constraint 1-R: There exists no distinct $r_{1},r_{2}\in[R]\setminus[R_{\mathrm{L}}]$ and distinct $i,j\in[N]$ such that ${\mathbb{H}}^{(r_{1})}_{i},{\mathbb{H}}^{(r_{2})}_{i},{\mathbb{H}}^{(r_{1})}_{j},{\mathbb{H}}^{(r_{2})}_{j}\in{\mathcal{R}}$ ;
•

Constraint 2-L: If $r\in[R_{\mathrm{L}}]$ , ${\mathbb{H}}^{(r)}_{j}\in{\mathcal{L}}$ for all $j\in{\mathcal{J}}_{r}(P_{R})$ ;
•

Constraint 2-R: If $r\in[R]\setminus[R_{\mathrm{L}}]$ , ${\mathbb{H}}^{(r)}_{j}\in{\mathcal{R}}$ for all $j\in{\mathcal{J}}_{r}(P_{R})$ .

We then turn our attention to the block matrices which satisfy above constraints without considering the repair process $P_{R}$ .

Given $R$ , $R_{\mathrm{L}}$ , and $\vec{\ell}=(\ell_{1},\ell_{2},\ldots,\ell_{R})$ such that $1\leq R\leq N$ , $0\leq R_{\mathrm{L}}\leq R$ , and $\sum_{r\in[R]}\ell_{r}=N$ , we define a set of block matrices as

{\mathcal{H}}(R,R_{\mathrm{L}},\vec{\ell}\,)=\left\{\begin{bmatrix}{{\mathbb{H}}}^{(1)}_{1}&{{\mathbb{H}}}^{(1)}_{2}&\cdots&{{\mathbb{H}}}^{(1)}_{N}\\ {{\mathbb{H}}}^{(2)}_{1}&{{\mathbb{H}}}^{(2)}_{2}&\cdots&{{\mathbb{H}}}^{(2)}_{N}\\ \vdots&\vdots&\ddots&\vdots\\ {{\mathbb{H}}}^{(R)}_{1}&{{\mathbb{H}}}^{(R)}_{2}&\cdots&{{\mathbb{H}}}^{(R)}_{N}\end{bmatrix}:\,\begin{matrix}{{\mathbb{H}}}^{(r)}_{i}\mbox{ is invertible if }i\in[\ell^{*}_{r}]\setminus[\ell^{*}_{r-1}];\\ \mbox{The first $R_{\mathrm{L}}$ rows satisfy Constraints \ref{thm:girth4free}-L and \ref{thm:sametype}-L};\\ \mbox{The last $R-R_{\mathrm{L}}$ rows satisfy Constraints \ref{thm:girth4free}-R and \ref{thm:sametype}-R}\end{matrix}\right\},

(72)

where $\ell^{*}_{r}=\sum_{k\in[r]}\ell_{k}$ . For every $\bar{{\mathbb{H}}}=\begin{bmatrix}\bar{{\mathbb{H}}}^{(r)}_{i}\end{bmatrix}_{r\in[R],i\in[N]}\in{\mathcal{H}}(R,R_{\mathrm{L}},\vec{\ell}\,)$ , similar with the definitions of ${\mathcal{J}}_{r}(P_{R})$ and $B(P_{R})$ , we define the ${\mathcal{J}}_{r}(\bar{{\mathbb{H}}})$ and $B(\bar{{\mathbb{H}}})$ as

	$\displaystyle{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})$	$\displaystyle\triangleq$	$\displaystyle\left\{j:j\in[N]\mbox{ and }\bar{{\mathbb{H}}}^{(r)}_{j}\in\{{\mathcal{L}}\cup{\mathcal{R}}\}\right\},$		(73)
	$\displaystyle B(\bar{{\mathbb{H}}})$	$\displaystyle\triangleq$	$\displaystyle N\times\Big{[}2(N-1)\Big{]}-\sum_{r\in[R]}\ell_{r}\times\|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})\|.$		(74)

It should be noted that, as discussed at the beginning of this section, we can construct a $P^{\prime}_{R}$ for any repair process $P_{R}$ such that $B(P_{R})=B(P^{\prime}_{R})$ and ${\mathbb{H}}_{P^{\prime}_{R}}\in{\mathcal{H}}(R,R_{\mathrm{L}},\vec{\ell}\,)$ ; on the contrary, there is no guarantee that we can construct a $P_{R}$ such that ${\mathbb{H}}_{P_{R}}=\bar{{\mathbb{H}}}$ for any $\bar{{\mathbb{H}}}\in{\mathcal{H}}(R,R_{\mathrm{L}},\vec{\ell}\,)$ . Therefore, we can conclude that

B^{*}(R)=\min_{P_{R}\in{\mathcal{P}}_{R}}B(P_{R})\geq\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}(R)}B(\bar{{\mathbb{H}}})\triangleq\bar{B}^{*}(R),

(75)

where

{\mathcal{H}}(R)\triangleq\bigcup_{\vec{\ell}\in\{(\ell_{1},\ldots,\ell_{R}):\sum_{r\in[R]}\ell_{r}=N\}}\,\bigcup_{0\leq R_{\mathrm{L}}\leq R}{\mathcal{H}}(R,R_{\mathrm{L}},\vec{\ell}\,).

(76)

We then turn our attention to $\bar{B}^{*}(R)$ rather than $B^{*}(R)$ .

Theorem 3.

$\bar{B}^{*}(R=2)=2N(N-1)-2\left\lfloor\frac{N^{2}}{2}\right\rfloor$ .

Proof:

Given an $\vec{\ell}=(\ell_{1},\ell_{2})$ , we consider an $\bar{{\mathbb{H}}}_{2}\in{\mathcal{H}}(2,1,\vec{\ell}\,)$ such that

\bar{{\mathbb{H}}}_{2}=\begin{bmatrix}{\mathbb{I}}^{(1)}_{1}&\cdots&{\mathbb{I}}^{(1)}_{\ell_{1}}&{\mathbb{L}}^{(1)}_{\ell_{1}+1}&\cdots&{\mathbb{L}}^{(1)}_{N}\\ {\mathbb{R}}^{(2)}_{1}&\cdots&{\mathbb{R}}^{(2)}_{\ell_{1}}&{\mathbb{I}}^{(2)}_{\ell_{1}+1}&\cdots&{\mathbb{I}}^{(2)}_{N}\end{bmatrix},

(77)

where ${\mathbb{I}}^{(r)}_{i}\in{\mathcal{M}}_{\mathrm{inv}}$ , ${\mathbb{L}}^{(r)}_{i}\in{\mathcal{L}}$ , and ${\mathbb{R}}^{(r)}_{i}\in{\mathcal{R}}$ . Since there must be at least $\ell_{r}^{\prime}$ invertible matrices in the $r$ th blockwise row of any $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R,R_{\mathrm{L}},\vec{\ell}^{\prime}=(\ell^{\prime}_{1},\ldots,\ell^{\prime}_{R})\,\right)$ , we have $|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|\leq N-\ell^{\prime}_{r}$ for any $\bar{{\mathbb{H}}}\in{\mathcal{H}}(R,R_{\mathrm{L}},\vec{\ell}^{\prime})$ . Since $|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}}_{2})|=N-\ell_{r}$ for all $r\in[2]$ , the total repair bandwidth of any $\bar{{\mathbb{H}}}\in{\mathcal{H}}(2,R_{\mathrm{L}},\vec{\ell}\,)$ is lower bounded as

