# A proof of the Etzion-Silberstein conjecture for monotone and MDS-
constructible Ferrers diagrams

Alessandro Neri  Department of Mathematics: Analysis, Logic and Discrete
Mathematics, Ghent University, Belgium Mima Stanojkovski  Dipartimento di
Matematica, Università di Trento, Italia

###### Abstract

Ferrers diagram rank-metric codes were introduced by Etzion and Silberstein in
2009. In their work, they proposed a conjecture on the largest dimension of a
space of matrices over a finite field whose nonzero elements are supported on
a given Ferrers diagram and all have rank lower bounded by a fixed positive
integer $d$. Since stated, the Etzion-Silberstein conjecture has been verified
in a number of cases, often requiring additional constraints on the field size
or on the minimum rank $d$ in dependence of the corresponding Ferrers diagram.
As of today, this conjecture still remains widely open. Using modular methods,
we give a constructive proof of the Etzion-Silberstein conjecture for the
class of strictly monotone Ferrers diagrams, which does not depend on the
minimum rank $d$ and holds over every finite field. In addition, we leverage
on the last result to also prove the conjecture for the class of MDS-
constructible Ferrers diagrams, without requiring any restriction on the field
size.

_Keywords_ : Rank-metric codes, Ferrers diagrams, Etzion-Silberstein
conjecture, monotone diagrams, convex diagrams, MDS-constructible diagrams.
2020 _Mathematics Subject Classification_ : 11T71, 15A03, 16S35, 94B05, 94B60.

##  1 Introduction

Let $\mathbb{F}$ be a finite field and let $1\leq d\leq n$ and $1\leq k\leq
n^{2}$ be integers. Let, moreover, $\mathbb{F}^{n\times n}$ denote the
collection of $n\times n$ matrices with coefficients in $\mathbb{F}$. Then the
map

$\mathbb{F}^{n\times n}\times\mathbb{F}^{n\times
n}\longrightarrow\mathbb{Z}_{\geq
0},\quad(A,B)\longmapsto\operatorname{rk}(A-B)$

defines a metric on $\mathbb{F}^{n\times n}$. An $[n\times
n,k,d]_{\mathbb{F}}$ _rank-metric code_ is a $k$-dimensional linear subspace
$\mathcal{C}$ of $\mathbb{F}^{n\times n}$ such that $d$ is the _minimum rank_
among the nonzero elements of $\mathcal{C}$. Rank-metric codes have a long
mathematical history [4, 7] and have become very popular in the field of
algebraic coding theory since the discovery of their applications to network
coding [15]. Thanks to the work of Delsarte [4], the parameters of an
$[n\times n,k,d]_{\mathbb{F}}$ rank-metric code are related by the following
Singleton-like Bound:

$k\leq n(n-d+1).$ (1)

Codes meeting the last bound with equality are called _maximum rank distance
codes_ (MRDs, for short) and constructions of such codes are known over finite
fields and can be found in [4, 7].

If rank-metric codes allow for codes within the entire matrix space
$\mathbb{F}^{n\times n}$, the more general _Ferrers diagram rank-metric codes_
are rank-metric codes with prescribed support. These were introduced in 2009
by Etzion and Silberstein [6] motivated by the application of subspace codes
in network coding and arise from subspace codes in a fixed Schubert cell. The
study of Ferrers diagram rank-metric codes has been carried on since by
various authors [2, 3, 5, 8, 10, 12, 13].

Let $[n]=\\{1,\ldots,n\\}$ and let $\mathcal{D}\subseteq[n]^{2}$ be a top-
right justified _Ferrers diagram_ (cf. Definition 2.2) outside of which the
elements of the code will have entries equal to $0$. A
$[\mathcal{D},k,d]_{\mathbb{F}}$ _Ferrers diagram rank-metric code_ is an
$[n\times n,k,d]_{\mathbb{F}}$ rank-metric code $\mathcal{C}$ such that if
$A=(a_{i,j})\in\mathcal{C}$ and $(x,y)\in[n]^{2}\setminus\mathcal{D}$, then
$a_{x,y}=0$. Two immediate examples are:

  1. $(1)$

If $\mathcal{D}=[n]^{2}$, then a $[\mathcal{D},k,d]_{\mathbb{F}}$ Ferrers
diagram rank-metric code is the same as an $[n\times n,k,d]_{\mathbb{F}}$
rank-metric code.

  2. $(2)$

If $\mathcal{D}=\mathcal{T}_{n}=\\{(i,j):i,j\in[n]\textup{ with }i\leq j\\}$,
then a $[\mathcal{T}_{n},k,d]_{\mathbb{F}}$ Ferrers diagram rank-metric code
is an $[n\times n,k,d]_{\mathbb{F}}$ rank-metric code that is contained in the
space of upper triangular $n\times n$ matrices with coefficients over
$\mathbb{F}$.

Adding constraints to the support of the matrices in the code clearly
influences the relation between its intrinsic parameters. In [6], the upper
bound on the dimension given in (1) is generalized to an explicit upper bound
$\nu_{\min}(\mathcal{D},d)$ taking into account the prescribed support
$\mathcal{D}$; cf. (6) for the exact value of $\nu_{\min}(\mathcal{D},d)$. For
instance:

  1. $(1)$

If $\mathcal{D}=[n]^{2}$, then $\nu_{\min}(\mathcal{D},d)=n(n-d+1)$ recovers
the bound in (1).

  2. $(2)$

If $\mathcal{D}=\mathcal{T}_{n}$, then $\nu_{\min}(\mathcal{D},d)$ is equal to

$\nu_{\min}(\mathcal{T}_{n},d)=\sum_{i=1}^{n-d+1}i=\binom{n-d+2}{2}=\frac{(n-d+1)(n-d+2)}{2}.$

A $[\mathcal{D},\nu_{\min}(\mathcal{D},d),d]_{\mathbb{F}}$ Ferrers diagram
rank-metric code is called a _maximum Ferrers diagram code_ (MFD, for short):
Etzion and Silberstein conjectured that these codes exist for every choice of
$d$ and $\mathcal{D}$, and over every finite field $\mathbb{F}$. This is now
known as the _Etzion-Silberstein conjecture_ and, as of today, it is still
open in its full generality. In particular and to the best of our knowledge,
not even the case of upper triangular matrices has been tackled completely.
The instances of existing
$[\mathcal{T}_{n},\nu_{\min}(\mathcal{T}_{n},d),d]_{\mathbb{F}}$ MFD codes
that are known to us and that do not impose any restriction on the field
$\mathbb{F}$ are the following:

  * •

The case where $d=1$, which consists of all the upper triangular matrices.

