$\mathcal{S}^{\prime}$. For $\Omega=[m]$, (33) also holds for
$\mathcal{S}^{\prime}$ since

$\displaystyle\dim\left(\bigcap_{i\in[m]}V^{\prime}_{i}\right)+\sum_{i\in[m]}r_{i}\leq\dim\left(\bigcap_{i\in[m]}V_{i}\right)+\sum_{i\in[m]}r_{i}$
$\displaystyle\leq\max_{i\in[m]}\\{\dim(V_{i})+r_{i}\\}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:attain-
max}}}{{=}}\dim(V_{m})+r_{m}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:unchanged}}}{{=}}\dim(V_{m}^{\prime})+r_{m}$
$\displaystyle\leq\max_{i\in[m]}\\{\dim(V_{i}^{\prime})+r_{i}\\},$

where the second inequality holds since $\mathcal{S}$ satisfies (33) with
$\Omega=[m]$. Finally, for nonempty $\Omega\subseteq[m]$ with
$2\leq|\Omega|\leq m-1$, (33) holds for $\mathcal{S}^{\prime}$ since

$\displaystyle\dim\left(\bigcap_{i\in\Omega}V^{\prime}_{i}\right)+\sum_{i\in\Omega}r_{i}\leq\dim\left(\bigcap_{i\in\Omega}V_{i}\right)+\sum_{i\in\Omega}r_{i}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:non-
tightness-2}}}{{\leq}}\max_{i\in\Omega}\\{\dim(V_{i})+r_{i}\\}-1$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:decrease-by-
one}}}{{\leq}}\max_{i\in[m]}\\{\dim(V_{i}^{\prime})+r_{i}\\}.$

By (52) and the definitions (30), (31), and (32), it holds that
$\det(M_{\mathcal{S}^{\prime}})=\det(M_{\mathcal{S}})|_{Z_{d}=0}$. And by the
induction hypothesis, we have $\det(M_{\mathcal{S}^{\prime}})\neq 0$. So
$\det(M_{\mathcal{S}})\neq 0$. ∎

##  5 Equivalence Between $\operatorname{GKP}(\ell)$ and
$\operatorname{MRD}(\ell)$

In this section, we will establish the equivalence between
$\operatorname{GKP}(\ell)$ and $\operatorname{MRD}(\ell)$ over a general field
$\mathbb{F}/\mathbb{F}_{q}$. As a key technical ingredient, we will prove our
formula for generic intersection dimension (see 1.16), generalizing a similar
formula established in [BGM23]. Our proofs closely follow the arguments of
Brakensiek–Gopi–Makam [BGM23].

###  5.1 Formula for Generic Intersection Dimension: Proof of 1.16

We first show that a generic linear code attains all generic kernel patterns.
Our proof of this fact follows closely that of [BGM23, Proposition 2.1].

######  Proposition 5.1.

Let $W$ be the $k\times n$ matrix $(Z_{i,j})_{i\in[k],j\in[n]}$ over
$\mathbb{F}:=\mathbb{F}_{q}(Z_{1,1},\dots,Z_{k,n})$. Let
$C\subseteq\mathbb{F}^{n}$ be the $[n,k]_{\mathbb{F}}$ code with $W$ as a
generator matrix. Then $C$ attains any generic kernel pattern. In other words,
$C$ is $\mathrm{GKP}(\ell)$ for all $\ell\geq 1$.

###### Proof.

Let $(V_{1},\dots,V_{k})$ be a generic kernel pattern, i.e.,
$\dim\left(\bigcap_{i\in\Omega}V_{i}\right)\leq k-|\Omega|$ for any nonempty
$\Omega\subseteq[k]$. By 3.2, we can find
$V^{\prime}_{i}\subseteq\mathbb{F}_{q}^{n}$ of dimension $k-1$ such that
$V_{i}\subseteq V^{\prime}_{i}$ and
$\dim(\bigcap_{i\in\Omega}V^{\prime}_{i})\leq k-|\Omega|$ for any nonempty
$\Omega\subseteq[k]$. By replacing $V_{i}$ with $V_{i}^{\prime}$, we may
assume that $\dim V_{i}=k-1$.

For $i\in[k]$, choose $A_{i}\in\mathbb{F}_{q}^{n\times\dim V_{i}}$ such that
$V_{i}=\langle A_{i}\rangle$. We want to show that there exists an invertible
matrix $M\in\mathbb{F}^{k\times k}$, whose $i$-th row is denoted by
$\bm{m}_{i}$, such that $\bm{m}_{i}WA_{i}=0$ for $i\in[k]$. Consider arbitrary
$i\in[k]$. Since $\dim V_{i}=k-1$, we have $\mathrm{rank}(WA_{i})=k-1$ by
Lemma 2.15. This implies that the solution space $\bm{m}_{i}$ is one-
dimensional, i.e., $\bm{m}_{i}$ is uniquely determined up to scaling. In fact,
we may write each entry of $\bm{m}_{i}$ as a polynomial in the entries of a
nonsingular maximal minor $P$ of $WA_{i}$ by applying Cramer’s rule and
clearing the common denominator $\det(P)$. As this holds for all $i\in[k]$,
$\det(M)$ may be expressed as a polynomial in $Z_{ij}$. By the GM-MRD theorem
(4.1), $\det(M)$ is nonzero even after assigning $Z_{i}^{q^{j-1}}$ to
$Z_{i,j}$ for $i\in[k]$ and $j\in[n]$. So $\det(M)\neq 0$, i.e., $M$ is
invertible.101010One can also prove $\det(M)\neq 0$ directly by adapting the
proof of 4.1, avoiding the use of symbolic Gabidulin codes. ∎

Next, we establish an intersection dimension formula for GKP($\ell$) codes.
Recall that for a $k\times n$ matrix $G$ over a field
$\mathbb{F}/\mathbb{F}_{q}$ and an $n\times\ell$ matrix $A$ over
$\mathbb{F}_{q}$, we denote $G_{A}:=GA\in\mathbb{F}^{k\times\ell}$ and define
$G_{V}\subseteq\mathbb{F}^{k}$ to be the column span of $G_{A}$.

######  Theorem 5.2.

Let $\mathbb{F}$ be an extension field of $\mathbb{F}_{q}$. Let $C$ be an
$[n,k]_{\mathbb{F}}$ code that is $\operatorname{GKP}(\ell)$ with a generator
matrix $G\in\mathbb{F}^{k\times n}$. Let $V_{1},\dots,V_{\ell}$ be subspaces
of $\mathbb{F}_{q}^{n}$, each of dimension at most $k$. Then

$\dim_{\mathbb{F}}\left(\bigcap_{i\in[\ell]}G_{V_{i}}\right)=\max_{P_{1}\sqcup
P_{2}\sqcup\cdots\sqcup
P_{s}=[\ell]}\left(\sum_{i\in[s]}\dim_{\mathbb{F}_{q}}\left(\bigcap_{j\in
P_{i}}V_{j}\right)-(s-1)k\right).$ (53)

###### Proof.

We first prove that LHS $\geq$ RHS. For any nonempty $S\subseteq[\ell]$, it
holds that $G_{V_{S}}\subseteq\bigcap_{i\in S}G_{V_{i}}$, where
$V_{S}:=\bigcap_{i\in S}V_{i}$. Therefore, for any partition $P_{1}\sqcup
P_{2}\sqcup\cdots\sqcup P_{s}=[\ell]$, we have

$\bigcap_{i\in[s]}G_{V_{P_{i}}}\subseteq\bigcap_{i\in[\ell]}G_{V_{i}}.$ (54)

Also note that by Lemma 2.1,

$\bigcap_{i\in[s]}G_{V_{P_{i}}}=\left(\sum_{i\in[s]}G_{V_{P_{i}}}^{\perp}\right)^{\perp}.$
(55)

As $C$ is a $\operatorname{GKP}(\ell)$ code, $C$ is a MRD code. So we have
$\dim G_{V_{P_{i}}}=\dim V_{P_{i}}$ for $i\in[s]$. It follows that

$\displaystyle\dim\left(\bigcap_{i\in[\ell]}G_{V_{i}}\right)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:containment}}}{{\geq}}\dim\left(\bigcap_{i\in[s]}G_{V_{P_{i}}}\right)\stackrel{{\scriptstyle\eqref{eq:intersection-
sum}}}{{=}}k-\dim\left(\sum_{i\in[s]}G_{V_{P_{i}}}^{\perp}\right)\geq
k-\sum_{i\in[s]}\dim\left(G_{V_{P_{i}}}^{\perp}\right)$
$\displaystyle=k-\sum_{i\in[s]}\left(k-\dim
V_{P_{i}}\right)=\sum_{i\in[s]}\dim\left(\bigcap_{j\in
P_{i}}V_{j}\right)-(s-1)k.$

So LHS $\geq$ RHS in (53).

Next, we prove that LHS $\leq$ RHS in (53). Denote RHS of (53) by $d$, i.e.,

$d=\max_{P_{1}\sqcup P_{2}\sqcup\cdots\sqcup
P_{s}=[\ell]}\left(\sum_{i\in[s]}\dim_{\mathbb{F}_{q}}\left(\bigcap_{j\in
P_{i}}V_{j}\right)-(s-1)k\right).$

Then for all partitions $P_{1}\sqcup P_{2}\sqcup\cdots\sqcup P_{s}=[\ell]$,

$\sum_{i=1}^{s}\dim\left(\bigcap_{j\in P_{i}}V_{j}\right)\leq(s-1)k+d.$

Thus, by Lemma 3.6, there exist integers $\delta_{1},\ldots,\delta_{\ell}\geq
0$ such that $\sum_{i=1}^{\ell}\delta_{i}=k-d$ and for all nonempty
$\Omega\subseteq[\ell]$,

$\dim\left(\bigcap_{i\in\Omega}V_{i}\right)\leq
k-\sum_{i\in\Omega}\delta_{i}.$

By 3.3, we know that the pattern $(T_{1},\dots,T_{k})$ with $\delta_{i}$
copies of $V_{i}$ for $i\in[\ell]$ and $d$ additional copies of $\\{0\\}$ is a
generic kernel pattern. Without loss of generality, assume the $d$ copies of
$\\{0\\}$ are $T_{k-d+1},\cdots,T_{k}$. For $i\in[k]$, choose
$A_{i}\in\mathbb{F}_{q}^{n\times\dim T_{i}}$ such that $T_{i}=\langle
A_{i}\rangle$.

As $C$ is $\operatorname{GKP}(\ell)$, there exists an invertible matrix
$M\in\mathbb{F}^{k\times k}$ such that $\bm{m}_{i}GA_{i}=0$ for all $i\in[k]$,
where $\bm{m}_{i}$ denotes the $i$-th row of $M$. As $M$ is invertible, we
have
$\dim_{\mathbb{F}}\left(\bigcap_{i\in[\ell]}(MG)_{V_{i}}\right)=\dim_{\mathbb{F}}\left(\bigcap_{i\in[\ell]}G_{V_{i}}\right)$.
Thus, to prove that LHS $\leq$ RHS in (53), i.e.,
$\dim_{\mathbb{F}}\left(\bigcap_{i\in[\ell]}G_{V_{i}}\right)\leq d$, it
suffices to show that for any $z\in\bigcap_{i\in[\ell]}(MG)_{V_{i}}$, only the
last $d$ coordinates of $z$ could be nonzero.

Let $z=(z_{1},\dots,z_{k})\in\bigcap_{i\in[\ell]}(MG)_{V_{i}}$. Consider
arbitrary $i\in[k-d]$. Then $T_{i}$ is a copy of $V_{j}$ for some
$j\in[\ell]$. As $z\in(MG)_{V_{j}}=(MG)_{T_{i}}$, we have
$z_{i}=\bm{m}_{i}GA_{i}\bm{u}$ for some $\bm{u}\in\mathbb{F}^{\dim V_{i}}$.
But $\bm{m}_{i}GA_{i}=0$. So $z_{i}=0$. This proves the claim that only the
last $d$ coordinates of $z$ could be nonzero, thereby completing the proof. ∎

Combining 5.1 and 5.2 yields our formula for generic intersection dimension.

######  Corollary 5.3 (1.16, restated).

Let $W$ be the $k\times n$ matrix $(Z_{i,j})_{i\in[k],j\in[n]}$ over
$\mathbb{F}:=\mathbb{F}_{q}(Z_{1,1},\dots,Z_{k,n})$. Let
$V_{1},\dots,V_{\ell}$ be subspaces of $\mathbb{F}_{q}^{n}$, each of dimension
at most $k$. Then

$\dim_{\mathbb{F}}\left(\bigcap_{i\in[\ell]}W_{V_{i}}\right)=\max_{P_{1}\sqcup
P_{2}\sqcup\cdots\sqcup
P_{s}=[\ell]}\left(\sum_{i\in[s]}\dim_{\mathbb{F}_{q}}\left(\bigcap_{j\in
P_{i}}V_{j}\right)-(s-1)k\right).$ (56)

###  5.2 Proof of 1.13

We are now ready to prove 1.13, which establishes an equivalence between
$\mathrm{GKP}(\ell)$ and $\mathrm{MRD}(\ell)$. The direction from
$\mathrm{GKP}(\ell)$ to $\mathrm{MRD}(\ell)$ follows easily from our
intersection dimension formula for $\mathrm{GKP}(\ell)$ codes.