$\displaystyle B(\bar{{\mathbb{H}}})$	$\displaystyle=$	$\displaystyle 2N(N-1)-\ell_{1}\|{\mathcal{J}}_{1}(\bar{{\mathbb{H}}})\|-\ell_{2}\|{\mathcal{J}}_{2}(\bar{{\mathbb{H}}})\|$	(78)
	$\displaystyle\geq$	$\displaystyle 2N(N-1)-\ell_{1}(N-\ell_{1})-\ell_{2}(N-\ell_{2})$	(79)
	$\displaystyle=$	$\displaystyle\min_{\bar{{\mathbb{H}}^{\prime}}\in{\mathcal{H}}(2,R_{\mathrm{L}},\vec{\ell}\,)}B(\bar{{\mathbb{H}}^{\prime}})=B(\bar{{\mathbb{H}}_{2}})$	(80)

for any $0\leq R_{\mathrm{L}}\leq 2$ . Therefore,

$\displaystyle\bar{B}^{*}(R)$	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2}):\,\ell_{1}+\ell_{2}=N\}}\,\min_{0\leq R_{\mathrm{L}}\leq 2}\,\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}(2,R_{\mathrm{L}}\vec{\ell}\,)}B(\bar{{\mathbb{H}}})$	(81)
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2}):\,\ell_{1}+\ell_{2}=N\}}B(\bar{{\mathbb{H}}}_{2})$	(82)
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2}):\,\ell_{1}+\ell_{2}=N\}}2N(N-1)-\ell_{1}(N-\ell_{1})-\ell_{2}(N-\ell_{2})$	(83)
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2}):\,\ell_{1}+\ell_{2}=N\}}2N(N-1)-\ell_{1}(N-\ell_{1})-(N-\ell_{1})\ell_{1}$	(84)
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2}):\,\ell_{1}+\ell_{2}=N\}}2N(N-1)+2\ell_{1}^{2}-2N\ell_{1}$	(85)
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2}):\,\ell_{1}+\ell_{2}=N\}}2N(N-1)+2\left(\ell_{1}-\frac{N}{2}\right)^{2}-\frac{N^{2}}{2}$	(86)
	$\displaystyle=$	$\displaystyle 2N(N-1)-2\left\lfloor\frac{N^{2}}{2}\right\rfloor.$	(87)

∎

Lemma 11.

$\bar{B}^{*}(2)\geq\bar{B}^{*}(3)$

Proof:

Given an

\bar{{\mathbb{H}}}=\begin{bmatrix}{\mathbb{I}}^{(1)}_{1}&{\mathbb{L}}^{(1)}_{2}&\cdots&{\mathbb{L}}^{(1)}_{\ell_{1}}&{\mathbb{B}}^{(1)}_{\ell_{1}+1}&\cdots&{\mathbb{B}}^{(1)}_{N-1}&{\mathbb{L}}^{(1)}_{N}\\ {\mathbb{L}}^{(2)}_{1}&{\mathbb{I}}^{(2)}_{2}&\cdots&{\mathbb{I}}^{(2)}_{\ell_{1}}&{\mathbb{L}}^{(2)}_{\ell_{1}+1}&\cdots&{\mathbb{L}}^{(2)}_{N-1}&{\mathbb{L}}^{(2)}_{N}\\ {\mathbb{R}}^{(3)}_{1}&{\mathbb{R}}^{(3)}_{2}&\cdots&{\mathbb{R}}^{(3)}_{\ell_{1}}&{\mathbb{I}}^{(3)}_{\ell_{1}+1}&\cdots&{\mathbb{I}}^{(3)}_{N-1}&{\mathbb{I}}^{(3)}_{N}\\ \end{bmatrix}\in{\mathcal{H}}(3,2,\vec{\ell}\,),

(88)

where $\vec{\ell}=(1,\ell_{1}-1,\ell_{2})$ , $\ell_{1}=\lceil\frac{N}{2}\rceil$ , and $\ell_{2}=N-\ell_{1}$ . We then have

$\displaystyle\bar{B}^{*}(3)$	$\displaystyle\leq$	$\displaystyle B(\bar{{\mathbb{H}}})$	(89)
	$\displaystyle=$	$\displaystyle 2N(N-1)-\ell_{1}-(\ell_{1}-1)(\ell_{2}+1)-\ell_{2}(\ell_{1})$	(90)
	$\displaystyle=$	$\displaystyle 2N(N-1)-\ell_{1}-(\ell_{1}-1)(N-\ell_{1}+1)-(N-\ell_{1})\ell_{1}$	(91)
	$\displaystyle=$	$\displaystyle 2N(N-1)-(N-\ell_{1}+1)(2\ell_{1}-1)$	(92)
	$\displaystyle=$	$\displaystyle\begin{cases}2N(N-1)-\frac{N^{2}}{2}-\frac{N}{2}+1&\mbox{if $N$ is even};\\ 2N(N-1)-\frac{N^{2}}{2}-\frac{N}{2}&\mbox{if $N$ is odd}\end{cases}$	(93)
	$\displaystyle\leq$	$\displaystyle 2N(N-1)-\frac{N^{2}}{2}$	(94)
	$\displaystyle\leq$	$\displaystyle\bar{B}^{*}(2).$	(95)

∎

Lemma 12.

Given an $\vec{\ell}=(\ell_{1},\ell_{2},\ell_{3})$ such that $\sum_{r\in[3]}\ell_{r}=N$ and $\ell_{1}\leq\ell_{2}$ . Let

\bar{{\mathbb{H}}}_{3}=\begin{bmatrix}{\mathbb{I}}^{(1)}_{1}&\cdots&{\mathbb{I}}^{(1)}_{\ell^{*}_{1}}&{\mathbb{L}}^{(1)}_{\ell^{*}_{1}+1}&\cdots&{\mathbb{L}}^{(1)}_{\ell^{*}_{2}}&{\mathbb{B}}^{(1)}_{\ell^{*}_{2}+1}&\cdots&{\mathbb{B}}^{(1)}_{N-1}&{\mathbb{L}}^{(1)}_{N}\\ {\mathbb{L}}^{(2)}_{1}&\cdots&{\mathbb{L}}^{(2)}_{\ell^{*}_{1}}&{\mathbb{I}}^{(2)}_{\ell^{*}_{1}+1}&\cdots&{\mathbb{I}}^{(2)}_{\ell^{*}_{2}}&{\mathbb{L}}^{(2)}_{\ell^{*}_{2}+1}&\cdots&{\mathbb{L}}^{(2)}_{N-1}&{\mathbb{L}}^{(2)}_{N}\\ {\mathbb{R}}^{(3)}_{1}&\cdots&{\mathbb{R}}^{(3)}_{\ell^{*}_{1}}&{\mathbb{R}}^{(3)}_{\ell^{*}_{1}+1}&\cdots&{\mathbb{R}}^{(3)}_{\ell^{*}_{2}}&{\mathbb{I}}^{(3)}_{\ell^{*}_{2}+1}&\cdots&{\mathbb{I}}^{(3)}_{N-1}&{\mathbb{I}}^{(3)}_{N}\\ \end{bmatrix}\in{\mathcal{H}}(3,2,\vec{\ell}\,),

(96)

where $\ell^{*}_{r}=\sum_{k\in[r]}\ell_{k}$ , ${\mathbb{I}}^{(r)}_{i}\in{\mathcal{M}}_{\mathrm{inv}}$ , ${\mathbb{B}}^{(r)}_{i}\in{\mathcal{M}}$ , ${\mathbb{L}}^{(r)}_{i}\in{\mathcal{L}}$ , and ${\mathbb{R}}^{(r)}_{i}\in{\mathcal{R}}$ . Then,

B(\bar{{\mathbb{H}}}_{3})=\min_{0\leq R_{\mathrm{L}}\leq R}\,\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}(3,R_{\mathrm{L}},\vec{\ell}\,)}B(\bar{{\mathbb{H}}}).