  * •

The case where $d=2$, for which an MFD code is given by the subspace of upper
triangular matrices where every diagonal sum is zero.

  * •

The case where $d=3$, which is settled by [2, Corollary 3.10].

  * •

The case where $d=n-1$, which is given by [2, Theorem 5.3] (for a constructive
proof see [1, Section 2.4]).

All other cases are settled with the additional assumption that
$|\mathbb{F}|\geq n-1$, as shown in [5]. This is due to the fact that the pair
$(\mathcal{T}_{n},d)$ is _MDS-constructible_ for every $d$ and $n$; cf. Remark
2.13.

In this paper, we fill the gap by providing a constructive proof of the
Etzion-Silberstein conjecture for the class of MDS-constructible Ferrers
diagrams without imposing any condition on the field size; cf. Theorem 4.22.
We do this by first proving the conjecture for the subclass of _strictly
monotone Ferrers diagrams_ ; cf Theorem 4.10. These are Ferrers diagram that
do not allow two non-empty columns to have the same number of elements, and
include therefore the upper triangular Ferrers diagrams $\mathcal{T}_{n}$ as a
special subclass. By adjunction, we also prove the Etzion-Silberstein
conjecture for _initially convex_ Ferrers diagrams; cf. Definition 4.4.

We conclude this introduction with a short account of our methods. To produce
MFD codes for strictly monotone Ferrers diagrams, we extend Delsarte-Gabidulin
constructions of MRD codes, which are realized through the identification of
$\mathbb{F}^{n\times n}$ with a skew algebra $\mathbb{L}[\sigma]$, where
$\mathbb{L}$ is an auxiliary cyclic field extension of $\mathbb{F}$ of degree
$n$ with Galois group generated by $\sigma$. In Theorem 3.4, for a given
finite field $\mathbb{F}$ of characteristic $p$ and $n=p^{m}$, we determine a
skew algebra representation of the space of matrices supported on a Ferrers
diagram $\mathcal{D}\subseteq[n]^{2}$, when $\mathcal{D}$ is _$p$ -monotone_;
cf. Definition 2.22. The crucial properties used are that $\mathbb{L}$ has
degree $p^{m}$ over $\mathbb{F}$ and that $\sigma-\operatorname{id}$ is a
nilpotent $\mathbb{F}$-linear endomorphism of $\mathbb{L}$. As a byproduct, we
also prove the Etzion-Silberstein conjecture for $p$-monotone and $p$-convex
Ferrers diagrams over finite fields of characteristic $p$ and where $n$ is a
power of $p$; cf. Theorem 4.2. For strictly monotone (and thus initially
convex) Ferrers diagrams, we are able to move away from the modular case by
showing that the property of being strictly monotone is preserved by
embeddings. Finally, we further extend our results to the class of MDS-
constructible Ferrers diagrams, relying on the combinatorial features of their
diagonals.

Organization of the paper. In Section 2 we set the notation for the whole
paper, properly define Ferrers diagram codes and state the Etzion-Silberstein
conjecture in its full generality. Additionally, we define MDS-constructible
and monotone Ferrers diagrams as well as the class of $p$-monotone Ferrers
diagrams and their adjoints. In Section 3, we concentrate on the case where
$n=p^{m}$ is a power of the characteristic $p$ and, given a field extension
$\mathbb{L}$ of $\mathbb{F}$ of degree $n$, we study the symmetries of a
natural full flag in $\mathbb{L}$. We exploit the results of Section 3 in
Section 4, where we prove the Etzion-Silberstein conjecture for $p$-monotone
Ferrers diagrams over finite fields of characteristic $p$. As corollaries we
derive that the Etzion-Silberstein conjecture holds true over any finite field
for the classes of strictly monotone, initially convex, and MDS-constructible
Ferrers diagrams.

Acknowledgements. Alessandro Neri is supported by the Research
Foundation–Flanders (FWO) grant 12ZZB23N. Mima Stanojkovski is funded by the
Italian program Rita Levi Montalcini, Edition 2020. They thank René Schoof for
a short proof of Proposition 3.3.

##  2 Notation and preliminaries

Let $n$ be a positive integer and let $\mathbb{F}$ be a field with a cyclic
Galois extension $\mathbb{L}$ of degree $n$. Fixing a generator $\sigma$ of
the Galois group $\operatorname{Gal}(\mathbb{L}/\mathbb{F})$, the
$\mathbb{F}$-algebra $(\operatorname{End}_{\mathbb{F}}(\mathbb{L}),+,\circ)$
can be represented as a skew group algebra $(\mathbb{L}[\sigma],+,\cdot)$.
This is defined as the set

$\mathbb{L}[\sigma]=\left\\{\sum_{i=0}^{n-1}a_{i}\sigma^{i}\,:\,a_{i}\in\mathbb{L}\right\\}$

endowed with usual addition, while the multiplication is defined on monomials
via

$(a\sigma^{i})\cdot(b\sigma^{j})=a\sigma^{i}(b)\sigma^{i+j},\qquad\mbox{ for
}a,b\in\mathbb{L},$

and extended by distributivity to $\mathbb{L}[\sigma]$. The map
$\mathbb{L}[\sigma]\rightarrow\operatorname{End}_{\mathbb{F}}(\mathbb{L})$
given by

$\begin{array}[]{ccc}\mathbb{L}[\sigma]&\longrightarrow&\operatorname{End}_{\mathbb{F}}(\mathbb{L})\\\
\sum\limits_{i=0}^{n-1}a_{i}\sigma^{i}&\longmapsto&\Big{(}\alpha\longmapsto\sum\limits_{i=0}^{n-1}a_{i}\sigma^{i}(\alpha)\Big{)}\end{array}$
(2)

is an $\mathbb{F}$-algebra isomorphism as a consequence of Artin’s Theorem of
linear independence of characters. By choosing any $\mathbb{F}$-basis
$\mathcal{B}$ of $\mathbb{L}$, one obtains an induced $\mathbb{F}$-algebra
isomorphism

$\mathbb{L}[\sigma]\ \ \longrightarrow\ \ \mathbb{F}^{n\times n}.$ (3)

This representation lies at the heart of the most prominent constructions of
families of optimal rank-metric codes, as we will see later in Section 2.1.