######  Theorem 5.4.

Let $C$ be an $[n,k]_{\mathbb{F}}$ code over a field
$\mathbb{F}/\mathbb{F}_{q}$. If $C$ is $\mathrm{GKP}(\ell)$, then it is
$\mathrm{MRD}(\ell)$.

###### Proof.

By 5.1, the “generic” linear code over $\mathbb{F}_{q}(Z_{1,1},\dots,Z_{k,n})$
defined by the generator matrix $W=(Z_{i,j})_{i\in[k],j\in[n]}$ is also
$\mathrm{GKP}(\ell)$. Let $G\in\mathbb{F}^{k\times n}$ be a generator matrix
of $C$. By 5.2, for any subspaces
$V_{1},\dots,V_{\ell}\subseteq\mathbb{F}_{q}^{n}$ of dimension at most $k$, we
have
$\dim(\bigcap_{i\in[\ell]}G_{V_{i}})=\dim(\bigcap_{i\in[\ell]}W_{V_{i}})$,
i.e., $C$ is $\mathrm{MRD}(\ell)$. ∎

Next, we prove the implication from $\mathrm{MRD}(\ell)$ to
$\mathrm{GKP}(\ell)$.

######  Theorem 5.5.

Let $\mathbb{F}$ be an extension field of $\mathbb{F}_{q}$. Let $C$ be an
$[n,k]_{\mathbb{F}}$ code that is $\operatorname{MRD}(\ell)$. Let
$\mathcal{T}=(T_{1},\dots,T_{k})$ be a generic kernel pattern of order at most
$\ell$. Then $C$ attains $\mathcal{T}$.

###### Proof.

By 3.5, by padding the subspaces if necessary, we may assume that there exist
subspaces $V_{1},\dots,V_{\ell}\subseteq\mathbb{F}_{q}^{n}$ and integers
$\delta_{1},\dots,\delta_{\ell}\geq 0$ such that $\dim V_{i}=k-\delta_{i}$ for
all $i\in[\ell]$ and $\mathcal{T}$ consists of $\delta_{i}$ copies of $V_{i}$
and $d:=k-\sum_{i\in[\ell]}\delta_{i}$ additional copies of $\\{0\\}$.

Since $\mathcal{T}$ is a generic kernel pattern of order at most $\ell$, by
Lemma 3.6, for all partitions $P_{1}\sqcup P_{2}\sqcup\dots\sqcup
P_{s}=[\ell]$,

$\sum_{i\in[s]}\dim\left(\bigcap_{j\in P_{i}}V_{j}\right)\leq(s-1)k+d.$ (57)

For the finest partition $\\{1\\}\sqcup\\{2\\}\sqcup\cdots\sqcup\\{\ell\\},$
the above inequality is indeed an equality because $\dim(V_{i})=k-\delta_{i}$
and thus

$\sum_{i=1}^{\ell}\dim(V_{i})=sk-\sum_{i=1}^{\ell}\delta_{i}=(s-1)k+d.$ (58)

Let $G\in\mathbb{F}^{k\times n}$ be a generator matrix of $C$. Then we have

$\dim\left(\bigcap_{i=1}^{\ell}G_{V_{i}}\right)=\dim\left(\bigcap_{i=1}^{\ell}W_{V_{i}}\right)=\sum_{i=1}^{\ell}\dim(V_{i})-(s-1)k=d,$

where the first equality holds since $C$ is $\mathrm{MRD}(\ell)$ and the other
equalities follow from 1.16, (57), and (58). So by Lemma 2.1,

$\dim\left(\sum_{i=1}^{\ell}G_{V_{i}}^{\perp}\right)=k-\dim\left(\bigcap_{i=1}^{\ell}G_{V_{i}}\right)=k-d.$
(59)

On the other hand, as $C$ is MRD, we have $\dim G_{V_{i}}^{\perp}=k-\dim
G_{V_{i}}=k-\dim V_{i}=\delta_{i}$ for $i\in[\ell]$. Therefore

$k-d\stackrel{{\scriptstyle\eqref{eq:dim-of-
sum}}}{{=}}\dim\left(\sum_{i=1}^{\ell}G_{V_{i}}^{\perp}\right)\leq\sum_{i=1}^{\ell}\dim\left(G_{V_{i}}^{\perp}\right)=\sum_{i=1}^{\ell}\delta_{i}=k-d$

which implies that
$\dim\left(\sum_{i=1}^{\ell}G_{V_{i}}^{\perp}\right)=\sum_{i=1}^{\ell}\dim\left(G_{V_{i}}^{\perp}\right)$.
Therefore,, the sum $\sum_{i=1}^{\ell}G_{V_{i}}^{\perp}$ is a direct sum,
i.e.,
$\sum_{i=1}^{\ell}G_{V_{i}}^{\perp}=\bigoplus_{i=1}^{\ell}G_{V_{i}}^{\perp}$.
For $i\in[\ell]$, let
$B_{i}=\left\\{\bm{v}_{1}^{(i)},\dots,\bm{v}_{\delta_{i}}^{(i)}\right\\}$ be a
basis of $G_{V_{i}}^{\perp}$. Then $B:=B_{1}\sqcup\dots\sqcup B_{\ell}$ is a
basis of $\sum_{i=1}^{\ell}G_{V_{i}}^{\perp}$. Extending $B$ to a basis
$B^{\prime}=B\sqcup\\{\bm{u}_{1},\dots,\bm{u}_{d}\\}$ of $\mathbb{F}^{k}$.

To show that $C$ attains $\mathcal{T}$, by permuting the subspaces $T_{i}$, we
may assume that

$\mathcal{T}=\left(V_{1}^{(1)},\dots,V_{1}^{(\delta_{1})},\dots,V_{\ell}^{(1)},\dots,V_{\ell}^{(\delta_{\ell})},\\{0\\},\dots\\{0\\}\right),$

where $V_{i}^{(j)}$ denotes the $j$-th copy of $V_{i}$. Choose
$M\in\mathbb{F}^{k\times k}$ such that the first $k-d$ columns of $M^{T}$ are
$\bm{v}^{(i)}_{j}$ for $i\in[\ell]$ and $j\in[\delta_{i}]$ (given in
lexicographic order of $(i,j)$), and the last $d$ columns of $M^{T}$ are
$\bm{u}_{1},\dots,\bm{u}_{d}$. Then $M$ is invertible since the columns of
$M^{T}$ form a basis of $\mathbb{F}^{k}$. By the construction of $M$, we have
$\bm{m}_{i}GA_{i}=0$ for all $i\in[k]$, where $\bm{m}_{i}$ denotes the $i$-th
row of $M$ and $A_{i}\in\mathbb{F}_{q}^{n\times\dim T_{i}}$ satisfies
$T_{i}=\langle A_{i}\rangle$. Therefore, $C$ attains the kernel pattern
$\mathcal{T}$. ∎

######  Corollary 5.6.

Let $C$ be an $[n,k]_{\mathbb{F}}$ code over a field
$\mathbb{F}/\mathbb{F}_{q}$. If $C$ is $\mathrm{MRD}(\ell)$, then it is
$\mathrm{GKP}(\ell)$.

Combining 5.4 and 5.6 proves 1.13.

##  6 Equivalence Between $\operatorname{MRD}(\ell)$ and $\operatorname{LD-
MRD}({\leq}\,\ell-1)$ up to Duality

In this section, we will show that a linear code $C$ is
$\operatorname{MRD}(\ell)$ if and only if its dual code is $\operatorname{LD-
MRD}({\leq}\,\ell-1)$. Our proofs resemble the arguments of
Brakensiek–Gopi–Makam [BGM23].

###  6.1 An Alternative Characterization of $\operatorname{MRD}(\ell)$ Over
$\mathbb{F}/\mathbb{F}_{q}$

We start by providing an alternative characterization of
$\operatorname{MRD}(\ell)$ codes. First, we need the following lemma.

######  Lemma 6.1.

Let $\mathbb{F}$ be an extension field of $\mathbb{F}_{q}$ and let
$G\in\mathbb{F}^{k\times n}$. For $i=1,\dots,\ell$, let $V_{i}$ be a subspace
of $\mathbb{F}_{q}^{n}$ and let $A_{i}\in\mathbb{F}_{q}^{n\times\dim V_{i}}$
such that $V_{i}=\langle A_{i}\rangle$. Then

$\dim\left(\bigcap_{i\in[\ell]}G_{V_{i}}\right)=\sum_{i\in[\ell]}\dim
G_{V_{i}}-\mathrm{rank}\left(G_{\\{A_{i}\\}_{i\in[\ell]}}\right),$ (60)

where we define the matrix
$G_{\\{A_{i}\\}_{i\in[\ell]}}:=\begin{pmatrix}GA_{1}&GA_{2}&&&\\\
GA_{1}&&GA_{3}&&\\\ \vdots&&&\ddots&\\\ GA_{1}&&&&GA_{\ell}\end{pmatrix}$.

###### Proof.

If $\dim G_{V_{i}}<\dim V_{i}$ for some $i\in[\ell]$, then some columns of
$GA_{i}$ are linear combinations of other columns. In this case, we may delete
the corresponding columns from $A_{i}$ and update $V_{i}$ correspondingly
without affecting the two sides of (60). By repeatedly performing the
deletions, we may assume that $\dim G_{V_{i}}=\dim V_{i}$ for all
$i\in[\ell]$. In particular, RHS of (60) equals the dimension of

$U:=\left\\{\bm{u}\in\mathbb{F}^{\sum_{i\in[\ell]}\dim V_{i}}\text{ such that
}G_{\\{A_{i}\\}_{i\in[\ell]}}\cdot\bm{u}=0\right\\}$

since the number of linearly independent constraints in the linear system
$G_{\\{A_{i}\\}_{i\in[\ell]}}\cdot\bm{u}=0$ is exactly
$\mathrm{rank}\left(G_{\\{A_{i}\\}_{i\in[\ell]}}\right)$.

It remains to find an isomorphism $\sigma:U\to\bigcap_{i\in[\ell]}G_{V_{i}}$
between the two $\mathbb{F}$-linear spaces. Let $\sigma$ send
$\bm{u}=(\bm{u}_{1},\bm{u}_{2},\dots,\bm{u}_{\ell})\in U$ with
$\bm{u}_{i}\in\mathbb{F}^{\dim V_{i}}$ to $-GA_{1}\cdot\bm{u}_{1}$. Note that
the definition of $U$ and that of $G_{\\{A_{i}\\}_{i\in[\ell]}}$ imply that
for $\bm{u}=(\bm{u}_{1},\bm{u}_{2},\dots,\bm{u}_{\ell})\in U$,

$-GA_{1}\cdot\bm{u}_{1}=GA_{2}\cdot\bm{u}_{2}=\cdots=GA_{\ell}\cdot\bm{u}_{\ell}.$
(61)

It follows that the image of $U$ under $\sigma$ is indeed contained in
$\bigcap_{i\in[\ell]}G_{V_{i}}$. Moreover, for any
$\bm{y}\in\bigcap_{i\in[\ell]}G_{V_{i}}$, by the fact that $\dim
G_{V_{i}}=\dim V_{i}$ for $i\in[\ell]$, there exists unique
$(\bm{u}_{1},\dots,\bm{u}_{\ell})$ with $\bm{u}_{i}\in\mathbb{F}^{\dim V_{i}}$
such that
$\bm{y}=-GA_{1}\cdot\bm{u}_{1}=GA_{2}\cdot\bm{u}_{2}=\cdots=GA_{\ell}\cdot\bm{u}_{\ell}$.
The map $\bm{y}\mapsto(\bm{u}_{1},\dots,\bm{u}_{\ell})$ is then the inverse of
$\sigma$. ∎

######  Corollary 6.2.

Let $\mathbb{F}$ be an extension field of $\mathbb{F}_{q}$. Let $C$ be an
$[n,k]_{\mathbb{F}}$ code that is $\operatorname{MRD}$, and let
$G\in\mathbb{F}^{k\times n}$ be a generator matrix of $C$. Let
$V_{1},V_{2},\dots,V_{\ell}$ be subspaces of $\mathbb{F}_{q}^{n}$, each of
dimension at most $k$. Then
$\dim\left(\bigcap_{i=1}^{\ell}G_{V_{i}}\right)\geq\dim\left(\bigcap_{i=1}^{\ell}W_{V_{i}}\right)$,
where
$W=(Z_{i,j})_{i\in[k],j\in[n]}\in\mathbb{F}(Z_{1,1},\dots,Z_{k,n})^{k\times
n}$.