(97)

Proof:

According to (72), the constraints (1-L and 2-L) on the first $R_{\mathrm{L}}$ rows of $\bar{{\mathbb{H}}}$ are different from the constraints (1-R and 2-R) on the last $R-R_{\mathrm{L}}$ rows of $\bar{{\mathbb{H}}}$ . Therefore, the design of the first $R_{\mathrm{L}}$ row is irrelevant to the design of the remaining $R-R_{\mathrm{L}}$ rows and vice versa. Also, for any $\bar{{\mathbb{H}}}\in{\mathcal{H}}(R,R_{\mathrm{L}},\vec{\ell}\,)$ , we can replace the matrices in ${\mathcal{L}}$ with the matrices in ${\mathcal{R}}$ and the matrices in ${\mathcal{R}}$ with the matrices in ${\mathcal{L}}$ without effecting the total repair bandwidth of $\bar{{\mathbb{H}}}$ , hence we have

\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}(R,R_{\mathrm{L}},\vec{\ell}\,)}B(\bar{{\mathbb{H}}})=\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}(R,R-R_{\mathrm{L}},\vec{\ell}^{\prime}\,)}B(\bar{{\mathbb{H}}}),

(98)

where $\vec{\ell}=(\ell_{1},\ell_{2},\ldots,\ell_{R})$ and $\vec{\ell}^{\prime}=(\ell_{R_{\mathrm{L}}+1},\ell_{R_{\mathrm{L}}+2},\ldots,\ell_{R},\ell_{1},\ell_{2},\ldots,\ell_{R_{\mathrm{L}}})$ . WLOG, we assume $R_{\mathrm{L}}\geq R/2$ .

We first prove that, for any $\bar{{\mathbb{H}}}\in{\mathcal{H}}(3,3,\vec{\ell}\,)$ , we can find an $\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}(3,2,\vec{\ell}\,)$ such that $B(\bar{{\mathbb{H}}})\geq B(\bar{{\mathbb{H}}}^{\prime})$ . If $\bar{{\mathbb{H}}}$ is with $R_{\mathrm{L}}=3$ , we can construct $\bar{{\mathbb{H}}}^{\prime}$ by replacing $\bar{{\mathbb{H}}}^{(3)}_{k}$ , for all $k\in[\ell^{*}_{2}]$ , with matrices in ${\mathcal{R}}$ without violating any constraint. Then

	$\displaystyle B(\bar{{\mathbb{H}}})$	$\displaystyle=$	$\displaystyle 2N(N-1)-\sum_{r\in[3]}\ell_{r}\|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})\|$		(99)
		$\displaystyle\geq$	$\displaystyle 2N(N-1)-\sum_{r\in[2]}\ell_{r}\|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})\|-\ell_{3}(N-\ell^{*}_{2})=B(\bar{{\mathbb{H}}}^{\prime}).$		(100)

Therefore, we only need to consider those $\bar{{\mathbb{H}}}\in{\mathcal{H}}(3,2,\vec{\ell}\,)$ while minimizing the total repair bandwidth.

Let $\bar{{\mathbb{H}}}$ be with $R_{\mathrm{L}}=2$ , as discussed before, we can have $\bar{{\mathbb{H}}}^{(3)}_{k}\in{\mathcal{R}}$ for all $k\in[\ell^{*}_{2}]$ to minimize $|{\mathcal{J}}_{3}(\bar{{\mathbb{H}}})|$ to be $N-\ell_{3}$ . We then consider the first two rows of $\bar{{\mathbb{H}}}$ . All those $\bar{{\mathbb{H}}}^{(1)}_{k}$ for all $k\in[\ell^{*}_{2}]\setminus[\ell^{*}_{1}]$ and $\bar{{\mathbb{H}}}^{(2)}_{k}$ for all $k\in[\ell^{*}_{1}]$ can be in ${\mathcal{L}}$ without violating Constraint 1-L. Hence, we have

\bar{{\mathbb{H}}}=\begin{bmatrix}{\mathbb{I}}^{(1)}_{1}&\cdots&{\mathbb{I}}^{(1)}_{\ell^{*}_{1}}&{\mathbb{L}}^{(1)}_{\ell^{*}_{1}+1}&\cdots&{\mathbb{L}}^{(1)}_{\ell^{*}_{2}}&\bar{{\mathbb{H}}}^{(1)}_{\ell^{*}_{2}+1}&\cdots&\bar{{\mathbb{H}}}^{(1)}_{N}\\ {\mathbb{L}}^{(2)}_{1}&\cdots&{\mathbb{L}}^{(2)}_{\ell^{*}_{1}}&{\mathbb{I}}^{(2)}_{\ell^{*}_{1}+1}&\cdots&{\mathbb{I}}^{(2)}_{\ell^{*}_{2}}&\bar{{\mathbb{H}}}^{(2)}_{\ell^{*}_{2}+1}&\cdots&\bar{{\mathbb{H}}}^{(2)}_{N}\\ {\mathbb{R}}^{(3)}_{1}&\cdots&{\mathbb{R}}^{(3)}_{\ell^{*}_{1}}&{\mathbb{R}}^{(3)}_{\ell^{*}_{1}+1}&\cdots&{\mathbb{R}}^{(3)}_{\ell^{*}_{2}}&{\mathbb{I}}^{(3)}_{\ell^{*}_{2}+1}&\cdots&{\mathbb{I}}^{(3)}_{N}\end{bmatrix},

(101)

where its submatrix

\begin{bmatrix}\bar{{\mathbb{H}}}^{(1)}_{\ell^{*}_{2}+1}&\cdots&\bar{{\mathbb{H}}}^{(1)}_{N}\\ \bar{{\mathbb{H}}}^{(2)}_{\ell^{*}_{2}+1}&\cdots&\bar{{\mathbb{H}}}^{(2)}_{N}\end{bmatrix}

(102)

must satisfy Constraint 1-L. Let $J_{r}\triangleq|\{k:k\in[N]\setminus[\ell^{*}_{2}]\mbox{ and }\bar{{\mathbb{H}}}^{(r)}_{k}\in{\mathcal{L}}\}|$ , if $J_{1}+J_{2}\geq(N-\ell^{*}_{2})+2=\ell_{3}+2$ , then the submatrix in (102) must violate Constraint 1-L according to the pigeonhole principle, which induces $J_{1}+J_{2}\leq\ell_{3}+1$ . Therefore,

$\displaystyle B(\bar{{\mathbb{H}}})$	$\displaystyle=$	$\displaystyle 2N(N-1)-\ell_{1}(\ell_{2}+J_{1})-\ell_{2}(\ell_{1}+J_{2})-\ell_{3}(N-\ell_{3})$	(103)
	$\displaystyle\geq$	$\displaystyle 2N(N-1)-\ell_{1}\left[\ell_{2}+(\ell_{3}+1-J_{2})\right]-\ell_{2}(\ell_{1}+J_{2})-\ell_{3}(N-\ell_{3})$	(104)
	$\displaystyle=$	$\displaystyle 2N(N-1)-2\ell_{1}\ell_{2}-\ell_{1}\ell_{3}-\ell_{1}-\ell_{3}(N-\ell_{3})+(\ell_{1}-\ell_{2})J_{2},$	(105)

where (105) can be minimized by maximizing $J_{2}$ when $\ell_{1}\leq\ell_{2}$ . Since $J_{2}\leq\ell_{3}$ by (102) and $J_{1}+J_{2}\leq\ell_{3}+1$ , we can minimize $\eqref{eq:H3J2}$ by assigning $J_{2}=\ell_{3}$ and $J_{1}=1$ as $\bar{{\mathbb{H}}}_{3}$ did in (88). ∎

Theorem 4.

\bar{B}^{*}(3)\geq\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3}):\,\ell_{1}+\ell_{2}+\ell_{3}=N\mbox{ and }\ell_{1}\leq\ell_{2}\}}2N(N-1)-N^{2}+\ell_{1}^{2}+\ell^{2}_{2}+\ell_{3}^{2}+\ell_{1}(\ell_{3}-1).