Throughout, we identify every element $p(\sigma)\in\mathbb{L}[\sigma]$ with
its image in $\operatorname{End}_{\mathbb{F}}(\mathbb{L})$ under (2): we
denote its evaluation in $\alpha\in\mathbb{L}$ by $p(\sigma)(\alpha)$. In
addition, the $\sigma$-degree (or simply degree) of a nonzero element
$p(\sigma)=a_{0}\operatorname{id}+a_{1}\sigma+\ldots+a_{n-1}\sigma^{n-1}\in\mathbb{L}[\sigma]$
is defined in the usual way as

$\deg p(\sigma)=\max\\{i\in\\{0,\ldots,n-1\\}\,:\,a_{i}\neq 0\\}.$

Artin’s theorem of linear independence of characters ensures then that, for
any nonzero $\sigma$-polynomial
$p(\sigma)=a_{0}\operatorname{id}+a_{1}\sigma+\ldots+a_{n-1}\sigma^{n-1}\in\mathbb{L}[\sigma]$,
the following holds:

$\dim_{\mathbb{F}}(\ker p(\sigma))\leq\deg p(\sigma).$ (4)

This last statement can be found for instance in [9, Theorem 5], although it
can also be derived from earlier works; see e.g. [4, 11].

###  2.1 Ferrers diagram rank-metric codes

Rank-metric codes were first introduced by Delsarte in [4] and then
reintroduced by Gabidulin in [7]. They have gained in popularity thanks to
their application to network coding [15] and for further applications in
distributed storage and cryptography.

######  Definition 2.1.

Let $\mathbb{F}$ be a field and $k,m,n,d$ positive integers. An $[n\times
m,k,d]_{\mathbb{F}}$ rank-metric code is a $k$-dimensional
$\mathbb{F}$-subspace $\mathcal{C}$ of $\mathbb{F}^{n\times m}$, where
$d=\min\\{\operatorname{rk}(A):A\in\mathcal{C}\setminus\\{0\\}\\}$ is called
the minimum rank distance of $\mathcal{C}$.

The parameters $k,m,n,d$ of a rank-metric code depend from each other through
the well-known Singleton-like bound, originally proved by Delsarte in [4] and
reading:

$k\leq\min\\{m(n-d+1),n(m-d+1)\\}.$ (5)

Codes meeting the bound of (5) with equality are called maximum rank distance
codes, or simply MRD codes. When $\mathbb{F}$ admits a degree $\max\\{n,m\\}$
cyclic extension field $\mathbb{L}$, there is a well-known construction of MRD
codes for every choice of $d,n,m$ with $1\leq d\leq\min\\{m,n\\}$, which is
due to Delsarte [4] for finite fields and to Guralnick [11] for general
fields. Given $\sigma$ a generator of
$\operatorname{Gal}(\mathbb{L}/\mathbb{F})$, this construction goes as
follows.

If $n=m$, then the $\mathbb{L}$-subspace

$\mathbb{L}[\sigma]_{n-d}=\left\\{\sum_{i=0}^{n-d}a_{i}\sigma^{i}\,:\,a_{i}\in\mathbb{L}\right\\}\subseteq\mathbb{L}[\sigma].$

has $\mathbb{F}$-dimension $n(n-d+1)$ and can be seen as an $[n\times
n,n(n-d+1)]_{\mathbb{F}}$ code, due to the isomorphism in (3). Furthermore, by
(4) it is clear that the minimum rank distance of $\mathcal{C}$ is $d$,
implying that $\mathbb{L}[\sigma]_{n-d}$ is an MRD code.

If $m<n$, then by the last case we know that $\mathbb{L}[\sigma]_{m-d}$ is an
$[n\times n,n(m-d+1),d+n-m]_{\mathbb{F}}$ MRD code. Projecting
$\mathbb{L}[\sigma]_{m-d}$ to $\mathbb{F}^{n\times m}$ by erasing any set of
$n-m$ columns, we then obtain an $[n\times m,n(m-d+1),d]_{\mathbb{F}}$ MRD
code.

If $m>n$, one reduces to the previous case by interchanging the values $n$ and
$m$ and transposing the matrices.

In 2009, motivated by the application of subspace codes to network coding,
Etzion and Silberstein initiated the study of Ferrers diagram rank-metric
codes [6]. These are rank-metric codes whose elements have entries which are
identically zero outside of a given Ferrers diagram, and arise from subspace
codes in a fixed Schubert cell.

######  Definition 2.2.

A (top-right justified) Ferrers diagram of order $n$ is a subset $\mathcal{D}$
of $[n]^{2}$ satisfying the following properties:

  * •

The set $\mathcal{D}$ is not empty.

  * •

If $(i,j)\in\mathcal{D}$ and $j^{\prime}\in\\{j,\ldots,n\\}$, then
$(i,j^{\prime})\in\mathcal{D}$.

  * •

If $(i,j)\in\mathcal{D}$ and $i^{\prime}\in\\{1,\ldots,i\\}$, then
$(i^{\prime},j)\in\mathcal{D}$.

Every Ferrers diagram $\mathcal{D}$ can be graphically represented as a top-
right justified grid of dots. We will often use this representation in order
to help the reader graphically catch the features of the Ferrers diagrams that
we will deal with in this paper. For compactness, however, we will also
identify a Ferrers diagram $\mathcal{D}$ with the vector
$(c_{1},\ldots,c_{n})$, where $c_{i}\in\\{0,\ldots,n\\}$ is the number of dots
in the column $i$ of the grid: in view of Definition 2.2 it then holds that
$0\leq c_{1}\leq\ldots\leq c_{n}\leq n$.