###### Proof.

For $i\in[\ell]$, choose $A_{i}\in\mathbb{F}_{q}^{n\times\dim V_{i}}$ such
that $V_{i}=\langle A_{i}\rangle$. As $C$ is MRD, we have $\dim G_{V_{i}}=\dim
V_{i}$ for $i\in[\ell]$. By Lemma 2.15, we also have $\dim W_{V_{i}}=\dim
V_{i}$ for $i\in[\ell]$. Then by Lemma 6.1, it suffices to prove that
$\operatorname{rank}\left(G_{\\{A_{i}\\}_{i\in[\ell]}}\right)<\operatorname{rank}\left(W_{\\{A_{i}\\}_{i\in[\ell]}}\right)$.
This holds since $G_{\\{A_{i}\\}_{i\in[\ell]}}$ can be obtained from
$W_{\\{A_{i}\\}_{i\in[\ell]}}$ by assigning the $(i,j)$-th entry of $G$ to
$Z_{i,j}$ for $(i,j)\in[k]\times[n]$. Such an assignment does not increase the
rank since if the determinant of a submatrix is nonzero after the assignment,
then it must have been nonzero before the assignment. ∎

The next lemma presents an alternative characterization of
$\operatorname{MRD}(\ell)$ codes, which appears weaker but is, in fact,
equivalent.

######  Lemma 6.3.

Let $\mathbb{F}$ be an extension field of $\mathbb{F}_{q}$. Let $C$ be an
$[n,k]_{\mathbb{F}}$ code with a generator matrix $G\in\mathbb{F}^{k\times
n}$. Let $\ell\geq 1$. Then the following are equivalent:

  1. 1.

$C$ is $\operatorname{MRD}(\ell)$.

  2. 2.

$C$ is $\operatorname{MRD}$ and for all subspaces
$V_{1},V_{2},\dots,V_{\ell}\subseteq\mathbb{F}_{q}^{n}$, each of dimension at
most $k$, we have $\bigcap_{i=1}^{\ell}G_{V_{i}}=0$ iff
$\bigcap_{i=1}^{\ell}W_{V_{i}}=0$, where
$W=(Z_{i,j})_{i\in[k],j\in[n]}\in\mathbb{F}(Z_{1,1},\dots,Z_{k,n})^{k\times
n}$.

###### Proof.

Item 1 implies Item 2 by the definition of $\operatorname{MRD}(\ell)$.
Conversely, we show that if Item 1 does not hold, neither does Item 2.

Suppose $C$ is not $\operatorname{MRD}(\ell)$. Further assume that $C$ is MRD
since otherwise Item 2 certainly does not hold. Then there exist subspaces
$V_{1},V_{2},\dots,V_{\ell}\subseteq\mathbb{F}_{q}^{n}$, each of dimension at
most $k$, such that
$\dim\left(\bigcap_{i=1}^{\ell}G_{V_{i}}\right)\neq\dim\left(\bigcap_{i=1}^{\ell}W_{V_{i}}\right)$.
By 6.2, we have

$\dim\left(\bigcap_{i=1}^{\ell}G_{V_{i}}\right)>d:=\dim\left(\bigcap_{i=1}^{\ell}W_{V_{i}}\right).$
(62)

We know by Lemma 6.1 and Lemma 2.15 that

$\operatorname{rank}\left(W_{\\{A_{i}\\}_{i\in[\ell]}}\right)=\left(\sum_{i\in[\ell]}\dim
W_{V_{i}}\right)-d=\left(\sum_{i\in[\ell]}\dim V_{i}\right)-d.$ (63)

This means we can find $d$ columns $\mathbf{c}_{1},\dots,\mathbf{c}_{d}$ of
$W_{\\{A_{i}\\}_{i\in[\ell]}}$ that can be written as linear combinations of
the remaining columns. Remove the corresponding $d$ columns in total from
$A_{1},\dots,A_{\ell}$ to obtain $A_{1}^{\prime},\dots,A_{\ell}^{\prime}$,
where $A_{i}^{\prime}$ is a submatrix of $A_{i}$, so that
$W_{\\{A_{i}^{\prime}\\}_{i\in[\ell]}}$ is obtained from
$W_{\\{A_{i}\\}_{i\in[\ell]}}$ by removing
$\mathbf{c}_{1},\dots,\mathbf{c}_{d}$. Then
$\mathrm{rank}\left(W_{\\{A_{i}^{\prime}\\}_{i\in[\ell]}}\right)=\mathrm{rank}\left(W_{\\{A_{i}\\}_{i\in[\ell]}}\right)$.
Let $V_{i}^{\prime}=\langle A_{i}^{\prime}\rangle$ for $i\in[\ell]$. Then
$\sum_{i\in[\ell]}\dim V_{i}^{\prime}=\left(\sum_{i\in[\ell]}\dim
V_{i}\right)-d$. By Lemma 6.1 and Lemma 2.15, we have

$\displaystyle\dim\left(\bigcap_{i=1}^{\ell}W_{V_{i}^{\prime}}\right)$
$\displaystyle=\left(\sum_{i\in[\ell]}\dim
W_{V_{i}^{\prime}}\right)-\mathrm{rank}\left(W_{\\{A_{i}^{\prime}\\}_{i\in[\ell]}}\right)=\left(\sum_{i\in[\ell]}\dim
V_{i}^{\prime}\right)-\mathrm{rank}\left(W_{\\{A_{i}^{\prime}\\}_{i\in[\ell]}}\right)$
$\displaystyle=\left(\sum_{i\in[\ell]}\dim
V_{i}\right)-d-\mathrm{rank}\left(W_{\\{A_{i}\\}_{i\in[\ell]}}\right)\stackrel{{\scriptstyle\eqref{ffo1}}}{{=}}0.$

On the other hand, removing a column from $A_{i}$ decreases $\dim
V_{i}=\dim\langle A_{i}\rangle$ by one and decreases
$\mathrm{rank}\left(G_{\\{A_{i}\\}_{i\in[\ell]}}\right)$ by zero or one. It
follows from (60) that

$\dim\left(\bigcap_{i\in[\ell]}G_{V_{i}^{\prime}}\right)\geq\dim\left(\bigcap_{i\in[\ell]}G_{V_{i}}\right)-d\stackrel{{\scriptstyle\eqref{eq:intersection-
lower-bound}}}{{>}}d-d=0.$

So Item 2 does not hold. ∎

###  6.2 Proof of Theorem 1.14

We now prove 1.14, which establishes an equivalence between $\operatorname{LD-
MRD}({\leq}\,\ell)$ and $\mathrm{MRD}(\ell+1)$ up to duality. First, we show
the implication from $\operatorname{LD-MRD}({\leq}\,\ell)$ to
$\mathrm{MRD}(\ell+1)$.

######  Theorem 6.4.

Let $C$ be an $[n,k]_{\mathbb{F}}$ code over a field
$\mathbb{F}/\mathbb{F}_{q}$. If $C$ is $\operatorname{LD-MRD}({\leq}\,\ell)$,
then $C^{\perp}$ is $\operatorname{MRD}(\ell+1)$.

###### Proof.

We will prove the contrapositive: If $C^{\perp}$ is not MRD$(\ell+1)$, then
$C$ is not $\operatorname{LD-MRD}({\leq}\,\ell)$. If $C^{\perp}$ is not MRD,
then $C$ is not either by Lemma 2.11. This, in turn, implies that $C$ is not
$\operatorname{LD-MRD}({\leq}\,\ell)$, since all $\operatorname{LD-
MRD}({\leq}\,\ell)$ codes are MRD. So we may assume that $C^{\perp}$ is MRD.

Let $H\in\mathbb{F}^{(n-k)\times n}$ be a parity check matrix of $C$. Then $H$
is also a generator matrix of $C^{\perp}$. By 6.2 and Lemma 6.3, the fact that
$C^{\perp}$ is MRD but not MRD$(\ell+1)$ implies that there exist subspaces
$V_{0},\ldots,V_{\ell}\subseteq\mathbb{F}_{q}^{n}$, each of dimension at most
$n-k$, such that

$\bigcap_{0\leq
i\leq\ell}H_{V_{i}}\neq\\{0\\}\quad\text{and}\quad\bigcap_{0\leq
i\leq\ell}W_{V_{i}}=\\{0\\}.$ (64)

For all partitions $P_{1}\sqcup P_{2}\sqcup\cdots\sqcup
P_{s}=\\{0,1,\ldots,\ell\\}$, by 1.16 and the fact that $\bigcap_{0\leq
i\leq\ell}W_{V_{i}}=\\{0\\}$, we have

$\sum_{i=1}^{s}\dim V_{P_{i}}\leq(s-1)(n-k),$ (65)

where $V_{P_{i}}=\bigcap_{j\in P_{i}}V_{j}$. Fix nonzero
$\bm{y}\in\bigcap_{0\leq i\leq\ell}H_{V_{i}}$, which is possible as
$\bigcap_{0\leq i\leq\ell}H_{V_{i}}\neq\\{0\\}$.

For $0\leq i\leq\ell$, choose $A_{i}\in\mathbb{F}_{q}^{n\times\dim V_{i}}$
such that $V_{i}=\langle A_{i}\rangle$. As $\bm{y}\in\bigcap_{0\leq
i\leq\ell}H_{V_{i}}$, there exist vectors $\bm{u}_{0},\dots,\bm{u}_{\ell}$
with $\bm{u}_{i}\in\mathbb{F}^{\dim V_{i}}$ such that
$HA_{0}\bm{u}_{0}=HA_{1}\bm{u}_{1}=\cdots=HA_{\ell}\bm{u}_{\ell}=\bm{y}$. Let
$\bm{v}_{i}=A_{i}\bm{u}_{i}$ for $0\leq i\leq\ell$, so that
$H\bm{v}_{i}=\bm{y}$. Define the partition $P_{1}\sqcup
P_{2}\sqcup\cdots\sqcup P_{s}=\\{0,1,\ldots,\ell\\}$ such that
$j,j^{\prime}\in\\{0,1,\dots,\ell\\}$ are in the same set of the partition iff
$\bm{v}_{j}=\bm{v}_{j^{\prime}}$. For $i\in[s]$, let
$\bm{v}_{P_{i}}:=\bm{v}_{j}$ for any $j\in P_{i}$. Then we have $s$ distinct
vectors $\bm{v}_{P_{1}},\dots,\bm{v}_{P_{s}}\in\mathbb{F}^{n}$.

Consider arbitrary $i\in[s]$. For all $j\in P_{i}$ and $\bm{x}\in
V_{j}^{\perp}\subseteq\mathbb{F}_{q}^{n}$, we have
$\bm{x}^{T}\bm{v}_{P_{i}}=\bm{x}^{T}\bm{v}_{j}=\bm{x}^{T}A_{j}\bm{u}_{j}=0$
since $\bm{x}^{T}A_{j}=0$. As this holds for all $j\in P_{i}$ and $\bm{x}\in
V_{j}^{\perp}$, we have that $\bm{x}^{T}\bm{v}_{P_{i}}=0$ for all
$\bm{x}\in\sum_{j\in P_{i}}V_{j}^{\perp}=\left(\bigcap_{j\in
P_{i}}V_{j}\right)^{\perp}=V_{P_{i}}^{\perp}$. (Here $\sum_{j\in
P_{i}}V_{j}^{\perp}=\left(\bigcap_{j\in P_{i}}V_{j}\right)^{\perp}$ holds by
Lemma 2.1.) So
$V_{P_{i}}^{\perp}\subseteq\ker_{\mathbb{F}_{q}}(\bm{v}_{P_{i}})$. It follows
by Lemma 2.7 that

$\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{P_{i}})=n-\dim(\ker_{\mathbb{F}_{q}}(\bm{v}_{P_{i}}))\leq
n-\dim\left(V_{P_{i}}^{\perp}\right)=\dim V_{P_{i}}.$ (66)

So we have

$\sum_{i=1}^{s}\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{P_{i}})\stackrel{{\scriptstyle\eqref{eq:rank-
bound-0}}}{{\leq}}\sum_{i=1}^{s}\dim
V_{P_{i}}\stackrel{{\scriptstyle\eqref{eq:sum-vpi}}}{{\leq}}(s-1)(n-k).$ (67)

For $i\in[s]$, let $\bm{z}_{i}=\bm{v}_{P_{i}}-\bm{v}_{P_{1}}$. Note that
$H\bm{z}_{i}=H\bm{v}_{P_{i}}-H\bm{v}_{P_{1}}=\bm{y}-\bm{y}=0$. So
$\bm{z}_{1},\dots,\bm{z}_{s}\in C$. And

$\sum_{i=1}^{s}d_{R}(-\bm{v}_{P_{1}},\bm{z}_{i})=\sum_{i=1}^{s}\mathrm{rank}_{\mathbb{F}_{q}}(\bm{z}_{i}-(-\bm{v}_{P_{1}}))=\sum_{i=1}^{s}\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{P_{i}})\stackrel{{\scriptstyle\eqref{eq:rank-
bound}}}{{\leq}}(s-1)(n-k).$ (68)

If $s=1,$ we get from (67) that
$\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{P_{1}})=0$, implying that
$\bm{v}_{P_{1}}=0$. But this is impossible since $\bm{y}=H\bm{v}_{P_{1}}$ is
nonzero. So $s\geq 2$. By (68), $C$ is not $\operatorname{LD-MRD}(s-1)$. As
$s\leq\ell+1$, we see that $C$ is not $\operatorname{LD-MRD}({\leq}\,\ell)$. ∎

Next, we show the other direction.