(106)

Proof:

We have learned from Lemma 12 that

\min_{0\leq R_{\mathrm{L}}\leq 3}\,\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(3,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ell_{2},\ell_{3})\right)}B(\bar{{\mathbb{H}}})

(107)

can be achieved by the $\bar{{\mathbb{H}}}_{3}$ in (88) while $\ell_{1}\leq\ell_{2}$ . For those $\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}(3,R_{\mathrm{L}},\vec{\ell}^{\prime})$ , where $\vec{\ell}^{\prime}=(\ell^{\prime}_{1},\ell^{\prime}_{2},\ell^{\prime}_{3})=(\ell_{2},\ell_{1},\ell_{3})$ , we can achieve

\min_{0\leq R_{\mathrm{L}}\leq 3}\,\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(3,R_{\mathrm{L}},\vec{\ell}^{\prime}=(\ell_{2},\ell_{1},\ell_{3})\right)}B(\bar{{\mathbb{H}}})

(108)

\bar{{\mathbb{H}}}^{\prime}_{3}=\begin{bmatrix}{\mathbb{I}}^{(1)}_{1}&\cdots&{\mathbb{I}}^{(1)}_{\ell^{\prime*}_{1}}&{\mathbb{L}}^{(1)}_{\ell^{\prime*}_{1}+1}&\cdots&{\mathbb{L}}^{(1)}_{\ell^{\prime*}_{2}}&{\mathbb{L}}^{(1)}_{\ell^{\prime*}_{2}+1}&\cdots&{\mathbb{L}}^{(1)}_{N-1}&{\mathbb{L}}^{(1)}_{N}\\ {\mathbb{L}}^{(2)}_{1}&\cdots&{\mathbb{L}}^{(2)}_{\ell^{\prime*}_{1}}&{\mathbb{I}}^{(2)}_{\ell^{\prime*}_{1}+1}&\cdots&{\mathbb{I}}^{(2)}_{\ell^{\prime*}_{2}}&{\mathbb{B}}^{(2)}_{\ell^{\prime*}_{2}+1}&\cdots&{\mathbb{B}}^{(2)}_{N-1}&{\mathbb{L}}^{(2)}_{N}\\ {\mathbb{R}}^{(3)}_{1}&\cdots&{\mathbb{R}}^{(3)}_{\ell^{\prime*}_{1}}&{\mathbb{R}}^{(3)}_{\ell^{\prime*}_{1}+1}&\cdots&{\mathbb{R}}^{(3)}_{\ell^{\prime*}_{2}}&{\mathbb{I}}^{(3)}_{\ell^{\prime*}_{2}+1}&\cdots&{\mathbb{I}}^{(3)}_{N-1}&{\mathbb{I}}^{(3)}_{N}\\ \end{bmatrix}

(109)

and $B(\bar{{\mathbb{H}}}^{\prime}_{3})=B(\bar{{\mathbb{H}}}_{3})$ .

Therefore, we conclude that

$\displaystyle\bar{B}^{*}(3)$	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3}):\,\ell_{1}+\ell_{2}+\ell_{3}=N\}}\,\min_{0\leq R_{\mathrm{L}}\leq R}\,\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}(3,R_{\mathrm{L}},\vec{\ell}\,)}B(\bar{{\mathbb{H}}})$	(110)
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3}):\,\ell_{1}+\ell_{2}+\ell_{3}=N\mbox{ and }\ell_{1}\leq\ell_{2}\}}B(\bar{{\mathbb{H}}}_{3})$	(111)
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3}):\,\ell_{1}+\ell_{2}+\ell_{3}=N\mbox{ and }\ell_{1}\leq\ell_{2}\}}2N(N-1)-\ell_{1}(\ell_{2}+1)-\ell_{2}(N-\ell_{2})-\ell_{3}(N-\ell_{3})$	(114)
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3}):\,\ell_{1}+\ell_{2}+\ell_{3}=N\mbox{ and }\ell_{1}\leq\ell_{2}\}}2N(N-1)$
		$\displaystyle\hfill-\ell_{1}(N-\ell_{1}-\ell_{3}+1)-\ell_{2}(N-\ell_{2})-\ell_{3}(N-\ell_{3})$
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3}):\,\ell_{1}+\ell_{2}+\ell_{3}=N\mbox{ and }\ell_{1}\leq\ell_{2}\}}2N(N-1)$
		$\displaystyle\hfill-(\ell_{1}+\ell_{2}+\ell_{3})N+\ell_{1}^{2}+\ell_{2}^{2}+\ell_{3}^{2}+\ell_{1}(\ell_{3}-1)$
	$\displaystyle=$	$\displaystyle\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3}):\,\ell_{1}+\ell_{2}+\ell_{3}=N\mbox{ and }\ell_{1}\leq\ell_{2}\}}2N(N-1)-N^{2}+\ell_{1}^{2}+\ell_{2}^{2}+\ell_{3}^{2}+\ell_{1}(\ell_{3}-1).$	(115)

∎

We then look into $\bar{B}^{*}(R)$ when $R>3$ . From Constraints 1-L, 1-R, 2-L, and 2-R we notice that, among all $\bar{{\mathbb{H}}}\in{\mathcal{H}}(R,R_{\mathrm{L}},\vec{\ell}\,)$ , we can minimize the total bandwidth by separately minimizing bandwidth of the first $R_{\mathrm{L}}$ rows in $\bar{{\mathbb{H}}}$ and the bandwidth of the remaining $R-R_{\mathrm{L}}$ rows in $\bar{{\mathbb{H}}}$ , i.e.,

	$\displaystyle\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ldots,\ell_{R})\right)}B(\bar{{\mathbb{H}}})$	(116)
$\displaystyle=$	$\displaystyle\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ldots,\ell_{R})\right)}2N(N-1)-\sum_{r\in[R]}\ell_{r}\|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})\|$	(117)
$\displaystyle=$	$\displaystyle\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ldots,\ell_{R})\right)}2R_{\mathrm{L}}(N-1)-\sum_{r\in[R_{\mathrm{L}}]}\ell_{r}\|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})\|$	(119)
	$\displaystyle+\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ldots,\ell_{R})\right)}2(R-R_{\mathrm{L}})(N-1)-\sum_{r\in[R]\setminus[R_{\mathrm{L}}]}\ell_{r}\|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})\|.$	(119)

Also as discussed in (62), we can turn the minimization problem in to a maximization program. Therefore, we first focus on the following problem,

\max_{\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ldots,\ell_{R})\right)}\sum_{r\in[R_{\mathrm{L}}]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|.

(120)

The following lemma further simplifies (120).

Lemma 13.

\max_{\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ldots,\ell_{R})\right)}\sum_{r\in[R_{\mathrm{L}}]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|=\max_{\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R_{\mathrm{L}}+1,R_{\mathrm{L}},\vec{\ell}_{\mathrm{L}}=(\ell_{1},\ldots,\ell_{R_{\mathrm{L}}},N-\ell^{*}_{R_{\mathrm{L}}})\right)}\sum_{r\in[R_{\mathrm{L}}]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|,

(121)

where $\ell^{*}_{r}=\sum_{i\in[r]}\ell_{i}$ .

Proof:

Since the summation of $\sum_{r\in[R_{\mathrm{L}}]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|$ only considers the first $R_{\mathrm{L}}$ rows, the maximization is relevant to $\sum_{r\in[R]\setminus[R_{\mathrm{L}}]}\ell_{r}$ but irrelevant of those individual $\ell_{r}$ for $r\in[R]\setminus[R_{\mathrm{L}}]$ . ∎

Lemma 14.