######  Remark 2.3.

We have already seen that, if a Ferrers diagram is given as a subset of
$[n]^{2}$, then its vector representation $(c_{1},\ldots,c_{n})$ is obtained
as

$c_{j}=|\\{i\in[n]\,:\,(i,j)\in\mathcal{D}\\}|.$

Conversely, if $\mathcal{D}$ is given by a vector $(c_{1},\ldots,c_{n})$, then
its representation as a subset of $[n]^{2}$ is derived as

$\mathcal{D}=\\{(i,j)\in[n]^{2}\,:\,i\leq c_{j}\\}.$

These maps are inverses to each other and we will use the two representations
of a Ferrers diagram interchangeably.

######  Example 2.4.

The Ferrers diagram of order $5$ given by $\mathcal{D}=(0,1,1,4,5)$ is
graphically represented by the following dotted grid.

There is a natural notion of adjunction for Ferrers diagrams, which
corresponds to the conjugate partition of an integer.

######  Definition 2.5.

Let $\mathcal{D}$ be a Ferrers diagram of order $n$. Then, the adjoint Ferrers
diagram $\mathcal{D}^{\top}$ is defined as

$\mathcal{D}^{\top}=\\{(n+1-j,n+1-i)\,:\,(i,j)\in\mathcal{D}\\}.$

In other words, the adjoint Ferrers diagram is obtained by transposing the
indices of the original Ferrers diagram with respect to the main antidiagonal.
We remark that, if $\mathcal{D}$ is a Ferrers diagram, then
$\mathcal{D}^{\top}$ is the Ferrers diagram obtained from the reflection, with
respect to the main antidiagonal, of a the graphical representation of
$\mathcal{D}$. This is illustrated in the following example.

######  Example 2.6.

The adjoint $\mathcal{D}^{\top}$ of the Ferrers diagram
$\mathcal{D}=(0,1,1,4,5)$ from Example 2.4 is $\mathcal{D}^{\top}=(1,2,2,2,4)$
and is graphically represented by the following.

For a given Ferrers diagram $\mathcal{D}=(c_{1},\ldots,c_{n})$ of order $n$,
denote by $\mathbb{F}^{\mathcal{D}}$ the space of $n\times n$ matrices over
$\mathbb{F}$ supported on $\mathcal{D}$, that is,

$\mathbb{F}^{\mathcal{D}}=\\{A=(a_{i,j})\in\mathbb{F}^{n\times
n}\,:\,i>c_{j}\textup{ implies }a_{i,j}=0\\}\subseteq\mathbb{F}^{n\times n}.$

######  Example 2.7.

If $\mathcal{D}=(1,2,3,\ldots,n)$, then $\mathbb{F}^{\mathcal{D}}$ is nothing
else than the collection of upper triangular $n\times n$ matrices over
$\mathbb{F}$. In this case, $\mathcal{D}$ satisfies
$\mathcal{D}=\mathcal{D}^{\top}$.

######  Definition 2.8.

Let $\mathcal{D}$ be a Ferrers diagram of order $n$. A
$[\mathcal{D},k,d]_{\mathbb{F}}$ Ferrers diagram rank-metric code is a
$k$-dimensional $\mathbb{F}$-subspace $\mathcal{C}$ of
$\mathbb{F}^{\mathcal{D}}$ endowed with the rank distance. The parameter $d$
is the minimum rank distance of $\mathcal{C}$ and is defined as

$d=\min\\{\operatorname{rk}(A)\,:\,A\in\mathcal{C},A\neq 0\\}.$

In their seminal paper [6], Etzion and Silberstein showed the following
relation between the parameters of a $[\mathcal{D},k,d]_{\mathbb{F}}$ code
$\mathcal{C}$, relying on the classical argument that $\mathcal{C}$ trivially
intersects any subspace of $\mathbb{F}^{\mathcal{D}}$ whose elements have rank
at most $d-1$.

######  Proposition 2.9 ([6, Theorem 1]).

Let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a Ferrers diagram of order $n$, and
let $\mathcal{C}$ be a $[\mathcal{D},k,d]_{\mathbb{F}}$ code. Then

$k\leq\min_{j=0,\ldots,d-1}\left\\{\sum_{i=1}^{n-j}\max\\{0,c_{i}-d+1+j\\}\right\\}.$

The quantity on the right-hand side of the bound in Proposition 2.9, which we
denote

$\nu_{\min}(\mathcal{D},d)=\min_{j=0,\ldots,d-1}\left\\{\sum_{i=1}^{n-j}\max\\{0,c_{i}-d+1+j\\}\right\\},$
(6)

represents, informally speaking, the minimum, among all the
$j\in\\{0,\ldots,d-1\\}$ of the number of dots remaining in the Ferrers
diagram $\mathcal{D}$ after removing the first $d-j-1$ rows and the last $j$
columns from $\mathcal{D}$. From this interpretation, it is immediately clear
that $\nu_{\min}(\mathcal{D},d)=\nu_{\min}(\mathcal{D}^{\top},d)$.

######  Example 2.10.

Let $\mathcal{D}=(0,1,1,4,5)$ be as in Example 2.4 and set $d=3$. Then

$\nu_{\min}(\mathcal{D},d)=\min_{j=0,1,2}\left\\{\sum_{i=1}^{5-j}\max\\{0,c_{i}-2+j\\}\right\\}=\min\\{2+3,3,1+1\\}=2,$

which is the same as the minimum among the number of dots remaining after
perfoming the following deletions:

In the same paper [6], the authors also conjectured that over finite fields
the bound given in Proposition 2.9 is tight for any Ferrers diagram. This is
now known as the Etzion-Silberstein conjecture and is stated below.

######  Conjecture 2.11 ([6, Conjecture 1]).

Let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a Ferrers diagram of order $n$, and
let $d\in\\{1,\ldots,n\\}$. If $\mathbb{F}$ is a finite field, then there
exists a $[\mathcal{D},k,d]_{\mathbb{F}}$ code $\mathcal{C}$ with

$k=\nu_{\min}(\mathcal{D},d).$

A code meeting the bound from Proposition 2.9 with equality is said to be a
maximum Ferrers diagram (MFD) code. Thus, the Etzion-Silberstein conjecture
(Conjecture 2.11) states that MFD codes exist over any finite field and for
every Ferrers diagram of order $n$ and minimum rank distance $1\leq d\leq n$.
Since $d=1$ is a trivial instance of the problem, we will often directly
consider $d\geq 2$.