######  Theorem 6.5.

Let $C$ be an $[n,k]_{\mathbb{F}}$ code over a field
$\mathbb{F}/\mathbb{F}_{q}$. If $C^{\perp}$ is $\operatorname{MRD}(\ell+1)$,
then $C$ is $\operatorname{LD-MRD}({\leq}\,\ell)$.

###### Proof.

We will instead prove the following equivalent statement: Suppose $C$ is MRD
but not $\operatorname{LD-MRD}({\leq}\,\ell)$. Then $C^{\perp}$ is not
$\operatorname{MRD}(\ell+1)$.

Let $H\in\mathbb{F}^{(n-k)\times n}$ be a parity check matrix of $C$. Then $H$
is also a generator matrix of $C^{\perp}$. Since $C$ is not $\operatorname{LD-
MRD}({\leq}\,\ell)$, there exist $L\leq\ell$, $\bm{z}\in\mathbb{F}^{n}$, and
distinct $\bm{z}_{0},\dots,\bm{z}_{L}\in\mathbb{F}^{n}$ such that

$\sum_{i=0}^{L}\mathrm{rank}_{\mathbb{F}_{q}}(\bm{z}_{i}-\bm{z})\leq
L(n-k)\quad\text{and}\quad H\bm{z}_{0}=\cdots=H\bm{z}_{L}=0.$

Let $\bm{v}_{i}=\bm{z}_{i}-\bm{z}\in\mathbb{F}^{n}$ for $0\leq i\leq L$. Then
$\bm{v}_{0},\ldots,\bm{v}_{L}$ are distinct vectors satisfying

$\sum_{i=0}^{L}\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{i})\leq
L(n-k)\quad\text{and}\quad H\bm{v}_{0}=\cdots=H\bm{v}_{L}.$ (69)

Let $\bm{y}:=H\bm{v}_{0}$, which equals $H\bm{v}_{i}$ for all $0\leq i\leq L$.
Assume that $\bm{y}=0$. Then $H\bm{v}_{i}=0$ for all $0\leq i\leq L$. So
$\bm{v}_{0},\dots,\bm{v}_{L}$ are codewords of $C$. As $C$ is MRD, all of
these codewords $\bm{v}_{i}$ except the zero codeword satisfy
$\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{i})\geq n-k+1$. But this contradicts
(69). So $\bm{y}\neq 0$. Moreover, we may assume that
$\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{i})\leq n-k$ for all $0\leq i\leq L$,
because if this were not the case, we could remove some $\bm{v}_{i}$ and (69)
would still hold (for smaller $L$).

Consider arbitrary $i\in\\{0,\dots,L\\}$. Write
$\bm{v}_{i}=(v_{i,1},\dots,v_{i,n})$. Let
$r_{i}=\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{i})$. We know
$r_{i}=\mathrm{span}_{\mathbb{F}_{q}}\\{v_{i,1},\dots,v_{i,n}\\}$ by
definition. Pick $u_{i,1},\dots,u_{i,r_{i}}\in\mathbb{F}$ that form a basis of
$\mathrm{span}_{\mathbb{F}_{q}}\\{v_{i,1},\dots,v_{i,n}\\}$ over
$\mathbb{F}_{q}$, and let
$\bm{u}_{i}=(u_{i,1},\dots,u_{i,r_{i}})\in\mathbb{F}^{r_{i}}$. Then there
exists a unique matrix $A_{i}\in\mathbb{F}_{q}^{n\times r_{i}}$ such that
$\bm{v}_{i}=A_{i}\bm{u}_{i}$. Let $V_{i}=\langle A_{i}\rangle$, i.e., $V_{i}$
is the column span of $A_{i}$ over $\mathbb{F}_{q}$. Let $V_{i}^{\prime}$ be
the column span of $A_{i}$ over $\mathbb{F}$. By definition, we have

  1. 1.

$\dim_{\mathbb{F}_{q}}V_{i}=\dim_{\mathbb{F}}V_{i}^{\prime}=r_{i}=\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{i})\leq
n-k$.

  2. 2.

$\bm{v}_{i}=A_{i}\bm{u}_{i}\in V_{i}^{\prime}$.

  3. 3.

$H_{V_{i}}=\\{HA_{i}\bm{u}:\bm{u}\in\mathbb{F}^{r_{i}}\\}=\\{H\bm{x}:\bm{x}\in
V_{i}^{\prime}\\}$. In particular, $H\bm{v}_{i}\in H_{V_{i}}$.

As $\bm{y}=H\bm{v}_{0}=\cdots=H\bm{v}_{L}$, we have $\bm{y}\in\bigcap_{0\leq
i\leq L}H_{V_{i}}$. Let $W=(Z_{i,j})_{i\in[n-k],j\in[n]}$. First consider the
case where $\bigcap_{0\leq i\leq L}W_{V_{i}}=\\{0\\}$. Note that,

$\dim\left(\bigcap_{0\leq i\leq L}H_{V_{i}}\right)>0=\dim\left(\bigcap_{0\leq
i\leq L}W_{V_{i}}\right)$

since we already know $\bm{y}\neq 0$ and $\bm{y}\in\bigcap_{0\leq i\leq
L}H_{V_{i}}$. So $C^{\perp}$ is not $\operatorname{MRD}(\ell+1)$.

Now consider the case where $\bigcap_{0\leq i\leq L}W_{V_{i}}\neq\\{0\\}$. By
1.16, we have

$\dim\left(\bigcap_{0\leq i\leq
L}W_{V_{i}}\right)=\sum_{i=1}^{s}\dim(V_{P_{i}})-(s-1)(n-k)>0$ (70)

for some partition $P_{1}\sqcup P_{2}\sqcup\cdots\sqcup
P_{s}=\\{0,1,\ldots,L\\}$, where $V_{P_{i}}=\bigcap_{j\in P_{i}}V_{j}$. Note
that this partition is not the finest partition of $\\{0,1\dots,L\\}$ since

$\sum_{i=0}^{L}\dim
V_{i}-L(n-k)=\sum_{i=0}^{L}\mathrm{rank}_{\mathbb{F}_{q}}(\bm{v}_{i})-L(n-k)\stackrel{{\scriptstyle\eqref{eq:rank-
bound-2}}}{{\leq}}0.$

So $s<L+1$.

Define the $\mathbb{F}$-subspace $V\subseteq\prod_{i=0}^{L}\mathbb{F}^{r_{i}}$
to be

$V=\left\\{(\bm{c}_{0},\ldots,\bm{c}_{L})\in\prod_{i=0}^{L}\mathbb{F}^{r_{i}}:HA_{0}\bm{c}_{0}=\cdots=HA_{L}\bm{c}_{L}\right\\}.$

Then $\sigma:(\bm{c}_{0},\dots,\bm{c}_{L})\mapsto HA_{0}\bm{c}_{0}$ maps $V$
to $\bigcap_{0\leq i\leq L}H_{V_{i}}$. As $C^{\perp}$ is MRD (since $C$ is)
and $\dim V_{i}\leq n-k$ for $0\leq i\leq L$, for every
$\bm{w}\in\bigcap_{0\leq i\leq L}H_{V_{i}}$, there exists unique
$\bm{x}=(\bm{c}_{0},\dots,\bm{c}_{L})\in V$ such that $\sigma(\bm{x})=\bm{w}$,
i.e., $\sigma:V\to\bigcap_{0\leq i\leq L}H_{V_{i}}$ is an isomorphism of
$\mathbb{F}$-linear spaces. In particular,

$\dim\left(\bigcap_{0\leq i\leq L}H_{V_{i}}\right)=\dim V.$ (71)

Define

$V^{\prime}=\left\\{(\bm{c}_{0},\ldots,\bm{c}_{L})\in
V:A_{j}\bm{c}_{j}=A_{j^{\prime}}\bm{c}_{j^{\prime}}\text{ for }i\in[s]\text{
and }j,j^{\prime}\in P_{i}\right\\}\subseteq V.$

Note that $(\bm{u}_{0},\dots,\bm{u}_{L})\in V\setminus V^{\prime}$ since
$\bm{v}_{0}=A_{0}\bm{u}_{0},\dots,\bm{v}_{L}=A_{L}\bm{u}_{L}$ are distinct and
$s<L+1$. So

$\dim V>\dim V^{\prime}.$ (72)

Consider $(\bm{c}_{0},\dots,\bm{c}_{L})\in V^{\prime}$ and $i\in[s]$. For
$j,j^{\prime}\in P_{i}$, we have
$A_{j}\bm{c}_{j}=A_{j^{\prime}}\bm{c}_{j^{\prime}}\in V^{\prime}_{j}\cap
V^{\prime}_{j^{\prime}}$. So for $j\in P_{i}$, it holds that

$\bm{c}_{j}\in\bigcap_{j^{\prime}\in
P_{i}}V_{j^{\prime}}^{\prime}=:V^{\prime}_{P_{i}}.$

As each $V_{j}^{\prime}$ is the $\mathbb{F}$-span of the
$\mathbb{F}_{q}$-linear space $V_{j}$, here $V^{\prime}_{P_{i}}$ is the
$\mathbb{F}$-span of the $\mathbb{F}_{q}$-linear space
$V_{P_{i}}=\bigcap_{j\in P_{i}}V_{j}$.111111Each $V_{j}^{\prime}$ may be
identified with $V_{j}\otimes_{\mathbb{F}_{q}}\mathbb{F}$. We are using the
fact that $\bigcap_{j\in
P_{i}}(V_{j}\otimes_{\mathbb{F}_{q}}\mathbb{F})=\left(\bigcap_{j\in
P_{i}}V_{j}\right)\otimes_{\mathbb{F}_{q}}\mathbb{F}$. See, e.g., [Ene10,
Proposition 1.6].  So $\sigma$ maps $V^{\prime}$ to
$\bigcap_{i\in[s]}H_{V_{P_{i}}}$. And for every
$\bm{w}\in\bigcap_{i\in[s]}H_{V_{P_{i}}}$, the inverse
$\bm{c}=(\bm{c}_{0},\dots,\bm{c}_{L})=\sigma^{-1}(\bm{w})\in V$ satisfies that
$A_{j}\bm{c}_{j}=A_{j^{\prime}}\bm{c}_{j^{\prime}}$ for $i\in[s]$ and
$j,j^{\prime}\in P_{i}$ by the uniqueness of $\bm{c}$. So $\sigma$ restricts
to an isomorphism from $V^{\prime}$ to $\bigcap_{i\in[s]}H_{V_{P_{i}}}$.
Therefore,

$\displaystyle\dim
V^{\prime}=\dim\left(\bigcap_{i\in[s]}H_{V_{P_{i}}}\right)\geq\dim\left(\bigcap_{i\in[s]}W_{V_{P_{i}}}\right)$
$\displaystyle\stackrel{{\scriptstyle\eqref{intersec1}}}{{\geq}}\sum_{i=1}^{s}\dim(V_{P_{i}})-(s-1)(n-k)$
(73) $\displaystyle\stackrel{{\scriptstyle\eqref{eq:intersect-
h-v}}}{{=}}\dim\left(\bigcap_{0\leq i\leq L}W_{V_{i}}\right)$

where the second step uses 6.2. Then $\dim\left(\bigcap_{0\leq i\leq
L}H_{V_{i}}\right)>\dim\left(\bigcap_{0\leq i\leq L}W_{V_{i}}\right)$ by (71),
(72), and (73). So $C^{\perp}$ is not $\operatorname{MRD}(\ell+1)$. ∎

Combining 6.4 and 6.5 proves 1.14.

##  7 Putting It Together

The last tool we need is the duality of Gabidulin codes, as stated by the
following theorem. For the proof, see [BHL+22, Lemma 2.7.2].121212[BHL+22,
Lemma 2.7.2] assumes $\mathbb{F}$ to be a finite extension over
$\mathbb{F}_{q}$. However, the same proof applies to any extension field
$\mathbb{F}$ of $\mathbb{F}_{q}$ without modification.