Given $R_{\mathrm{L}}\geq 3$ and $\vec{\ell}=\{\ell_{1},\ell_{2},\ldots,\ell_{R_{\mathrm{L}}},N-\ell^{*}_{R_{\mathrm{L}}}\}$ , for any $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R_{\mathrm{L}}+1,R_{\mathrm{L}},\vec{\ell}\,\right)$ , we can always construct a $\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}\left(3,2,\vec{\ell}^{\prime}=(\ell^{\prime}_{1},\ell^{\prime}_{2},N-\ell^{*}_{R_{\mathrm{L}}})\right)$ such that

\sum_{r\in[R_{\mathrm{L}}]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|\leq\sum_{r\in[2]}\ell^{\prime}_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}}^{\prime})|.

(122)

Proof:

To simplify the problem formulation, for any $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R_{\mathrm{L}}+1,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ldots,\ell_{R_{\mathrm{L}}+1})\right)$ , we define $J^{(r)}_{i}(\bar{{\mathbb{H}}})$ as

J^{(r)}_{i}(\bar{{\mathbb{H}}})\triangleq\left|\left\{j:j\in[\ell^{*}_{i}]\setminus[\ell^{*}_{i-1}]\mbox{ and }\bar{{\mathbb{H}}}^{(r)}_{j}\in{\mathcal{L}}\right\}\right|,

(123)

which denotes the number of block matrices in $\begin{bmatrix}\bar{{\mathbb{H}}}^{(r)}_{\ell^{*}_{i-1}+1}&\bar{{\mathbb{H}}}^{(r)}_{\ell^{*}_{i-1}+2}&\cdots&\bar{{\mathbb{H}}}^{(r)}_{\ell^{*}_{i}}\end{bmatrix}$ belongs to ${\mathcal{L}}$ . By definition, we have $J^{(r)}_{r}(\bar{{\mathbb{H}}})=0$ and $|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|=\sum_{i\in[R+1]}J^{(r)}_{i}(\bar{{\mathbb{H}}})$ .

Table I:

J^{(r)}_{i}(\bar{{\mathbb{H}}})

for

\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(4,3,\vec{\ell}=(\ell_{1},\ell_{2},\ell_{3},N-\ell^{*}_{3})\right)

$\ell_{1}$	$\ell_{2}$	$\ell_{3}$	$N-\ell^{*}_{3}$
$J^{(1)}_{1}(\bar{{\mathbb{H}}})=0$	$J^{(1)}_{2}(\bar{{\mathbb{H}}})$	$J^{(1)}_{3}(\bar{{\mathbb{H}}})$	$J^{(1)}_{4}(\bar{{\mathbb{H}}})$
$J^{(2)}_{1}(\bar{{\mathbb{H}}})$	$J^{(2)}_{2}(\bar{{\mathbb{H}}})=0$	$J^{(2)}_{3}(\bar{{\mathbb{H}}})$	$J^{(2)}_{4}(\bar{{\mathbb{H}}})$
$J^{(3)}_{1}(\bar{{\mathbb{H}}})$	$J^{(3)}_{2}(\bar{{\mathbb{H}}})$	$J^{(3)}_{3}(\bar{{\mathbb{H}}})=0$	$J^{(3)}_{4}(\bar{{\mathbb{H}}})$

Let us consider the case of $R_{\mathrm{L}}=3$ , i.e., $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(4,3,\vec{\ell}=(\ell_{1},\ell_{2},\ell_{3},N-\ell^{*}_{3})\right)$ . WLOG, we asumme $\ell_{1}\leq\ell_{2}\leq\ell_{3}$ . The corresponding $J^{(r)}_{i}(\bar{{\mathbb{H}}})$ for all $r\in[3]$ and $i\in[4]$ are listed in Table I, and we have

\sum_{r\in[3]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|=\ell_{1}\left[J^{(1)}_{2}(\bar{{\mathbb{H}}})+J^{(1)}_{3}(\bar{{\mathbb{H}}})+J^{(1)}_{4}(\bar{{\mathbb{H}}})\right]\\ +\ell_{2}\left[J^{(2)}_{1}(\bar{{\mathbb{H}}})+J^{(2)}_{3}(\bar{{\mathbb{H}}})+J^{(2)}_{4}(\bar{{\mathbb{H}}})\right]+\ell_{3}\left[J^{(3)}_{1}(\bar{{\mathbb{H}}})+J^{(3)}_{2}(\bar{{\mathbb{H}}})+J^{(3)}_{4}(\bar{{\mathbb{H}}})\right].

(124)

Combining Constraint 1-L and pigeonhole principle, we can have the following inequalities from Table I,

$\displaystyle J^{(1)}_{2}(\bar{{\mathbb{H}}})+J^{(1)}_{4}(\bar{{\mathbb{H}}})+J^{(3)}_{2}(\bar{{\mathbb{H}}})+J^{(3)}_{4}(\bar{{\mathbb{H}}})$	$\displaystyle\leq$	$\displaystyle\ell_{2}+(N-\ell^{*}_{3})+1$	(125)
$\displaystyle J^{(2)}_{1}(\bar{{\mathbb{H}}})+J^{(2)}_{4}(\bar{{\mathbb{H}}})+J^{(3)}_{1}(\bar{{\mathbb{H}}})+J^{(3)}_{4}(\bar{{\mathbb{H}}})$	$\displaystyle\leq$	$\displaystyle\ell_{1}+(N-\ell^{*}_{3})+1$	(126)
$\displaystyle J^{(1)}_{3}(\bar{{\mathbb{H}}})+J^{(1)}_{4}(\bar{{\mathbb{H}}})+J^{(2)}_{3}(\bar{{\mathbb{H}}})+J^{(2)}_{4}(\bar{{\mathbb{H}}})$	$\displaystyle\leq$	$\displaystyle\ell_{3}+(N-\ell^{*}_{3})+1$	(127)
$\displaystyle J^{(1)}_{3}(\bar{{\mathbb{H}}})+J^{(2)}_{3}(\bar{{\mathbb{H}}})$	$\displaystyle\leq$	$\displaystyle\ell_{3}+1$	(128)
$\displaystyle J^{(1)}_{2}(\bar{{\mathbb{H}}})+J^{(3)}_{2}(\bar{{\mathbb{H}}})$	$\displaystyle\leq$	$\displaystyle\ell_{2}+1$	(129)
$\displaystyle J^{(2)}_{1}(\bar{{\mathbb{H}}})+J^{(3)}_{1}(\bar{{\mathbb{H}}})$	$\displaystyle\leq$	$\displaystyle\ell_{1}+1.$	(130)

Then we can have an upper bound of (124) by applying (126), (128), and (129) as

	$\displaystyle\sum_{r\in[3]}\ell_{r}\|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})\|$	(132)
$\displaystyle\leq$	$\displaystyle\ell_{1}\left[(\ell_{2}+1)-J^{(3)}_{2}(\bar{{\mathbb{H}}})+(\ell_{3}+1)-J^{(2)}_{3}(\bar{{\mathbb{H}}})+J^{(1)}_{4}(\bar{{\mathbb{H}}})\right]$
	$\displaystyle+\ell_{2}\left[\Big{(}\ell_{1}+N-\ell^{*}_{3}+1\Big{)}-J^{(3)}_{1}(\bar{{\mathbb{H}}})-J^{(3)}_{4}(\bar{{\mathbb{H}}})+J^{(2)}_{3}(\bar{{\mathbb{H}}})\right]$
	$\displaystyle+\ell_{3}\left[J^{(3)}_{1}(\bar{{\mathbb{H}}})+J^{(3)}_{2}(\bar{{\mathbb{H}}})+J^{(3)}_{4}(\bar{{\mathbb{H}}})\right]$
$\displaystyle=$	$\displaystyle\ell_{1}\left[\ell_{2}+\ell_{3}+2+J^{(1)}_{4}(\bar{{\mathbb{H}}})\right]+\ell_{2}\Big{[}\ell_{1}+N-\ell^{*}_{3}+1\Big{]}$
	$\displaystyle+(\ell_{3}-\ell_{1})J^{(3)}_{2}(\bar{{\mathbb{H}}})+(\ell_{2}-\ell_{1})J^{(2)}_{3}(\bar{{\mathbb{H}}})+(\ell_{3}-\ell_{2})J^{(3)}_{1}(\bar{{\mathbb{H}}})+(\ell_{3}-\ell_{2})J^{(3)}_{4}(\bar{{\mathbb{H}}})$
$\displaystyle\leq$	$\displaystyle\ell_{1}\left[\ell_{2}+\ell_{3}+2+J^{(1)}_{4}(\bar{{\mathbb{H}}})\right]+\ell_{2}\Big{[}\ell_{1}+N-\ell^{*}_{3}+1\Big{]}$	(133)
	$\displaystyle+(\ell_{3}-\ell_{1})\ell_{2}+(\ell_{2}-\ell_{1})\ell_{3}+(\ell_{3}-\ell_{2})\ell_{1}+(\ell_{3}-\ell_{2})(N-\ell^{*}_{3})$	(133)
$\displaystyle=$	$\displaystyle\ell_{1}\left[2+J^{(1)}_{4}(\bar{{\mathbb{H}}})\right]+\ell_{2}\Big{[}\ell_{3}+1\Big{]}+\ell_{3}\Big{[}\ell_{1}+\ell_{2}+N-\ell^{*}_{3}\Big{]},$	(134)