Though Conjecture 2.11 is still widely open, there is a number of cases in
which it has been confirmed; see [12, Section 2.3] for a summary. Two main
techniques have been used for this: the construction of MFD codes as subcodes
of MRD codes and the concept of MDS-constructibility.

The first one allows to completely settle the case where $d=2$: see for
instance [6, Theorem 2], [8, Corollary 19]), and [2, Theorem III.1]. For
$d\geq 3$, the structure of MRD codes has been exploited in various ways for
the construction of MFD codes, often with additional restrictions on the
minimum rank $d$ depending on the Ferrers diagram $\mathcal{D}$. This method
can already be found in Etzion and Silberstein’s seminal paper [6], and it has
been improved and refined ever since (see e.g. [2, 5, 8, 12, 13]). A summary
of the MRD-subcodes constructions can be found in [13, Theorem 2.8].

The second general method uses the theory of MDS codes in the Hamming metric,
but only allows to prove Conjecture 2.11 over “large enough” finite fields and
for Ferrers diagrams with the property of being _MDS-constructible_ ; cf.
Definition 2.12. In the following, for $i\in[n]$, let

$\Delta_{i}^{n}=\\{(j,j+i-1):j\in[n-i+1]\\}$

be the $i$-th diagonal of an $n\times n$ grid.

######  Definition 2.12.

Let $\mathcal{D}\subseteq[n]^{2}$ be a Ferrers diagram of order $n$, and let
$d\in[n]$ be a positive integer. The pair $(\mathcal{D},d)$ is MDS-
constructible if

$\nu_{\min}(\mathcal{D},d)=\sum_{i=1}^{n}\max\\{0,|\mathcal{D}\cap\Delta_{i}^{n}|-d+1\\}.$

We remark that, since $|\Delta_{i}^{n}|=n-i+1$, for an MDS-constructible pair
$(\mathcal{D},d)$ one has

$\nu_{\min}(\mathcal{D},d)=\sum_{i=1}^{n-d+1}\max\\{0,|\mathcal{D}\cap\Delta_{i}^{n}|-d+1\\}.$
(7)

The notion of MDS-constructible pairs was introduced in [2], but it was
already shown in [5, Theorem 7] that
$[\mathcal{D},\nu_{\min}(\mathcal{D},d),d]_{\mathbb{F}}$ MFD codes exists (and
can be explicitly constructed) whenever
$|\mathbb{F}|\geq\max\\{|\mathcal{D}\cap\Delta_{i}^{n}|-d\,:\,i\in[n]\\}$. The
rough idea is to embed MDS codes of length $|\mathcal{D}\cap\Delta_{i}^{n}|$
and Hamming distance $d$ into $\mathcal{D}\cap\Delta_{i}^{n}$ and then take
the direct sum of these spaces. For a detailed description of this
construction we refer the reader to [5, Construction 1]. We remark that a
similar diagonal construction for rank metric codes was already given in [14,
Theorem 3]. Recently, existence results for MFD codes over some MDS-
constructible pairs were derived in [10], with partial improvements on the
field size.

######  Remark 2.13.

It is readily seen that the upper triangular Ferrers diagrams fall into this
category. In other words, for positive integers $n$ and $1\leq d\leq n$, if
$\mathcal{T}_{n}=\\{(i,j)\,:\,i,j\in[n],i\leq j\\}$ denotes the upper-
triangular $n\times n$ Ferrers diagram, then the pair $(\mathcal{T}_{n},d)$ is
MDS-constructible.

In Section 4 we prove 2.11 for the family of MDS-constructible Ferrers
diagrams over any finite field. To get rid of the restriction on the field
size, we exploit MRD-subcode constructions. The novelty of our approach is the
use of an auxiliary extension field whose degree is a power of the
characteristic.

###  2.2 Monotone and convex Ferrers diagrams

Two special families of Ferrers diagrams which we will focus on, are those of
$p$-monotone and $p$-convex Ferrers diagrams, where $p$ is a prime number. We
define them both in this section.

######  Definition 2.14.

A Ferrers diagram $\mathcal{D}=(c_{1},\ldots,c_{n})$ of order $n$ is called
monotone if, whenever $i\in\\{1,\ldots,n-1\\}$ is such that $0<c_{i}<n$, one
has $c_{i+1}>c_{i}$.

######  Example 2.15.

The following are examples of monotone Ferrers diagrams of order $n=5$.

$\mathcal{D}=(0,0,1,3,4),\;\;\mathcal{D}=(1,2,4,5,5),\;\;\mathcal{D}=(2,3,5,5,5),\;\;\mathcal{D}=(0,1,4,5,5)$

######  Remark 2.16.

Let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a monotone Ferrers diagram of order
$n$, and define

$\ell=\min\\{i\,:\,c_{i}\neq 0\\}\ \ \textup{ and }\ \
r=\max\\{i\,:\,c_{i}\neq n\\}.$

Then, for every $i,j\in\\{\ell,\ell+1,\ldots,r+1\\}$ with $i<j$, one has
$c_{j}-c_{i}\geq j-i$.

######  Remark 2.17.

Let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a monotone Ferrers diagram of order
$n$. If $i,j<n$ are such that $(i,j)\in\mathcal{D}$, then monotonicity yields
$(i+1,j+1)\in\mathcal{D}$.

######  Definition 2.18.

A Ferrers diagram $\mathcal{D}=(c_{1},\ldots,c_{n})$ of order $n$ is called
convex if, for all $i\in\\{1,\ldots,n-1\\}$, one has that $c_{i+1}-c_{i}\leq
1$.

######  Example 2.19.

The following are examples of convex Ferrers diagrams of order $n=5$.