######  Theorem 7.1 (Duality of Gabidulin codes).

Let $\mathbb{F}$ be an extension field of $\mathbb{F}_{q}$, and let
$\alpha_{1},\dots,\alpha_{n}\in\mathbb{F}$ be linearly independent over
$\mathbb{F}_{q}$. Then there exists
$(\beta_{1},\dots,\beta_{n})\in\mathbb{F}^{n}\setminus\\{0\\}$ such that

$\sum_{i=1}^{n}\alpha_{i}^{q^{j-1}}\beta_{i}^{q^{h-1}}=0\qquad\text{for
$(j,h)\in[k]\times[n-k]$}.$ (74)

The choice of $(\beta_{1},\dots,\beta_{n})$ satisfying (74) is unique up to a
scalar in $\mathbb{F}\setminus\\{0\\}$. Moreover, $\beta_{1},\dots,\beta_{n}$
are linearly independent over $\mathbb{F}_{q}$, and
$\left(\beta_{j}^{q^{i-1}}\right)_{i\in[n-k],j\in[n]}$ is a parity check
matrix of $\mathcal{G}_{n,k}(\alpha_{1},\dotsc,\alpha_{n})$, i.e.,

$\mathcal{G}_{n,k}(\alpha_{1},\dotsc,\alpha_{n})^{\perp}=\mathcal{G}_{n,n-k}(\beta_{1},\dotsc,\beta_{n}).$

We are now ready to prove our main theorems (1.3 and 1.4) on the list
decodability of random Gabidulin codes:

######  Theorem 7.2.

Let $\delta>0$. Let $1\leq k\leq n\leq m$ be integers such that $q^{m}\geq
3(n-k)q^{nk\cdot\min\\{\ell+1,k\\}+k}/\delta$. Let $\ell\in[k]$. Let
$(\alpha_{1},\dots,\alpha_{n})$ be uniformly distributed over the set of all
vectors in $\mathbb{F}_{q^{m}}^{n}$ whose coordinates are linearly independent
over $\mathbb{F}_{q}$. Then it holds with probability at least $1-\delta$ that
the Gabidulin code $\mathcal{G}_{n,k}(\alpha_{1},\dots,\alpha_{n})$ over
$\mathbb{F}_{q^{m}}$ is $\operatorname{LD-MRD}({\leq}\,\ell)$, i.e., it is
$\left(\frac{L}{L+1}\left(1-{k}/{n}\right),L\right)$-average radius list
decodable (and hence also
$\left(\frac{L}{L+1}\left(1-{k}/{n}\right),L\right)$-list decodable) for all
$L\in[\ell]$.

###### Proof.

Let $S$ be the set of all $(\alpha_{1},\dots,\alpha_{n})\in\mathbb{F}^{n}$
whose coordinates are linearly independent over $\mathbb{F}_{q}$. Define an
equivalence relation $\sim$ on $S$ by letting $\alpha\sim\alpha^{\prime}$ iff
$\alpha=c\alpha^{\prime}$ for some $c\in\mathbb{F}\setminus\\{0\\}$. Denote
the equivalence class of $\alpha$ by $[\alpha]$ and denote the set of the
equivalence classes by $S/\sim$. By 7.1, for
$\alpha=(\alpha_{1},\dots,\alpha_{n})\in S$, we can find its dual basis
$\beta=(\beta_{1},\dots,\beta_{n})\in S$ by solving (74), and $[\beta]$ is
uniquely determined by $\alpha$. Also note that scaling $\alpha$ does not
affect $[\beta]$. So we obtain a map from $S/\sim$ to itself that sends
$[\alpha]$ to $[\beta]$, where $\alpha$ and $\beta$ satisfy 7.1. Moreover, by
applying 7.1 again, but with $k$ replaced by $n-k$, we can solve $\alpha$ from
$\beta$ using (74) and hence get the map $[\beta]\mapsto[\alpha]$. The two
maps are inverse to each other by definition. So the map
$[\alpha]\mapsto[\beta]$ is a permutation of $S/\sim$.

Now let $\alpha$ be uniformly distributed over $S$. From the discussion above,
we see that $[\beta]$, which is uniquely determined by $\alpha$ via (74), is
uniformly distributed over $S/\sim$. Moreover, the Gabidulin code
$\mathcal{G}_{n,n-k}(\beta_{1},\dots,\beta_{n})$ depends only on the
equivalence class $[\beta]$ of $\beta=(\beta_{1},\dots,\beta_{n})$, since
scaling $\beta$ corresponds to scaling the rows of the generator matrix
$\left(\beta_{j}^{q^{i-1}}\right)_{i\in[n-k],j\in[n]}$, which does not change
the code.

By 1.15, the Gabidulin code $\mathcal{G}_{n,k}(\alpha_{1},\dots,\alpha_{n})$
is $\operatorname{LD-MRD}({\leq}\,\ell)$ iff its dual code
$\mathcal{G}_{n,n-k}(\beta_{1},\dots,\beta_{n})$ is
$\operatorname{GKP}(\ell+1)$. As $[\beta]$ is uniformly distributed over
$S/\sim$, by 4.2, $\mathcal{G}_{n,n-k}(\beta_{1},\dots,\beta_{n})$ is
$\operatorname{GKP}(\ell+1)$ with probability at least
$1-3(n-k)q^{nk\cdot\min\\{\ell+1,k\\}+k-m}\geq 1-\delta$. Therefore,
$\mathcal{G}_{n,k}(\alpha_{1},\dots,\alpha_{n})$ is $\operatorname{LD-
MRD}({\leq}\,\ell)$ with the same probability. ∎

Setting
$\ell=\lceil\frac{1-R-\varepsilon}{\varepsilon}\rceil\leq\frac{1-R}{\varepsilon}$,
by 7.2, we obtain the following corollary, which shows that random Gabidulin
codes achieve list decoding capacity in the rank metric with high probability.

######  Corollary 7.3.

Let $\delta>0$. Let $1\leq k\leq n\leq m$ with $q^{m}\geq
3(n-k)q^{nk\cdot\min\\{(1-k/n)/\varepsilon+1,k\\}+k}/\delta$. Let
$\varepsilon>0$ and let $(\alpha_{1},\dots,\alpha_{n})$ be uniformly
distributed over the set of all vectors in $\mathbb{F}_{q^{m}}^{n}$ whose
coordinates are linearly independent over $\mathbb{F}_{q}$. Then it holds with
probability at least $1-\delta$ that the Gabidulin code
$\mathcal{G}_{n,k}(\alpha_{1},\dots,\alpha_{n})$ over $\mathbb{F}_{q^{m}}$ is
$\left(1-R-\varepsilon,\frac{1-R}{\varepsilon}\right)$-average radius list
decodable (and hence also
$\left(1-R-\varepsilon,\frac{1-R}{\varepsilon}\right)$-list decodable) in the
rank metric, where $R=k/n$ is the rate of the code.

We conclude this section with a discussion about the symbolic Gabidulin code
$\mathcal{G}_{n,k}(Z_{1},\dots,Z_{n})$. By 4.1 and 1.13,
$\mathcal{G}_{n,k}(Z_{1},\dots,Z_{n})$ is $\operatorname{GKP}(\ell)$ and
$\operatorname{MRD}(\ell)$ for all $\ell$. It is also $\operatorname{LD-
MRD}({\leq}\,\ell)$ for all $\ell$, which can be shown via a proof strategy
similar to that of 7.2. The key difference is that we need to argue that
certain elements $Y_{1},\dots,Y_{n}\in\mathbb{F}_{q}(Z_{1},\dots,Z_{n})$,
which form a dual basis of $Z_{1},\dots,Z_{n}$, are algebraically independent
over $\mathbb{F}_{q}$, so that no nonzero polynomial over $\mathbb{F}_{q}$ can
vanish at $(Y_{1},\dots,Y_{n})$. For this, we need to pass to the projective
space $\mathbb{P}^{n}$ over $\mathbb{F}_{q}$ and view the maps
$[\alpha]\mapsto[\beta]$ and $[\beta]\mapsto[\alpha]$ as well-defined
morphisms that are inverse to each other (over dense open subsets of
$\mathbb{P}^{n}$). These morphisms must preserve dimension and, consequently,
algebraic independence. The details are omitted.

##  8 Conclusions and Future Directions

In this paper, we have proved that, with high probability, a random Gabidulin
code $C\subseteq\mathbb{F}_{q^{m}}^{n}$ is list decodable, and even average-
radius list decodable, up to the optimal generalized Singleton bound when $m$
is large enough. Our result requires that $m=\Omega_{\ell}(n^{2})$, which is
optimal up to a factor depending only on $\ell$, as demonstrated by our lower
bound. In achieving this, we have formulated various notions of higher-order
MRD codes and established their equivalence, analogous to the work of
Brakensiek, Gopi, and Makam [BGM23]. We have also proved the GM-MRD theorem,
which is an essential ingredient for our main results.

We conclude with the following potential future directions: (1) Are there
explicit constructions of Gabidulin codes whose parameters match or come close
to those achieved in 1.3? (2) Is it possible to reduce the parameter $m$ by
slightly compromising the rate of the code, in a manner analogous to the work
of Guo and Zhang [GZ23], and Alrabiah, Guruswami, and Li [AGL23]? (3) Inspired
by the work of [BDG23], we propose a similar conjecture below, which we call
the “ultimate GM-MRD conjecture.”

######  Conjecture 8.1 (Ultimate GM-MRD conjecture).

Let $C\in\mathbb{F}_{q^{m}}^{N}$ be any $[N,k]$ MRD code with a generator
matrix $G\in\mathbb{F}_{q^{m}}^{k\times N}$. Let $C^{\prime}$ be the $[n,k]$
code defined by the generator matrix $G^{\prime}:=GA$, where
$A\in\mathbb{F}_{q}^{N\times n}$ is a randomly sampled full rank matrix. Then,
with probability $1-o_{N}(1)$, the code $C^{\prime}$ is
$\operatorname{GKP}(\ell)$ for all $\ell\geq 1$.

## References

  * [AGL23] Omar Alrabiah, Venkatesan Guruswami, and Ray Li.  Randomly punctured Reed–Solomon codes achieve list-decoding capacity over linear-sized fields.  arXiv preprint arXiv:2304.09445, 2023.  To appear in STOC 2024.
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##  Appendix A Field Size Lower Bound for $\operatorname{LD-MRD}(\ell)$

We prove a lower bound on the field size of $\operatorname{LD-MRD}(\ell)$
codes by adapting the argument in [AGL24].

######  Theorem A.1.

Let $\ell\geq 2$. For any $r\in[0,1]$, any $\operatorname{MRD}$ code
$C\subseteq\mathbb{F}_{q^{m}}^{n}$ of rate $R$ that is
$\left(\frac{\ell\left(1-R\right)}{\ell+1},\ell\right)$-avearge-radius list-
decodable must have $m=\Omega_{\ell}((nR-1)(n-\ell-nR+1))$, which is
$\Omega_{\ell}(n^{2})$ if the rate $R$ of $C$ is in $[c,1-c-\ell/n]$ for some
constant $c>0$.

###### Proof.

Fix a subspace $V_{0}\subseteq\mathbb{F}_{q}^{n}$ of dimension $\ell$. Choose
a subspace $\overline{V}_{0}$ such that
$V_{0}\oplus\overline{V}_{0}=\mathbb{F}_{q}^{n}$. Assume the minimum distance
of $C$ is $d$. As $C$ is MRD, the size of $C$ is $q^{m(n-d+1)}$ [Del78,
Theorem 5.4]. Let $k=n-d+1=nR$. For any two distinct codewords $M_{1},M_{2}\in
C$ (viewed as matrices in $\mathbb{F}_{q}^{m\times n}$), we have
$\mathrm{rank}(M_{1}-M_{2})\geq d=n-k+1$. Let $\mathcal{F}$ be the collection
of subspaces $V\subseteq\overline{V}_{0}$ of dimension $k-1$. The size of
$\mathcal{F}$ is the number of subspaces of dimension $k-1$ contained in a
subspace of dimension $n-\ell$, which is at least $q^{(n-\ell-k+1)(k-1)}$. It
suffices to prove that $\ell q^{\ell m}\geq|\mathcal{F}|/2$, as this would
imply $m=\Omega_{\ell}((k-1)(n-\ell-k+1))$.

Assume to the contrary that $\ell q^{\ell m}<|\mathcal{F}|/2$. Let $M$ be
uniformly distributed from $C$. For a fixed subspace $V\in\mathcal{F}$, let
$A\in\mathbb{F}_{q}^{n\times(k-1)}$ such that $\langle A\rangle=V$. Let
$E_{V}$ be the event that there exists a codeword $M_{1}\in C$ different from
$M$ such that $MA=M_{1}A$, i.e., $(M-M_{1})A=0$. If $E_{V}$ does not hold,
then $M$ is uniquely determined by $MA\in\mathbb{F}_{q}^{m\times{(k-1)}}$. As
the number of possible values of $MA$ is at most $q^{(k-1)m}$ and
$|C|=q^{mk}$, we have

$\Pr[\lnot E_{V}]\leq\frac{q^{m(k-1)}}{q^{mk}}=q^{-m}.$

Therefore, over random $M\in C$, the expected number of $V\in\mathcal{F}$ such
that $E_{V}$ happens is $\sum_{V\in\mathcal{F}}(1-\Pr[\lnot
E_{V}])\geq|\mathcal{F}|/2$. Then, we can fix a codeword $M\in C$ such that
the size of the set

$\mathcal{F}_{M}:=\\{V\in\mathcal{F}:E_{V}\text{ happens}\\}$

is at least $|\mathcal{F}|/2$.