where (133) is due to $\ell_{1}\leq\ell_{2}\leq\ell_{3}$ , $J^{(3)}_{2}(\bar{{\mathbb{H}}})\leq\ell_{2}$ , $J^{(2)}_{3}(\bar{{\mathbb{H}}})\leq\ell_{3}$ , $J^{(3)}_{1}(\bar{{\mathbb{H}}})\leq\ell_{1}$ , and $J^{(3)}_{4}(\bar{{\mathbb{H}}})\leq N-\ell^{*}_{3}$ . We then investigate the $\bar{{\mathbb{H}}}$ achieves (134) as Table II shown.

Table II:

J^{(r)}_{i}(\bar{{\mathbb{H}}})

for the

\bar{{\mathbb{H}}}

achieves (134).

$\ell_{1}$	$\ell_{2}$	$\ell_{3}$	$N-\ell^{*}_{3}$
$J^{(1)}_{1}(\bar{{\mathbb{H}}})=0$	$J^{(1)}_{2}(\bar{{\mathbb{H}}})=1$	$J^{(1)}_{3}(\bar{{\mathbb{H}}})=1$	$J^{(1)}_{4}(\bar{{\mathbb{H}}})$
$J^{(2)}_{1}(\bar{{\mathbb{H}}})=1$	$J^{(2)}_{2}(\bar{{\mathbb{H}}})=0$	$J^{(2)}_{3}(\bar{{\mathbb{H}}})=\ell_{3}$	$J^{(2)}_{4}(\bar{{\mathbb{H}}})=0$
$J^{(3)}_{1}(\bar{{\mathbb{H}}})=\ell_{1}$	$J^{(3)}_{2}(\bar{{\mathbb{H}}})=\ell_{2}$	$J^{(3)}_{3}(\bar{{\mathbb{H}}})=0$	$J^{(3)}_{4}(\bar{{\mathbb{H}}})=N-\ell^{*}_{3}$

Due to inequality (125), we have $J^{(1)}_{4}(\bar{{\mathbb{H}}})=0$ . Therefore, we upper bound (124) as

\sum_{r\in[3]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|\leq 2\ell_{1}+\ell_{2}(\ell_{3}+1)+\ell_{3}\Big{(}\ell_{1}+\ell_{2}+(N-\ell^{*}_{3})\Big{)}.

(135)

Table III:

J^{(r)}_{i}(\bar{{\mathbb{H}}}^{\prime})

for

\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}\left(3,2,\vec{\ell}=(\ell_{1}+\ell_{2},\ell_{3},N-\ell^{*}_{3})\right)

$\ell_{1}+\ell_{2}$	$\ell_{3}$	$N-\ell^{*}_{3}$
$J^{(1)}_{1}(\bar{{\mathbb{H}}}^{\prime})=0$	$J^{(1)}_{2}(\bar{{\mathbb{H}}}^{\prime})=\ell_{3}$	$J^{(1)}_{3}(\bar{{\mathbb{H}}}^{\prime})=1$
$J^{(2)}_{1}(\bar{{\mathbb{H}}}^{\prime})=\ell_{1}+\ell_{2}$	$J^{(2)}_{2}(\bar{{\mathbb{H}}}^{\prime})=0$	$J^{(2)}_{3}(\bar{{\mathbb{H}}}^{\prime})=N-\ell^{*}_{3}$

Next, we construct a $\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}\big{(}3,2,\vec{\ell}^{\prime}=(\ell_{1}+\ell_{2},\ell_{3},N-\ell^{*}_{3})\big{)}$ with the corresponding $J^{(r)}_{i}(\bar{{\mathbb{H}}}^{\prime})$ as shown in Table III. Therefore

\sum_{r\in[2]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}}^{\prime})|=\ell_{1}(\ell_{3}+1)+\ell_{2}(\ell_{3}+1)+\ell_{3}\Big{(}\ell_{1}+\ell_{2}+(N-\ell^{*}_{3})\Big{)},

(136)

By subtracting (135) by (136), we have

\sum_{r\in[3]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|-\sum_{r\in[2]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}}^{\prime})|\leq\ell_{1}(1-\ell_{3})\leq 0.

(137)

Therefore, for any $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(4,3,\vec{\ell}=(\ell_{1},\ell_{2},\ell_{3},N-\ell^{*}_{3})\right)$ , we can always construct an $\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}\left(3,2,\vec{\ell}^{\prime}\right)$ such that $\vec{\ell}^{\prime}_{\mathrm{L}}=(\ell_{1}+\ell_{2},\ell_{3},N-\ell^{*}_{3})$ and

\sum_{r\in[3]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|\leq\sum_{r\in[2]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}}^{\prime})|.

(138)

Next we consider an $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R_{\mathrm{L}}+1,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ell_{2},\ldots,\ell_{R_{\mathrm{L}}},N-\ell^{*}_{R_{\mathrm{L}}})\,\right)$ for $R_{\mathrm{L}}>3$ , which is with the corresponding $J^{(r)}_{i}(\bar{{\mathbb{H}}})$ in Table IV.

Table IV:

J^{(r)}_{i}(\bar{{\mathbb{H}}})

for

\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R_{\mathrm{L}}+1,R_{\mathrm{L}},\vec{\ell}=(\ell_{1},\ell_{2},\ldots,\ell_{R_{\mathrm{L}}},N-\ell^{*}_{R_{\mathrm{L}}})\,\right)