$\mathcal{D}=(0,1,2,2,3),\;\;\mathcal{D}=(2,3,3,4,5),\;\;\mathcal{D}=(3,3,4,5,5),\;\;\mathcal{D}=(2,3,3,3,4)$

We remark that convex Ferrers diagrams are adjoints of monotone Ferrers
diagrams and the other way around, as one can see from Example 2.15 and
Example 2.19.

######  Definition 2.20.

Let $p$ be a prime number and let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a
Ferrers diagram of order $n$. Let $h$ be the largest non-negative integer with
the following properties:

  1. (1)

The power $p^{h}$ divides $n$.

  2. (2)

For every $i\in\\{1,\ldots,n\\}$, the power $p^{h}$ divides $c_{i}$.

  3. (3)

If $r,s\in\\{0,\ldots,n-1\\}$ and $Q$ is a non-negative integer such that
$0\leq r-Qp^{h},s-Qp^{h}\leq p^{h}-1$, then $c_{n-r}=c_{n-s}$.

The number $h$ is called the $p$-height of $\mathcal{D}$ and the
$p$-contraction of $\mathcal{D}$ is the Ferrers diagram
$\mathcal{D}^{(p)}=(c_{1}^{\prime},\ldots,c_{n/p^{h}}^{\prime})$ of order
$n/p^{h}$ that is defined by

$c_{i}^{\prime}=\frac{c_{p^{h}i}}{p^{h}},\quad i\in\\{1,\ldots,n/p^{h}\\}.$

We remark that, in the context of Definition 2.20, the value $h=0$ satisfies
(1), (2), and (3) so the $p$-height of $\mathcal{D}$ is well defined.
Moreover, if $\mathcal{D}$ is a Ferrers diagram of order $n$ of $p$-height
$h=0$, then the $p$-contraction of $\mathcal{D}$ is equal to $\mathcal{D}$.
Roughly speaking, as the following example suggests, if $h$ is the $p$-height
of a Ferrers diagram $\mathcal{D}$, then $[n^{2}]$ can be partitioned into
blocks of size $p^{h}\times p^{h}$ where the intersection of $\mathcal{D}$
with any block is either full or empty.

######  Example 2.21.

The Ferrers diagram $\mathcal{D}=(4,4,4,4,8,8,8,8)$ of order $8$ has
$2$-height equal to $2$. Its $2$-contraction is the Ferrers diagram
$\mathcal{D}^{(2)}=(1,2)$ of order $2$.

$\overrightarrow{\ 2\mbox{-contraction }}$

######  Definition 2.22.

Let $p$ be a prime number and let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a
Ferrers diagram of order $n$. The diagram $\mathcal{D}$ is called $p$-monotone
if its $p$-contraction is monotone. Similarly, $\mathcal{D}$ is $p$-convex if
its $p$-contraction is convex.

We remark that, with this new definition, monotone Ferrers diagrams are a
special instance of the larger family of $p$-monotone Ferrers diagrams.

######  Example 2.23.

The following are examples of $2$-monotone Ferrers diagrams of order $8$.

$\mathcal{D}=(0,0,2,2),\qquad\mathcal{D}=(2,2,4,4,6,6),\qquad\mathcal{D}=(0,0,4,4,8,8,8,8,8)$

The following straightforward result asserts that $p$-convex Ferrers diagrams
and $p$-monotone Ferrers diagrams are adjoint to each other.

######  Lemma 2.24.

Let $p$ be a prime number and let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a
Ferrers diagram of order $n$. Then $\mathcal{D}$ is $p$-monotone if and only
if $\mathcal{D}^{\top}$ is $p$-convex.

In the next section, we will describe a very useful representation of Ferrers
diagram matrix spaces $\mathbb{F}^{\mathcal{D}}$ when $\mathcal{D}$ is
$p$-monotone. Furthermore, by adjunction – that is, transposing with respect
to the antidiagonal – we automatically obtain also a representation of
$\mathbb{F}^{\mathcal{D}}$ when $\mathcal{D}$ is $p$-convex.

The following remark generalizes Remark 2.16 to the case of $p$-monotone
Ferrers diagrams.

######  Remark 2.25.

Let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a $p$-monotone Ferrers diagram of
order $n$ with $p$-height equal to $h$. Set $\ell=\min\\{i\,:\,c_{i}\neq 0\\}$
and $r=\max\\{i\,:\,c_{i}\neq n\\}.$ It follows from $p$-monotonicity that

$\ell=1+sp^{h}\ \ \textup{and}\ \ r=tp^{h},\ \ \textup{ for some
}s,t\in\\{0,\ldots,n/p^{h}\\}.$

The Ferrers diagram $\mathcal{D}$ being monotone, we derive that, if
$a,b\in\\{s,\ldots,t+1\\}$, then

$a>b\ \ \Longrightarrow\ \ c_{ap^{h}}-c_{bp^{h}}\geq(a-b)p^{h}.$

######  Remark 2.26.

We note that, in the special case where $\mathcal{D}$ is a $p$-monotone
Ferrers diagram, for every choice of an integer $1\leq d\leq n$, the expected
maximum dimension is

$\nu_{\min}(\mathcal{D},d)=\sum_{i=1}^{n-d+1}c_{i}.$

In other words, $\nu_{\min}(\mathcal{D},d)$ can be computed by exclusively
deleting columns from the Ferrers diagram $\mathcal{D}$. This is a direct
consequence of Remark 2.25.

##  3 Flags and modular skew algebras

In this section we fix a prime $p$, a positive integer $n=p^{m}$, and a field
$\mathbb{F}$ of characteristic $p$ with a cyclic Galois extension $\mathbb{L}$
of degree $n$. We fix a generator $\sigma$ of
$\operatorname{Gal}(\mathbb{L}/\mathbb{F})$, and define
$\bar{\sigma}=\sigma-\operatorname{id}\in\mathbb{L}[\sigma]$. Note that, in
this case the endomorphism $\bar{\sigma}$ is nilpotent of order $n$, since

$\bar{\sigma}^{n}=(\sigma-\operatorname{id})^{n}=(\sigma-\operatorname{id})^{p^{m}}=\sigma^{p^{m}}-\operatorname{id},$

which maps every element of $\mathbb{L}$ to $0$. We define

$\mathcal{F}_{i}=\ker(\bar{\sigma}^{i}),\quad\mbox{ for every
}i\in\\{0,\ldots,n\\}.$ (8)

In particular, we have that $\mathcal{F}_{0}=\\{0\\}$,
$\mathcal{F}_{1}=\mathbb{F}$, and $\mathcal{F}_{n}=\mathbb{L}$. We work under
these assumptions until the end of the present section. The following result
collects some immediate properties of the endomorphisms $\bar{\sigma}^{i}$ and
of the $\mathbb{F}$-subspaces $\mathcal{F}_{i}$ of $\mathbb{L}$.