Let $A_{0}\in\mathbb{F}_{q}^{n\times\ell}$ such that $\langle
A_{0}\rangle=V_{0}$. By the definition of $\mathcal{F}_{M}$, for each
$V\in\mathcal{F}_{M}$, there exists a codeword $M_{V}\neq M$ such that the
kernel subspace of $M-M_{V}$ contains $V$. Since
$M_{V}A_{0}\in\mathbb{F}_{q}^{m\times\ell}$ for any codeword $M_{V}$ and $\ell
q^{\ell m}<|\mathcal{F}|/2\leq|\mathcal{F}_{M}|$, by the pigeonhole principle,
there exists distinct $V_{1},\ldots,V_{\ell}\in\mathcal{F}_{M}$ such that
$M_{V_{1}}A_{0}=\dots=M_{V_{\ell}}A_{0}$. Moreover, by the definition of
$\mathcal{F}_{M}$, for $i=1,\dots,\ell$, there exists
$A_{i}\in\mathbb{F}_{q}^{n\times(k-1)}$ with $\langle A_{i}\rangle=V_{i}$ such
that $(M-M_{V_{i}})A_{i}=0$.

Assume $M_{V_{i}}=M_{V_{j}}$ for some $i\neq j$. Then $(M-M_{V_{i}})A_{i}=0$
and $(M-M_{V_{i}})A_{j}=0$. Let
$A\in\mathbb{F}_{q}^{n\times\dim(V_{i}+V_{j})}$ such that $\langle
A\rangle=V_{i}+V_{j}$. As the columns of $A$ are in $V_{i}+V_{j}=\langle
A_{i}\rangle+\langle A_{j}\rangle$, we have $(M-M_{V_{i}})A=0$, i.e.,
$V_{i}+V_{j}$ is contained in the kernel subspace of $M-M_{V_{i}}$. Since $M$
and $M_{V_{i}}$ are in the MRD code $C$, we have
$\mathrm{rank}(M-M_{V_{i}})\geq n-k+1$. This implies that the kernel subspace
of $M-M_{V_{i}}$ is at most $k-1$. So $\dim(V_{i}+V_{j})\leq k-1$. However, as
$V_{i}\neq V_{j}$ and $\dim V_{i}=\dim V_{j}=k-1$, we have
$\dim(V_{i}+V_{j})\geq k$, which yields a contradiction. Thus, we conclude
that $M_{V_{1}},\ldots,M_{V_{\ell}}$ are all distinct.

Since $\overline{V}_{0}\cap V_{0}=\\{0\\}$, there exists
$B_{0}\in\mathbb{F}_{q}^{n\times(n-\ell)}$ such that $\langle
B_{0}\rangle=\overline{V}_{0}$ and
$\begin{pmatrix}A_{0}&B_{0}\end{pmatrix}\in\mathbb{F}_{q}^{n\times n}$ has
full rank. Let $Y\in\mathbb{F}_{q}^{m\times n}$ such that
$(M_{V_{1}}-Y)A_{0}=\cdots=(M_{V_{\ell}}-Y)A_{0}=0$ and $(M-Y)B_{0}=0$. This
can be achieved by choosing
$Y=\begin{pmatrix}M_{V_{1}}A_{0}&MB_{0}\end{pmatrix}\begin{pmatrix}A_{0}&B_{0}\end{pmatrix}^{-1}$.

For $i\in[\ell]$, we have $(M-Y)A_{i}=0$ since $\langle A_{i}\rangle=V_{i}$,
$V_{i}\subseteq\overline{V}_{0}$, $\overline{V}_{0}=\langle B_{0}\rangle$, and
$(M-Y)B_{0}=0$. And for $i\in[\ell]$, we know $(M-M_{V_{i}})A_{i}=0$, which
implies

$(M_{V_{i}}-Y)A_{i}=(M_{V_{i}}-M)A_{i}+(M-Y)A_{i}=0\quad\text{and}\quad(M_{V_{i}}-Y)A_{0}=0.$

Since $V_{0}\cap\langle V_{i}\rangle\subseteq
V_{0}\cap\overline{V}_{0}=\\{0\\}$ for $i\in[\ell]$, we have
$\dim(V_{0}+V_{i})=\dim V_{0}+\dim V_{i}=\ell+k-1$ and hence

$\mathrm{rank}(M_{V_{i}}-Y)\leq n-(\ell+k-1)\leq n-k-1,$

where we use the fact that $\ell\geq 2.$ As $(M-Y)B_{0}=0$, we have
$\mathrm{rank}(M-Y)\leq n-\dim(\overline{V}_{0})=\ell$. It follows that

$\mathrm{rank}(M-Y)+\sum_{i=1}^{\ell}\mathrm{rank}(M_{V_{i}}-Y)\leq\ell+\ell(n-k-1)=\ell(n-k)$

which contradicts the claim that $C$ is
$\left(\frac{\ell\left(1-k/n\right)}{\ell+1},\ell\right)$-avearge-radius list-
decodable. ∎


Destroying Kaluza-Klein and Kerr-Newman black holes

Haryanto M<EMAIL_ADDRESS>and Paulus C.
<EMAIL_ADDRESS>
Center for Theoretical Physics,

Department of Physics, Parahyangan Catholic University,

Jalan Ciumbuleuit 94, Bandung 40141, Indonesia

We investigate the destroying of charged and rotating black holes in Einstein-
Maxwell-(dilaton) theory. We show that a test particle with some appropriate
properties in the black hole background can turn the black hole into a naked
singularity. In this work, we neglect the self-force, self-energy, and
radiative effects as considered by some others in literature. As the result,
we are able to show that the Kerr-Newman and Kaluza-Klein black holes can be
destroyed by the test particle. Furthermore, for Kaluza-Klein black hole, we
consider the overspinning of the black hole by a neutral test scalar field.

##  1 Introduction

Black hole is a singularity covered by an event horizon with an extremely
strong gravity around its surrounding. The advent of black hole image reported
by the Event Horizon Telescope collaboration [1, 2] confirms the prediction of
black holes in Einstein gravity. This finding is a solid evidence of black
hole existence which then invites further related studies on black hole [3, 4,
5]. Nevertheless, as the horizon ceases to exist, the singularity is then
visible to an outside observer as a naked singularity. Penrose in 1969
proposed that no naked singularity can exist in our universe [6], and the
proposal is known as the weak cosmic censorship conjecture (WCCC). This
conjecture had been tested repeatedly, including the quite famous one by Wald
in his 1974 gedanken experiment [7]. In this Wald experiment of thought, one
consider a black hole capturing a test particle which opens a possibility to
turn into a naked singularity. However, in Wald’s original scenario, it was
shown that naked such naked singularity production is impossible.

About two decades after the Wald’s gedanken experiment proposal [7], the
conjecture by Penrose was again tested in the work by Hubeny [8] where she
showed that a particular set up of a charged test object may turn a Reissner-
Nordstrom black hole into a naked singularity. The similar idea to Hubeny’s in
violating WCCC for Kerr black hole was proposed Jacobson and Sotiriou [9], for
Kerr-Newman black hole were discussed in [10, 11], and for magnetized Kerr
black hole is investigated in [12]. The reconsideration of WCCC in the fashion
of Hubeny [8] is not limited just to black holes in Einstein-Maxwell theory,
but also beyond [13, 14, 15, 16, 17, 18, 19, 20]. However, a newer version of
checking WCCC by Sorce and Wald [21] shows that if the self-force and finite
size effects are considered, then no such black hole destruction as
illustrated in [8, 9, 11, 12] can occur.

As we know, there exists gravitational theories beyond Einstein-Maxwell. For
example, considering an extra dilaton field in addition to the graviton and
Maxwell fields, namely the Einstein-Maxwell-dilaton theory. An effective
action corresponding to the four dimensional Einstein-Maxwell-dilaton theory
can be obtained from the five dimensional vacuum Einstein after performing a
Kaluza-Klein compactification. A four dimensional black hole solution in such
theory is known as the Kaluza-Klein black hole [22, 23, 24] which has been
studied quite extensively in literature, ranging from those works which may be
relevant for some astrophysical purposes [25, 26, 27] to some holography
related [28, 29, 30, 31].

Interests in these black hole solutions motivate us to test the WCCC
associated to them. In the Einstein-Maxwell theory, we investigate the
possibility to destroy the near extremal Kerr-Newman black hole in the way
that slightly different compared to the one presented in [11, 10]. The same
way in testing WCCC is also applied to the rotating and charged black hole in
Einstein-Maxwell-dilaton theory, in particular the Kaluza-Klein black hole
case [22, 24, 23]. We show that, with the help of numerical results, both
black holes can be destroyed by letting them capture test object from far
away. Indeed, similar to the presentations in [11, 33, 12, 10, 14, 13], here
we neglect the self-energy or backreaction factors which recently were used by
some authors in proving the WCCC violation proposed in [8, 9]. For Kaluza-
Klein black hole discussion, we also add an analysis on the naked singularity
production by considering the neutral scalar field absorption which leads to
an overspinning of the black hole. This consideration has been performed for
the Kerr-Newman case in [32].

The organization of this paper is as follows. In section 2, we provide a brief
review on some features of black hole solutions studied in this paper. In the
next section, we obtain the required maximum and minimum energies for the test
object which may destroy these black holes. Then in section 4, we evaluate
some numerical examples which illustrate the naked singularity production. To
support the result for Kaluza-Klein case in section 4, overspinning an
extremal Kaluza-Klein black hole by a test neutral scalar field is considered
in section 5. Finally, we give conclusions. We consider the natural units
$c={\hbar}=k_{B}=G_{4}=1$.

##  2 Black hole solutions

Let us first review the Kaluza-Klein solution of Einstein-Maxwell-dilaton
theory. Consider the action for vacuum Einstein in five dimension

$S=\int{{d^{5}}x\sqrt{\left|{\tilde{g}}\right|}}\tilde{R}\,,$ (2.1)

where $\tilde{R}$ is the five dimensional Ricci scalar, $\tilde{g}$ is the
determinant of five dimensional spacetime metric $g_{MN}$. Here we used the
coordinate $x^{M}=\left\\{{x^{\mu},y}\right\\}$ where the four dimensional
spacetime is denoted by coordinate $x^{\mu}$, and $y$ is the coordinate for
fifth dimension. This fifth dimension is compact and has a period of $2\pi
R_{5}$. We also assume that the five dimensional spacetime to have the Killing
vector $\partial_{y}$. By using the five dimensional line element

$ds_{5}^{2}=e^{{{4\Phi}\mathord{\left/{\vphantom{{4\Phi}{\sqrt{3}}}}\right.\kern-1.2pt}{\sqrt{3}}}}\left({dy+2A_{\mu}dx^{\mu}}\right)^{2}+e^{-{{2\Phi}\mathord{\left/{\vphantom{{2\Phi}{\sqrt{3}}}}\right.\kern-1.2pt}{\sqrt{3}}}}g_{\mu\nu}dx^{\mu}dx^{\nu}\,,$
(2.2)

the five dimensional action (2.1) can be written as

$S=2\pi
R_{5}\int{d^{4}x\sqrt{\left|g\right|}\left[{R-2\left({\nabla\Phi}\right)^{2}-e^{-2\sqrt{3}\Phi}F_{\mu\nu}F^{\mu\nu}}\right]}\,,$
(2.3)

where $R$ is now the four dimensional Ricci scalar and $g$ is the determinant
of four dimensional metric tensor $g_{\mu\nu}$. Above,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field-strength
tensor of electromagnetic field.

Taking a variation to the action (2.3) gives us the equation of motions

$\nabla_{\mu}\left({e^{-2\sqrt{3}\Phi}F^{\mu\nu}}\right)=0\,,$ (2.4)
$\nabla^{2}\Phi+\frac{{\sqrt{3}}}{2}e^{-2\sqrt{3}\Phi}F_{\mu\nu}F^{\mu\nu}=0\,,$
(2.5)

and

$R_{\mu\nu}-2\nabla_{\mu}\Phi\nabla_{\nu}\Phi-2e^{-2\sqrt{3}\Phi}\left[{F_{\mu\lambda}F_{\nu}^{\lambda}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}}\right]=0\,.$
(2.6)

A rotating and charged black hole solution which solves the equations of
motion above is [24]

$ds_{4}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-\frac{{1-Z}}{\beta}dt^{2}-\frac{{2aZ\Delta_{x}}}{{\beta\Delta_{v}}}dtd\phi+\beta\Sigma\left({\frac{{dr^{2}}}{{\Delta_{r}}}+\frac{{dx^{2}}}{{\Delta_{x}}}}\right)$
$+\left[{\beta\left({\Delta_{r}-2Mr}\right)+\frac{{a^{2}\Delta_{x}Z}}{\beta}}\right]\Delta_{x}d\phi^{2}\,,$
(2.7)

where

$\beta=\sqrt{1+\frac{{v^{2}Z}}{{\Delta_{v}}}}\,,$ (2.8)
$Z=\frac{2Mr}{\Sigma}\,,$ (2.9)

$\Delta_{r}=r^{2}-2Mr+a^{2}$, $\Sigma=r^{2}+a^{2}x^{2}$, $\Delta_{x}=1-x^{2}$,
and $\Delta_{v}=1-v^{2}$. Here $v$ is normally referred as the boost velocity
parameter [34]. The accompanying non gravitational fields are

$A_{\mu}dx^{\mu}=\frac{{vZ}}{{2\beta^{2}\Delta_{v}}}\left({dt-a\Delta_{x}\sqrt{\Delta_{v}}d\phi}\right)$
(2.10)

for the vector field, and

$\Phi=-\frac{\sqrt{3}}{2}\beta\,,$ (2.11)

for the dilaton.