$\ell_{1}$	$\ell_{2}$	$\cdots$	$\ell_{R_{\mathrm{L}}-1}$	$\ell_{R_{\mathrm{L}}}$	$N-\ell^{*}_{R_{\mathrm{L}}}$
$J^{(1)}_{1}(\bar{{\mathbb{H}}})=0$	$J^{(1)}_{2}(\bar{{\mathbb{H}}})$	$\cdots$	$J^{(1)}_{R_{\mathrm{L}}-1}(\bar{{\mathbb{H}}})$	$J^{(1)}_{R_{\mathrm{L}}}(\bar{{\mathbb{H}}})$	$J^{(1)}_{R_{\mathrm{L}}+1}(\bar{{\mathbb{H}}})$
$J^{(2)}_{1}(\bar{{\mathbb{H}}})$	$J^{(2)}_{2}(\bar{{\mathbb{H}}})=0$	$\cdots$	$J^{(2)}_{R_{\mathrm{L}}-1}(\bar{{\mathbb{H}}})$	$J^{(2)}_{R_{\mathrm{L}}}(\bar{{\mathbb{H}}})$	$J^{(2)}_{R_{\mathrm{L}}+1}(\bar{{\mathbb{H}}})$
$\vdots$	$\vdots$	$\ddots$	$\vdots$	$\vdots$	$\vdots$
$J^{(R_{\mathrm{L}}-1)}_{1}(\bar{{\mathbb{H}}})$	$J^{(R_{\mathrm{L}}-1)}_{2}(\bar{{\mathbb{H}}})$	$\cdots$	$J^{(R_{\mathrm{L}}-1)}_{R_{\mathrm{L}}-1}(\bar{{\mathbb{H}}})=0$	$J^{(R_{\mathrm{L}}-1)}_{R_{\mathrm{L}}}(\bar{{\mathbb{H}}})$	$J^{(R_{\mathrm{L}}-1)}_{R_{\mathrm{L}}+1}(\bar{{\mathbb{H}}})$
$J^{(R_{\mathrm{L}})}_{1}(\bar{{\mathbb{H}}})$	$J^{(R_{\mathrm{L}})}_{2}(\bar{{\mathbb{H}}})$	$\cdots$	$J^{(R_{\mathrm{L}})}_{R_{\mathrm{L}}-1}(\bar{{\mathbb{H}}})$	$J^{(R_{\mathrm{L}})}_{R_{\mathrm{L}}}(\bar{{\mathbb{H}}})=0$	$J^{(R_{\mathrm{L}})}_{R_{\mathrm{L}}+1}(\bar{{\mathbb{H}}})$

As discussed in (135), we have

\sum_{r\in\{r^{\prime},R_{\mathrm{L}}-1,R_{\mathrm{L}}\}}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|\leq 2\ell_{r^{\prime}}+\ell_{R_{\mathrm{L}}-1}(\ell_{R_{\mathrm{L}}}+1)+\ell_{R_{\mathrm{L}}}(N-\ell_{R_{\mathrm{L}}}),

(139)

for all $r^{\prime}\in[R_{\mathrm{L}}-2]$ . Hence,

\sum_{r\in[R_{\mathrm{L}}]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|\leq 2\left(\sum_{r\in[R_{\mathrm{L}}-2]}\ell_{r}\right)+\ell_{R_{\mathrm{L}}-1}(\ell_{R_{\mathrm{L}}}+1)+\ell_{R_{\mathrm{L}}}(N-\ell_{R_{\mathrm{L}}})=\sum_{r\in[3]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}}^{\prime})|,

(140)

where $\bar{{\mathbb{H}}}^{\prime}$ is with the corresponding $J^{(r)}_{i}(\bar{{\mathbb{H}}}^{\prime})$ in Table V.

Table V:

J^{(r)}_{i}(\bar{{\mathbb{H}}}^{\prime})

for the

\bar{{\mathbb{H}}}^{\prime}

in (140).

$\sum_{r\in[R_{\mathrm{L}}-2]}\ell_{r}$	$\ell_{R_{\mathrm{L}}-1}$	$\ell_{R_{\mathrm{L}}}$	$N-\ell^{*}_{R_{\mathrm{L}}}$
$J^{(1)}_{1}(\bar{{\mathbb{H}}})=0$	$J^{(1)}_{2}(\bar{{\mathbb{H}}})=1$	$J^{(1)}_{3}(\bar{{\mathbb{H}}})=1$	$J^{(1)}_{4}(\bar{{\mathbb{H}}})=0$
$J^{(2)}_{1}(\bar{{\mathbb{H}}})=1$	$J^{(2)}_{2}(\bar{{\mathbb{H}}})=0$	$J^{(2)}_{3}(\bar{{\mathbb{H}}})=\ell_{R_{\mathrm{L}}}$	$J^{(2)}_{4}(\bar{{\mathbb{H}}})=0$
$J^{(3)}_{1}(\bar{{\mathbb{H}}})=\sum_{r\in[R_{\mathrm{L}}-2]}\ell_{r}$	$J^{(3)}_{2}(\bar{{\mathbb{H}}})=\ell_{R_{\mathrm{L}}-1}$	$J^{(3)}_{3}(\bar{{\mathbb{H}}})=0$	$J^{(3)}_{4}(\bar{{\mathbb{H}}})=N-\ell^{*}_{R_{\mathrm{L}}}$

Since

\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}\left(4,3,\bigg{(}\sum_{r\in[R_{\mathrm{L}}-2]}\ell_{r},\ell_{R_{\mathrm{L}}-1},\ell_{R_{\mathrm{L}}},N-\ell^{*}_{R_{\mathrm{L}}}\bigg{)}\right),

(141)

as proven in (138), we can find an $\bar{{\mathbb{H}}}^{\prime\prime}\in{\mathcal{H}}\Big{(}3,2,(\ell^{\prime\prime}_{1},\ell^{\prime\prime}_{2},N-\ell^{*}_{R_{\mathrm{L}}})\Big{)}$ such that

\sum_{r\in[R_{\mathrm{L}}+2]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|\leq\sum_{r\in[3]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}}^{\prime})|\leq\sum_{r\in[2]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}}^{\prime\prime})|.

(142)

∎

Next, for those remaining $R-R_{\mathrm{L}}$ rows of $\bar{{\mathbb{H}}}$ , we can have a similar lemma below.

Lemma 15.

Given $R-1\geq 3$ and $\vec{\ell}=\{\ell^{*}_{R-1},\ell_{1},\ell_{2},\ldots,\ell_{R-1}\}$ , for any $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(R,R-1,\vec{\ell}\,\right)$ , we can always construct a $\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}\left(3,2,\vec{\ell}^{\prime}=(\ell^{*}_{R-1},\ell^{\prime}_{1},\ell^{\prime}_{2}\right)$ such that

\sum_{r\in[R]\setminus[1]}\ell_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})|\leq\sum_{r\in\{2,3\}}\ell^{\prime}_{r}|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}}^{\prime})|.

(143)

Proof:

The proof is similar to the proof of Lemma 14, hence we omit it here. ∎

Combining Theorems 3, 4 and Lemmas 14, 15, we can lower bound $\bar{B}^{*}(R)$ for all $R\in[N]$ in the following theorem.

Theorem 5.

Let

\Delta_{3}\triangleq\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3}):\,\sum_{r\in[3]}\ell_{r}=N\mbox{ and }\ell_{1}\leq\ell_{2}\}}2N(N-1)-N^{2}+\ell_{1}^{2}+\ell^{2}_{2}+\ell_{3}^{2}+\ell_{1}(\ell_{3}-1)

(144)

and

\Delta_{4}\triangleq\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3},\ell_{4}):\,\sum_{r\in[4]}\ell_{r}=N\mbox{, }\ell_{1}\leq\ell_{2}\mbox{, and }\ell_{3}\leq\ell_{4}\}}2N(N-1)-\\ \ell_{1}(\ell_{2}+1)-\ell_{2}(N-\ell_{2})-\ell_{3}(\ell_{4}+1)-\ell_{4}(N-\ell_{4}),

(145)

then $\bar{B}^{*}(R)\geq\min\{\Delta_{3},\Delta_{4}\}$ for all $R\in[N]$ .

Proof:

As discussed in (98), we only need to consider the case of $R_{\mathrm{L}}\geq R-R_{\mathrm{L}}\geq 1$ .

Lemma 11 and Theorem 4 have proven $\bar{B}^{*}(1)\geq\bar{B}^{*}(2)\geq\bar{B}^{*}(3)\geq\Delta_{3}$ for the case of $R=2$ . We next consider the case of $R\geq 3$ .