######  Lemma 3.1.

The following hold:

  1. $(1)$

The set $\\{\bar{\sigma}^{i}:i\in\\{0,\ldots,n-1\\}\\}$ is an
$\mathbb{L}$-basis of $\mathbb{L}[\sigma]$.

  2. $(2)$

For each $i\in\\{0,\ldots,n\\}$, one has $\dim_{\mathbb{F}}\mathcal{F}_{i}=i$.

  3. $(3)$

For each $i\in\\{1,\ldots,n\\}$, one has
$\mathcal{F}_{i-1}\subseteq\mathcal{F}_{i}$.

  4. $(4)$

For every $i,j\in\\{0,\ldots,n\\}$, one has
$\bar{\sigma}^{j}(\mathcal{F}_{i})=\mathcal{F}_{\max\\{0,i-j\\}}$.

A direct consequence of Lemma 3.1 is that
$\mathcal{F}=(\mathcal{F}_{0},\ldots,\mathcal{F}_{n})$ is a full flag of the
$\mathbb{F}$-vector space $\mathbb{L}$, that is,
$\\{0\\}=\mathcal{F}_{0}\subset\mathcal{F}_{1}\subset\ldots\subset\mathcal{F}_{n-1}\subset\mathcal{F}_{n}=\mathbb{L}$
is a chain of $\mathbb{F}$-vector spaces and the $\mathbb{F}$-dimension of
each quotient $\mathcal{F}_{i}/\mathcal{F}_{i-1}$ is equal to $1$.

######  Definition 3.2.

An $\mathbb{F}$-basis $\mathcal{B}=(\beta_{1},\ldots,\beta_{n})$ of
$\mathbb{L}$ is called $\mathcal{F}$-compatible if, for every index
$i\in\\{1,\ldots,n\\}$, the $\mathbb{F}$-span
$\langle\beta_{1},\ldots,\beta_{i}\rangle_{\mathbb{F}}$ is equal to
$\mathcal{F}_{i}$.

The following result on the multiplicative behaviour of the flag $\mathcal{F}$
will be crucial in the proof of Theorem 3.4.

######  Proposition 3.3.

Let $a,b,h\geq 0$ be integers satisfying $p^{h}(a+b)\leq n$. Then one has

$\mathcal{F}_{p^{h}a}\cdot\mathcal{F}_{p^{h}b}\subseteq\mathcal{F}_{p^{h}(a+b-1)}.$

###### Proof.

We work by induction on $a+b$. If $a=0$ or $b=0$, then the claim is true since
$\mathcal{F}_{0}=\\{0\\}$. We assume thus that $a,b\geq 1$. Let
$x\in\mathcal{F}_{p^{h}a}$ and $y\in\mathcal{F}_{p^{h}b}$ be arbitrary and
write

$u=\bar{\sigma}^{p^{h}}(x)=\sigma^{p^{h}}(x)-x\in\mathcal{F}_{p^{h}(a-1)}\ \
\textup{ and }\ \
v=\bar{\sigma}^{p^{h}}(y)=\sigma^{p^{h}}(y)-y\in\mathcal{F}_{p^{h}(b-1)}.$

We compute

$\sigma^{p^{h}}(xy)=\sigma^{p^{h}}(x)\sigma^{p^{h}}(y)=(x+u)(y+v)=xy+xv+yu+uv,$

from which we derive that

$\bar{\sigma}^{p^{h}}(xy)=\sigma^{p^{h}}(xy)-xy=xv+yu+uv\in\mathcal{F}_{p^{h}a}\cdot\mathcal{F}_{p^{h}(b-1)}+\mathcal{F}_{p^{h}b}\cdot\mathcal{F}_{p^{h}(a-1)}+\mathcal{F}_{p^{h}(a-1)}\cdot\mathcal{F}_{p^{h}(b-1)}.$

It follows from the induction hypothesis that
$\bar{\sigma}^{p^{h}}(xy)\in\mathcal{F}_{p^{h}(a+b-2)}$ and so
$xy\in\mathcal{F}_{p^{h}(a+b-1)}$. ∎

Let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a Ferrers diagram of order
$n=p^{m}$. Throughout we write $\mathbb{L}[\sigma;\mathcal{D}]$ for the
$\mathbb{F}$-subspace

$\mathbb{L}[\sigma;\mathcal{D}]=\bigoplus_{i=1}^{n}\mathcal{F}_{c_{i}}\bar{\sigma}^{i-1}=\left\\{\sum_{i=1}^{n}\lambda_{i}\bar{\sigma}^{i-1}\,:\,\lambda_{i}\in\mathcal{F}_{c_{i}}\right\\}$

of $\mathbb{L}[\sigma]$ defined by $\mathcal{D}$ with respect to the flag
$\mathcal{F}$.

######  Theorem 3.4.

Let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a $p$-monotone Ferrers diagram of
order $n=p^{m}$. Let $\mathbb{L}$ be a cyclic Galois extension of $\mathbb{F}$
of order $n$, whose Galois group is generated by $\sigma$. Let $\mathcal{B}$
be any $\mathbb{F}$-basis of $\mathbb{L}$ associated with the flag
$\mathcal{F}=(\mathcal{F}_{0},\ldots,\mathcal{F}_{n})$. Then, the isomorphism
in (3) induces by restriction an isomorphism
$\mathbb{L}[\sigma;\mathcal{D}]\rightarrow\mathbb{F}^{\mathcal{D}}$.