Note that the black hole spacetime (2.7) possesses the $\partial_{t}$ and
$\partial_{\phi}$ Killing symmetries, which is also the case for Kerr-Newman
spacetime. This allows us to employ the Komar integral to compute the
following conserved quantities, namely the mass

${\tilde{M}}=\frac{{M\left({2-v^{2}}\right)}}{{2\Delta_{v}}}\,,$ (2.12)

and the angular momentum

${\tilde{J}}=\frac{Ma}{\sqrt{\Delta_{v}}}\,.$ (2.13)

For the conserved electric charge, the standard textbook formula gives us

${\tilde{Q}}=\frac{Mv}{\Delta_{v}}\,.$ (2.14)

Here we consider the positive electric charge for the black hole, therefore
$0\leq v\leq 1$.

As we have mentioned that the spacetime (2.7) can contain a black hole, there
exists horizon with the radius is given by the larger roots of $\Delta_{r}$,
i.e.

$r_{+}=M+\sqrt{M^{2}-a^{2}}\,.$ (2.15)

It turns out that this is just the outer horizon of Kerr black hole, where
$M\geq a$ is the corresponding black hole condition. Accordingly, the
extremality is achieved at $a=M$ which yields $r_{\rm ext.}=M$ for a Kaluza-
Klein black hole. However, note that mass $M$ and rotational parameter $a$
above are not the proper conserved quantities that associate to a Kaluza-Klein
black hole, due to the results in (2.12), (2.13), and (2.14). Using the
inverse of these relations for a positive total charge $\tilde{Q}$, namely

$v=\frac{{2\tilde{Q}}}{{\tilde{M}+\sqrt{\tilde{M}^{2}+2\tilde{Q}^{2}}}}\,,$
(2.16)
$M=\frac{{\tilde{M}}}{2}\left({3-\sqrt{1+\frac{{2\tilde{Q}^{2}}}{{\tilde{M}^{2}}}}}\right)\,,$
(2.17)

and

$a=\frac{{\sqrt{2}\tilde{J}}}{{\tilde{M}\sqrt{1-\frac{{\tilde{Q}^{2}}}{{\tilde{M}^{2}}}+\left({1+\frac{{2\tilde{Q}^{2}}}{{\tilde{M}^{2}}}}\right)^{1/2}}}}\,,$
(2.18)

the horizon radius in extremal case can be rewritten as

$r_{\rm{ext.}}=\frac{{3\tilde{M}-\sqrt{\tilde{M}^{2}+2\tilde{Q}^{2}}}}{2}\,,$
(2.19)

accordingly.

Now let us turn to the Einstein-Maxwell theory whose action can be obtained by
setting $\Phi\to 0$ in the Einstein-Maxwell-dilaton action (2.3) above. The
corresponding equations of motion then read

${\nabla_{\mu}}{F^{\mu\nu}}=0\,,$ (2.20)

and

${R_{\mu\nu}}=2{F_{\mu\alpha}}F_{\nu}^{\alpha}-\frac{1}{2}{g_{\mu\nu}}{F_{\alpha\beta}}{F^{\alpha\beta}}\,.$
(2.21)

The rotating and charged black hole spacetime satisfying these equations of
motion is known as the Kerr-Newman black hole solution whose line element can
be written as

$d{s^{2}}=-\frac{{{\Delta_{KN}}}}{\Sigma}{\left({dt-a{\Delta_{x}}}\right)^{2}}+\Sigma\left({\frac{{d{r^{2}}}}{{{\Delta_{KN}}}}+\frac{{d{x^{2}}}}{{{\Delta_{x}}}}}\right)+\frac{{{\Delta_{x}}}}{\Sigma}{\left({adt-\left({{r^{2}}+{a^{2}}}\right)d\phi}\right)^{2}}\,,$
(2.22)

where

${\Delta_{KN}}={r^{2}}-2Mr-{a^{2}}-{Q^{2}}\,.$ (2.23)

Note that the functions $\Sigma$ and $\Delta_{x}$ appearing in the Kerr-Newman
metric above are just the same to those in Kaluza-Klein solution (2.7). On the
other hand, the accompanying vector solution in solving the Einstein-Maxwell
equations (2.20) and (2.21) reads

${A_{\mu}}d{x^{\mu}}=-\frac{{{\tilde{Q}}r}}{\Sigma}\left({dt-a{\Delta_{x}}d\phi}\right)\,.$
(2.24)

Again, as a black hole solution, the Kerr-Newman singularity is covered by an
event horizon located at $r_{+}=M+\sqrt{M^{2}-a^{2}-Q^{2}}$. The extremal
state for Kerr-Newman black hole is denoted by $M^{2}=a^{2}+Q^{2}$, and in
this condition we have $r_{+}=M$. Moreover, as we have mentioned that the
Kerr-Newman metric enjoy the $\partial_{t}$ and $\partial_{\phi}$ Killing
symmetries, the black hole mass $M$ and angular momentum $J=Ma$ can be
verified by using the Komar integral associated with each of these Killing
symmetries. Furthermore, the surface integral to obtain total electric charge
in the spacetime (2.22) gives us $Q$ as the charge of black hole333Here we use
the tilde notation for Kerr-Newman black hole charge to make it distinguished
to the charge of Kaluza-Klein black hole above..

##  3 Energies of test particle

In this section, we would like to obtain the upper and lower limits for the
energy of a test particle that may destroy the black holes under
consideration. We imagine that a test particle released from far away and
moving towards the black hole. Definitely, the properties of black hole will
change after capturing this test object which is parameterized by some
physical properties such as energy $E$, charge $q$, and angular momentum $L$.
Indeed, since both of the Kerr-Newman and Kaluza-Klein black hole spacetimes
possess the $\partial_{t}$ and $\partial_{\phi}$ Killing symmetries, $E$ and
$L$ are normally known as the two constants of motion in the spacetime. If the
change of black hole parameter suggest the production of naked singularity
after capturing the test particle, then we can infer that the black hole is
already destroyed by the test particle.

As a start, let us first discuss the geodesics of a charged test particle in a
general rotating charged black hole stationary spacetime with axial symmetry.
This applies to the cases we consider in this paper, namely the Kerr-Newman
and Kaluza-Klein black hole spacetimes. In a general curved background, the
motion of a massive charged test particle is dictated by the geodesic equation

${\ddot{x}}^{\mu}+\Gamma_{\alpha\beta}^{\mu}{\dot{x}}^{\alpha}{\dot{x}}^{\beta}=\frac{q}{\mu}F^{\mu\nu}{\dot{x}}_{\nu}\,,$
(3.1)

where $\mu$ and $q$ are the mass and electric charge of the particle,
respectively. In this geodesic equation, the “dot” stands for the derivative
with respect to some affine parameter $s$, ${\dot{(})}=\tfrac{d()}{ds}$. As we
know, it can be shown that the geodesic equation above is associated with the
Lagrangian

${\cal L}=\frac{1}{2}\mu
g_{\alpha\beta}{\dot{x}}^{\alpha}{\dot{x}}^{\beta}+qA_{\mu}{\dot{x}}^{\mu}\,.$
(3.2)

From this Lagrangian, the energy $E$ and angular momentum $L$ as the constants
of motion for the particle in a stationary background can be obtained as

$E=-\frac{{\partial{\cal
L}}}{\partial{\dot{t}}}=-\mu\left({g_{tt}\dot{t}+g_{t\phi}\dot{\phi}}\right)-qA_{t}\,,$
(3.3)

and

$L=\frac{{\partial{\cal
L}}}{\partial{\dot{\phi}}}=\mu\left({g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}}\right)+qA_{\phi}\,.$
(3.4)

Furthermore, since ${\dot{x}}_{\mu}{\dot{x}}^{\mu}=-1$ for the timelike
object, one can obtain a relation between $E$ and $L$ from the last two
equations above that reads

$E=\frac{{g_{t\phi}}}{{g_{\phi\phi}}}\left({qA_{\phi}-L}\right)-qA_{t}+{\sqrt{{\left({\frac{{g_{t\phi}^{2}-g_{\phi\phi}g_{tt}}}{{g_{\phi\phi}^{2}}}}\right)\left({\left({L-qA_{\phi}}\right)^{2}+\mu^{2}g_{\phi\phi}\left({1+g_{rr}\dot{r}^{2}+g_{\theta\theta}\dot{\theta}^{2}}\right)}\right)}}}\,.$
(3.5)

Note that, in getting eq. (3.5) we have considered the solution that implies
${\dot{t}}>0$ only. Subsequently, inserting the corresponding components of
$g_{\mu\nu}$ into (3.5) followed by the $r=r_{+}$ evaluation, the minimum
energy for a test particle to reach the event horizon can be obtained. In
Kaluza-Klein spacetime, this minimum energy reads444Note that the mass $m$
appearing in this minimum energy is the one from Kerr seed solution, as one
can view Kaluza-Klein black hole spacetime is a product of some transformation
with boost velicity parameter $v$ with Kerr metric as the seed [34]. The
proper conserved mass for Kaluza-Klein black hole is given in (2.12).

$E_{\min}=\frac{{aL\sqrt{\Delta_{v}}-vqMr_{+}}}{{2Mr_{+}}}\,,$ (3.6)

while for the Kerr-Newman case is [11]

${E_{\min}}=\frac{{aL+qQ{r_{+}}}}{{{a^{2}}+r_{+}^{2}}}\,.$ (3.7)

Therefore, for a test particle to destroy the black hole, its energy cannot be
lower than the minimum energy above for each particular case.

On the other hand, there exist an upper bound for a test particle energy which
could destroy a black hole. In other words, if this energy is bigger than a
maximum value that is allowed, the black hole destruction can never occur. The
maximum energy for the test particle itself comes from an inequality that
violates the black hole condition. For the Kaluza-Klein case, the black hole
condition reads

$\frac{\tilde{M}}{2}\left({3-\sqrt{1+\frac{{2{\tilde{Q}^{2}}}}{{{\tilde{M}^{2}}}}}}\right)\geq\sqrt{2}\frac{\tilde{J}}{\tilde{M}}{\left({1-\frac{{{\tilde{Q}^{2}}}}{{{\tilde{M}^{2}}}}+\sqrt{1+\frac{{2{\tilde{Q}^{2}}}}{{{\tilde{M}^{2}}}}}}\right)^{-\frac{1}{2}}}\,.$
(3.8)

At this point, for a future benefit, we can define

${\delta_{KK}}\equiv
M-a=\frac{\tilde{M}}{2}\left({3-\sqrt{1+\frac{{2{\tilde{Q}^{2}}}}{{{\tilde{M}^{2}}}}}}\right)-\sqrt{2}\frac{\tilde{J}}{\tilde{M}}{\left({1-\frac{{{\tilde{Q}^{2}}}}{{{\tilde{M}^{2}}}}+\sqrt{1+\frac{{2{\tilde{Q}^{2}}}}{{{\tilde{M}^{2}}}}}}\right)^{-\frac{1}{2}}}\,,$
(3.9)

as the near to extremal parameter in Kaluza-Klein black hole case. Obviously,
the extremal condition is represented by $\delta_{KK}=0$, and the black hole
destruction is denoted by $\delta_{KK}<0$. Moreover, it will be useful for us
later to show the angular momentum of Kaluza-Klein black hole at extremal
state, which can be written as

${\tilde{J}}=\frac{{\sqrt{2}\left({3\tilde{M}-\sqrt{\tilde{M}^{2}+2\tilde{Q}^{2}}}\right)\left({\tilde{M}\sqrt{\tilde{M}^{2}+2\tilde{Q}^{2}}+\tilde{M}^{2}-\tilde{Q}^{2}}\right)}}{4}\,.$
(3.10)

Obviously, as the black hole captures a test particle with energy $E$, angular
momentum $L$, and electric charge $q$, the black hole physical parameters
change as ${\tilde{M}}\to{\tilde{M}}+E$, ${\tilde{J}}\to{\tilde{J}}+L$, and
${\tilde{Q}}\to{\tilde{Q}}+q$. Consequently the corresponding black hole
condition changes as well, and the related inequality for a naked singularity
production reads

${({\tilde{M}}+E)^{2}}\left({3-\sqrt{1+\frac{{2{({\tilde{Q}}+q)^{2}}}}{{{({\tilde{M}}+E)^{2}}}}}}\right)<2\sqrt{2}({\tilde{J}}+L){\left({1-\frac{{{({\tilde{Q}}+q)^{2}}}}{{{({\tilde{M}}+E)^{2}}}}+\sqrt{1+\frac{{2{({\tilde{Q}}+q)^{2}}}}{{{({\tilde{M}}+E)^{2}}}}}}\right)^{-\frac{1}{2}}}\,.$
(3.11)

However, solving Eq. (3.11) to obtain the exact form of $E_{\rm max}$ is
troublesome. Thence we omit to do so and rely on some numerical results in the
next section to tell whether the Kaluza-Klein black hole can be destroyed or
not.