For the case of $R\geq 3$ and $R-R_{\mathrm{L}}=1$ , any $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left\{R,R-1,(\ell_{1},\ell_{2},\ldots,\ell_{R})\right\}$ can be replaced by some $\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}\left\{3,2,(\ell^{\prime}_{1},\ell^{\prime}_{2},\ell_{R})\right\}$ such that $B({\mathbb{H}})\geq B({\mathbb{H}}^{\prime})$ due to Lemma 14. Therefore, $\bar{B}^{*}(R)\geq\bar{B}^{*}(3)\geq\Delta_{3}$ when $R\geq 3$ and $R-R_{\mathrm{L}}=1$ .

The remaining case is of $R\geq 3$ and $R-R_{\mathrm{L}}\geq 2$ . Since $R_{\mathrm{L}}\geq R-R_{\mathrm{L}}$ , this case is equivalent to the case of $R_{\mathrm{L}}\geq R-R_{\mathrm{L}}\geq 2$ . For the case of $R_{\mathrm{L}}\geq R-R_{\mathrm{L}}\geq 2$ , any $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left\{R,R_{\mathrm{L}},(\ell_{1},\ell_{2},\ldots,\ell_{R})\right\}$ can be replaced by some $\bar{{\mathbb{H}}}^{\prime}\in{\mathcal{H}}\left\{4,2,(\ell^{\prime}_{1},\ell^{\prime}_{2},\ell^{\prime}_{3},\ell^{\prime}_{4})\right\}$ such that $\ell^{\prime}_{1}+\ell^{\prime}_{2}=\sum_{r\in[R_{\mathrm{L}}]}\ell_{r}$ , $\ell^{\prime}_{3}+\ell^{\prime}_{4}=\sum_{r\in[R]\setminus[R_{\mathrm{L}}]}\ell_{r}$ , $B({\mathbb{H}})\geq B({\mathbb{H}}^{\prime})$ . Therefore, we can conclude that

\bar{B}^{*}(R)\geq\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3},\ell_{4}):\sum_{r\in[4]}\ell_{r}=N\}}\min_{\bar{{\mathbb{H}}}\in{\mathcal{H}}(4,2,\vec{\ell}\,)}B(\bar{{\mathbb{H}}}),

(146)

when $R_{\mathrm{L}}\geq R-R_{\mathrm{L}}\geq 2$ . We then prove that (146) can be further lower bounded by $\Delta_{4}$ .

As discussed in the beginning of this section, we swap rows or columns without effecting the repairing bandwidth of $\bar{{\mathbb{H}}}$ . Hence, WLOG, we can assume $\ell_{1}\leq\ell_{2}$ and $\ell_{3}\leq\ell_{4}$ for any $\bar{{\mathbb{H}}}\in{\mathcal{H}}\left\{4,2,(\ell_{1},\ell_{2},\ell_{3},\ell_{4})\right\}$ . By following the same trick in proving Lemma 12, we can have

	$\displaystyle\sum_{r\in[4]}\ell_{r}\|{\mathcal{J}}_{r}(\bar{{\mathbb{H}}})\|$	$\displaystyle=$	$\displaystyle\sum_{r\in[4]}\ell_{r}\left(\sum_{i\in[4]\setminus\{r\}}J^{(r)}_{i}(\bar{{\mathbb{H}}})\right)$		(147)
		$\displaystyle\leq$	$\displaystyle\ell_{1}(\ell_{2}+1)+\ell_{2}(N-\ell_{2})+\ell_{3}(\ell_{4}+1)+\ell_{4}(N-\ell_{4}),$		(148)

as illustrated in Table VI.

Table VI:

J^{(r)}_{i}(\bar{{\mathbb{H}}})

for

\bar{{\mathbb{H}}}\in{\mathcal{H}}\left(4,2,(\ell_{1},\ell_{2},\ell_{3},\ell_{4})\right)

$\ell_{1}$	$\ell_{2}$	$\ell_{3}$	$\ell_{4}$
$J^{(1)}_{1}(\bar{{\mathbb{H}}})=0$	$J^{(1)}_{2}(\bar{{\mathbb{H}}})=\ell_{2}$	$J^{(1)}_{3}(\bar{{\mathbb{H}}})=1$	$J^{(1)}_{4}(\bar{{\mathbb{H}}})=0$
$J^{(2)}_{1}(\bar{{\mathbb{H}}})=\ell_{1}$	$J^{(2)}_{2}(\bar{{\mathbb{H}}})=0$	$J^{(2)}_{3}(\bar{{\mathbb{H}}})=\ell_{3}$	$J^{(2)}_{4}(\bar{{\mathbb{H}}})=\ell_{4}$
$J^{(3)}_{1}(\bar{{\mathbb{H}}})=0$	$J^{(3)}_{2}(\bar{{\mathbb{H}}})=1$	$J^{(3)}_{3}(\bar{{\mathbb{H}}})=0$	$J^{(3)}_{4}(\bar{{\mathbb{H}}})=\ell_{4}$
$J^{(4)}_{1}(\bar{{\mathbb{H}}})=\ell_{1}$	$J^{(4)}_{2}(\bar{{\mathbb{H}}})=\ell_{2}$	$J^{(4)}_{3}(\bar{{\mathbb{H}}})=\ell_{3}$	$J^{(4)}_{4}(\bar{{\mathbb{H}}})=0$

Therefore, we can rewrite (146) as

\bar{B}^{*}(R)\geq\min_{\vec{\ell}\in\{(\ell_{1},\ell_{2},\ell_{3},\ell_{4}):\sum_{r\in[4]}\ell_{r}=N,\,\ell_{1}\leq\ell_{2},\,\ell_{3}\leq\ell_{4}\}}2N(N-1)-\\ \ell_{1}(\ell_{2}+1)-\ell_{2}(N-\ell_{2})-\ell_{3}(\ell_{4}+1)-\ell_{4}(N-\ell_{4})=\Delta_{4},

(149)

when $R_{\mathrm{L}}\geq R-R_{\mathrm{L}}\geq 2$ .

We finally conclude that $\bar{B}^{*}(R)\geq\min\{\Delta_{3},\Delta_{4}\}$ for all $R\in[N]$ . ∎

IV Numerical results

The derived lower bound, $\min\{\Delta_{3},\Delta_{4}\}$ for $4\leq N\leq 200$ , is plotted in Figure 1. The values at $N=26$ and $N=32$ are achieved by the codes provided by the department in Israel, which implies that the derived bound is so tight that it can be achieved. It should be noted that $\min\{\Delta_{3},\Delta_{4}\}=\Delta_{3}$ for all $4\leq N\leq 200$ , except when $N=4$ and $6$ .

Refer to caption — Figure 1: The lower bound $\min\{\Delta_{3},\Delta_{4}\}$ from $N=4$ to $N=200$ .

References

[1] John R. Silvester, “Determinants of Block Matrices,” The Mathematical Gazette, vol. 84, no. 501, pp. 460–467, Nov. 2000.

Lower Bound on the Optimal Access Bandwidth of (K+2,K,2K+2,K,2)-MDS Array Code with Degraded Read Friendly

I Preliminary

Definition 1.

Definition 2.

Lemma 1.

Proof:

Lemma 2.

Proof:

Definition 3.

Lemma 3.

Proof:

Example.

II Some properties

Lemma 4.

Proof:

Example.

Lemma 5.

Note.

Proof:

Lemma 6.

Note.

Proof:

Lemma 7.

Proof:

Theorem 1.

Proof:

Lemma 8.

Proof:

Lemma 9.

Proof:

Lemma 10.

Proof:

Theorem 2.

Proof:

III Lower bound on B​(PR)B(P_{R})

Theorem 3.

Proof:

Lemma 11.

Proof:

Lemma 12.

Proof:

Theorem 4.

Proof:

Lemma 13.

Proof:

Lemma 14.

Proof:

Lemma 15.

Proof:

Theorem 5.

Proof:

IV Numerical results

References

Lower Bound on the Optimal Access Bandwidth of ( $K+2,K,2$ )-MDS Array Code with Degraded Read Friendly

III Lower bound on $B(P_{R})$