Now let us turn to the Kerr-Newman case where the maximum energy for the test
particle is given by the inequality

${\left({M+E}\right)^{2}}<{\left({Q+q}\right)^{2}}+\frac{{{{\left({J+L}\right)}^{2}}}}{{{{\left({M+E}\right)}^{2}}}}\,,$
(3.12)

i.e.

${E_{\max}}=\frac{1}{2}{\left({2{{\left({Q+q}\right)}^{2}}+2\sqrt{{{\left({Q+q}\right)}^{4}}+4{{\left({J+L}\right)}^{2}}}}\right)^{\frac{1}{2}}}-M\,.$
(3.13)

The corresponding near to extremal parameter for Kerr-Newman black hole is

$\delta_{KN}^{2}={{M}^{2}}-{{Q}^{2}}-\frac{{{{J}^{2}}}}{{{{M}^{2}}}}\,,$
(3.14)

and the extremal state is denoted by $\delta_{KN}=0$. Accordingly, solving eq.
(3.14) for $J$ and insert the result into eq. (3.13) gives us

$E_{\max}=\frac{(q+{Q})^{2}+\sqrt{(q+{Q})^{4}+4\left(L+{M}\sqrt{{M}^{2}-{Q}^{2}-\delta_{KN}^{2}}\right)^{2}}}{\sqrt{2}}-{M}\,.$
(3.15)

In the next section we will examine some numerical examples for these $E_{\rm
max}$ and $E_{\rm min}$ above to seek a possibility for the black holes to be
destroyed. The existence of $E_{\rm max}>E_{\rm min}$ for a range of $\delta$
suggests that the black hole can be destroyed by a test particle.

##  4 Destroying the black holes

Here we provide some numerical evaluations of eqs. (3.11) and (3.15) by using
some setups which allow the production of a naked singularity from a near
extremal black holes. In getting the plots for maximum and minimum energies,
we can use the numerical values for $M={\tilde{M}}=100$, $Q={\tilde{Q}}=96$,
$L=0.5$, and $q=10^{-3}$. The results are given in figs. 4.1 and 4.3 where in
both plots we can observe that there is a range of small $\delta$ where
$E_{\rm max}>E_{\rm min}$. From this fact we can infer that the near extremal
Kaluza-Klein and Kerr-Newman black holes can be destroyed by using a test
particle.

Figure 4.1: Plot of $E_{\rm max}$ and $E_{\rm min}$ for a KK black hole with a
neutral test particle. Figure 4.2: Effective potential for the test particle
discussed in Fig. 4.1. Figure 4.3: Plot of $E_{\rm max}$ and $E_{\rm min}$ for
a KN black hole with a neutral test particle. Figure 4.4: Effective potential
for the test particle discussed in fig. 4.3.

Now we need to confirm that the test particle that fits the description in
figs. 4.1 and 4.3 can really reach the horizon. For this purpose, we can use
the effective potential $V_{eff}=-{\dot{r}}^{2}$ method, namely if we can show
that $V_{eff}<0$ along the path then we can assure that the particle can
really fall into the black hole from far distance. To proceed, we can pick an
appropriate case based on the result in figs. 4.1 and 4.3, for example
$\delta_{KK}=\delta_{KN}=5\times 10^{-4}$. For the Kaluza-Klein black hole
case, this implies $E_{\min}=0.002293799446$ and $E_{max}=0.00276891218$.
Therefore we are allowed to consider a test particle with $E=0.0024$ released
from far away whose effective potential is given in fig. 4.2. On the other
hand, the associated energies for the Kerr-Newman case are
$E_{\min}=0.002188411458$ and $E_{max}=0.0021885$. Accordingly we can consider
the test particle with $E=0.218845$, which leads to fig. 4.4. Both effective
potential plots 4.2 and 4.4 suggest that the test particle under
considerations can really touch the horizon after released from far away.

Furthermore, we can also consider the destroying of black holes by a neutral
test particle. We can consider the same setup for test object in both
backgrounds of Kaluza-Klein and Kerr-Newman black holes, namely $L=0.2$ and
$q=0$, and set for the black hole parameters as $M={\tilde{M}}=100$ and
$Q={\tilde{Q}}=94.5$. We find using this numerical values, both near extremal
black holes still can be destroyed by the test particle, as indicated in figs
4.5 and 4.7, where the corresponding effective potentials are given in figs.
4.6 and 4.8.

Figure 4.5: Plot of $E_{\rm max}$ and $E_{\rm min}$ for a KK black hole.
Figure 4.6: Effective potential for the test particle discussed in Fig. 4.1.
Figure 4.7: Plot of $E_{\rm max}$ and $E_{\rm min}$ for a KN black hole.
Figure 4.8: Effective potential for the test particle discussed in fig. 4.3.

##  5 Scalar field perturbation

To support the results in previous section where we have showed that both
black holes under consideration can be destroyed, here we provide an
investigation on extremal Kaluza-Klein black hole destruction by using a
neutral test scalar field. Similar study for Kerr-Newman has been reported in
[32], where Düztaş has shown that neutral test scalar field can destroy an
extremal Kerr-Newman black hole. The argument is based on the fact that
superradiance can occur for a range of frequency $0<\omega<m\Omega_{H}^{\rm
ext.}$ in the background of extremal rotating black holes, including Kerr-
Newman and Kaluza-Klein. Here, $m$ is an integer related to the spheroidal
harmonics of the scalar field, and $\Omega_{H}^{\rm ext.}$ is the angular
velocity at horizon. If the frequency is outside of this range, i.e.
$\omega>m\Omega_{H}^{\rm ext.}$, then the field is absorbed by the black hole
leading to some changes to the black holes properties. Moreover, one can also
show that there exist an upper bound for the frequency of scalar field, namely
$\omega_{\rm{max-ext.}}$, to allow the overspinning of an extremal black hole
leading to a naked singularity production. Therefore, if we can show that
$m\Omega_{H}^{\rm ext.}<\omega_{\rm{max-ext.}}$ for some particular
considerations, then a scalar field with the frequency $m\Omega_{H}^{\rm
ext.}<\omega<\omega_{\rm{max-ext.}}$ can destroy the black hole.

The superradiance effect around Kaluza-Klein black hole has been investigated
in [35]. It was shown that superradiance can occur for Kaluza-Klein black hole
for the incoming neutral test scalar field with frequency
$0<\omega<\omega_{\rm{sl-ext.}}$, where $\omega_{\rm{sl-
ext.}}=m\Omega_{H}^{\rm ext.}$ and the extremal angular velocity at horizon is
given by [35]

$\Omega_{H}^{\rm
ext.}={\frac{\sqrt{2}\left(1-{k}^{2}+\alpha\right)^{3/2}\left(3-\alpha\right)}{2{\tilde{M}}\left(1+\alpha\right)\left(2+2\alpha-10{k}^{2}+4\alpha{k}^{2}-{k}^{4}\right)}}\,.$
(5.1)

Above, we have expressed $\Omega_{H}^{\rm ext.}$ in the conserved mass
$\tilde{M}$, rotational parameter $a$, and charge $\tilde{Q}$. Moreover, we
have used the constant $k$ as the ratio of total charge $\tilde{Q}$ to the
black hole mass $\tilde{M}$, i.e. ${\tilde{Q}}=k{\tilde{M}}$, and
$\alpha=\sqrt{1+2k^{2}}$ for simplicity. To be explicit, a neutral test scalar
field with frequency $0<\omega<\omega_{\rm{sl-ext.}}$ in the background of
extremal Kaluza-Klein black hole can gain some energy from the black hole, and
those with $\omega>\omega_{\rm{sl-ext.}}$ will be absorbed by the black hole.

Now we can obtain an upper bound for the frequency of scalar field which allow
the production of naked singularity from an extremal Kaluza-Klein black hole
using the inequality555Note that this inequality is just eq. (3.11) with
$q=0$.

${({\tilde{M}}+\delta{\tilde{M}})^{2}}\left({3-\sqrt{1+\frac{{2{{\tilde{Q}}^{2}}}}{{{({\tilde{M}}+\delta{\tilde{M}})^{2}}}}}}\right)<\frac{2\sqrt{2}({\tilde{J}}+\delta{\tilde{J}})}{\sqrt{{1-\frac{{{{\tilde{Q}}^{2}}}}{{{({\tilde{M}}+\delta{\tilde{M}})^{2}}}}+\sqrt{1+\frac{{2{{\tilde{Q}}^{2}}}}{{{({\tilde{M}}+\delta{\tilde{M}})^{2}}}}}}}}\,.$
(5.2)

Note that we have set $\delta{\tilde{Q}}=0$ in the inequality above since the
incoming scalar field is neutral. Following [32], we can set
$\delta{\tilde{J}}=m\delta{\tilde{M}}\omega^{-1}$ and
$\delta{\tilde{M}}=\epsilon{\tilde{M}}$ for $\epsilon\ll 1$ as the angular
momentum and energy of incoming test scalar field, respectively. Solving the
last inequality for $\omega$ gives us

$\omega<\omega_{\rm{max-
ext.}}=\frac{{2\sqrt{2}m\epsilon}}{{{\tilde{M}}\left[{\left({\alpha-3}\right){\cal
Y}+\left({3\epsilon-\delta+3}\right){\cal Z}}\right]}}$ (5.3)

where $\delta=\sqrt{1+2\epsilon+\epsilon^{2}+2k^{2}}$, ${\cal
Y}=\sqrt{1-k^{2}+\beta}$, ${\cal
Z}=\sqrt{\epsilon^{2}+2\epsilon+\delta\epsilon+1+\delta-k^{2}}$, and
$\beta=\sqrt{1+2k}$.

Obviously, the incorporating expressions for $\omega_{\rm{max-ext.}}$ and
$\omega_{\rm{sl-ext.}}$ in eqs. (5.3) and (5.1) above are considerably more
complicated compared to that of Kerr-Newman case reported in [32]. This
hinders us to obtain some exact bound for $k$ allowing $\omega_{\rm{max-
ext.}}>\omega_{\rm{sl-ext.}}$ in justifying the naked singularity production.
Nevertheless, we can define a quantity

$\Delta_{\omega}=\frac{{\tilde{M}}\left(\omega_{\rm{max-ext.}}-\omega_{\rm{sl-
ext.}}\right)}{\sqrt{2}{m}}\,,$ (5.4)

which measures the gap between $\omega_{\rm{max-ext.}}$ and $\omega_{\rm{sl-
ext.}}$, and check it numerically for some appropriate numerical examples. The
result of $\Delta_{\omega}>0$ tells us that the absorption of a test scalar
field leading to naked singularity production is possible for an extremal
Kaluza-Klein black hole. A numerical example is given in fig. 5.1 where we
find there exist $\Delta_{\omega}>0$ for a range of $k$. This result support
our previous claim that a Kaluza-Klein black hole can be destroyed.

Figure 5.1: Plot of $\Delta_{\omega}$ vs. $k$ for $\epsilon=10^{-9}$.

##  6 Conclusions

In this paper, we have studied the naked singularities production from Kerr-
Newman and Kaluza-Klein black holes. Both black holes are charged and
rotating, hence the destruction can be done by overspinning or overcharging
them. In section 4, we provide some numerical results for the maximum and
minimum of a test body which may lead to horizon destruction. Indeed, further
numerical evaluations of the effective potentials for some possible setups
leading to black hole destruction are needed, as we have provided in that
section. The effective potentials that we obtain confirm that the test body
which can destroy the black hole can really fall into the horizon from far
away.

To support the results in section 4, section 5 is devoted to study the
overspinning of an extremal Kaluza-Klein black hole by a neutral scalar field.
This section is motivated by the work in [32] where the author showed how to
overspin an extremal Kerr-Newman black hole by using a neutral scalar field.
However, due to the complexity of incorporated properties for the Kaluza-Klein
black holes, we cannot provide an analytic expression for the ratio between
black hole charge $\tilde{Q}$ to its mass $\tilde{M}$ which allows the
destruction. The results presented in section 5 is limited to some numerical
examples which suggest that an extremal Kaluza-Klein black hole can be
overspun by a scalar field.

Nowadays, there is a growing interest in checking the weak cosmic censorship
conjecture by using the method developed by Wald and Sorce [21] and it has
been applied to some various cases [36, 37, 38]. Using this approach, it can
be shown that a black hole cannot be overspun or overcharged leading to the
violation of weak cosmic censorship conjecture. Performing this method by Wald
and Sorce to Kaluza-Klein black hole should be an interesting project and we
address it to our future work.

## Acknowledgement

This work is supported by Lembaga Penelitian Dan Pengabdian Kepada Masyarakat,
Parahyangan Catholic University.

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