###### Proof.

Since
$\dim_{\mathbb{F}}(\mathbb{L}[\sigma;\mathcal{D}])=c_{1}+\ldots+c_{n}=\dim_{\mathbb{F}}(\mathbb{F}^{\mathcal{D}})$,
proving that the isomorphism in (3) induces by restriction an isomorphism
$\mathbb{L}[\sigma;\mathcal{D}]\rightarrow\mathbb{F}^{\mathcal{D}}$ is
equivalent to proving that

$f(\mathcal{F}_{i})\subseteq\mathcal{F}_{c_{i}}\qquad\quad\mbox{ for every
}f\in\mathbb{L}[\sigma;\mathcal{D}]\textup{ and }i\in\\{1,\ldots,n\\}.$ (9)

We proceed thus to prove (9). To this end, let $h$ be the $p$-height of
$\mathcal{D}$. This means that $p^{h}$ divides every entry $c_{k}$ of
$\mathcal{D}$ and, for every $k\in\\{0,\ldots,n/p^{h}-1\\}$, that
$c_{1+kp^{h}}=c_{2+kp^{h}}=\ldots=c_{(k+1)p^{h}}$. Since, for each index $k$,
one has

$\mathcal{F}_{1+kp^{h}}\subset\ldots\subset\mathcal{F}_{(k+1)p^{h}}\ \
\textup{ and }\ \
\mathcal{F}_{c_{1+kp^{h}}}=\ldots=\mathcal{F}_{c_{(k+1)p^{h}}},$

checking (9) is equivalent to checking (9) for indices $i$ of the form
$i=ap^{h}$ with $a\in\\{1,\ldots n/p^{h}\\}$. Let thus $a\in\\{1,\ldots
n/p^{h}\\}$ and write $i=ap^{h}$. Let, moreover,
$f\in\mathbb{L}[\sigma;\mathcal{D}]$: leveraging on linearity, we assume
without loss of generality that $f=\lambda_{j}\bar{\sigma}^{j-1}$, for some
$j\in\\{1,\ldots,n\\}$ and $\lambda_{j}\in\mathcal{F}_{c_{j}}$. Applying the
same argument as before, we assume without loss of generality that $j=bp^{h}$
with $b\in\\{1,\ldots,n/p^{h}\\}$. Let $s$ and $t$ be as in Remark 2.25.

If $b<s$, then $\mathcal{F}_{j}=\\{0\\}$ and we derive trivially
$f(\mathcal{F}_{i})=\\{0\\}\subseteq\mathcal{F}_{c_{i}}$. In a dual way, if
$a>t$, then $\mathcal{F}_{c_{i}}=\mathbb{L}$ and thus we have
$f(\mathcal{F}_{i})\subseteq\mathcal{F}_{c_{i}}$. Moreover, if $a\leq b$, then
Lemma 3.1$(4)$ yields that $\bar{\sigma}^{j-1}(\mathcal{F}_{i})=\\{0\\}$ and,
once again, we get $f(\mathcal{F}_{i})\subseteq\mathcal{F}_{c_{i}}$.

We assume now that $s\leq b<a\leq t$ and write $c_{i}=cp^{h}$ and
$c_{j}=dp^{h}$. Remark 2.25 ensures that $c-d\geq a-b$ and so we have
$p^{h}(d+a-b+1)\leq p^{h}(c+1)\leq n$. With the aid of Lemma 3.1$(4)$ and
Proposition 3.3 we compute

$f(\mathcal{F}_{i})=\lambda_{j}\bar{\sigma}^{j-1}(\mathcal{F}_{i})=\lambda_{j}\mathcal{F}_{i-j+1}=\lambda_{j}(\mathcal{F}_{ap^{h}-bp^{h}+1})\subseteq\mathcal{F}_{dp^{h}}\mathcal{F}_{(a-b+1)p^{h}}\subseteq\mathcal{F}_{(a-b+d)p^{h}}.$

and thus
$f(\mathcal{F}_{i})\subseteq\mathcal{F}_{(a-b+d)p^{h}}\subseteq\mathcal{F}_{cp^{h}}=\mathcal{F}_{c_{i}}$.
∎

We conclude this section by providing a few concrete examples.

######  Example 3.5.

Let $\mathbb{F}=\mathbb{F}_{5}$ and $n=5$. Consider the monotone Ferrers
diagram $\mathcal{D}=(1,3,4,5,5)$ and the space $\mathbb{F}^{\mathcal{D}}$
representing the matrices supported on $\mathcal{D}$ over
$\mathbb{F}=\mathbb{F}_{5}$. Furthermore, consider the degree $5$ extension
field $\mathbb{L}=\mathbb{F}_{5^{5}}=\mathbb{F}_{5}(\gamma)$, where
$\gamma^{5}+4\gamma+3=0$. Let $\sigma$ be the $5$-Frobenius automorphism of
$\mathbb{F}_{5^{5}}$ defined as $\sigma(\alpha)=\alpha^{5}$ and set
$\bar{\sigma}=\sigma-\operatorname{id}$. Take the full flag
$\mathcal{F}=(\mathcal{F}_{0},\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3},\mathcal{F}_{4},\mathcal{F}_{5})$
of $\mathbb{F}_{5^{5}}$ over $\mathbb{F}_{5}$ given by

$\mathcal{F}_{i}=\ker(\bar{\sigma}^{i}).$

It is easy to see that
$\mathcal{B}=(1,\gamma^{2968},\gamma^{1531},\gamma^{1556},\gamma^{1566})$ is
an $\mathcal{F}$-compatible $\mathbb{F}_{5}$-basis of $\mathbb{F}_{5^{5}}$. By
Theorem 3.4, the algebra $\mathbb{F}_{5}^{\mathcal{D}}$ is then isomorphic to

$\mathbb{F}_{5^{5}}[\sigma;\mathcal{D}]=\mathcal{F}_{1}\operatorname{id}\oplus\,\mathcal{F}_{3}\bar{\sigma}\oplus\mathcal{F}_{4}\bar{\sigma}^{2}\oplus\mathcal{F}_{5}\bar{\sigma}^{3}\oplus\mathcal{F}_{5}\bar{\sigma}^{4},$

where the isomorphism is with respect to the chosen $\mathbb{F}_{5}$-basis
$\mathcal{B}$.

######  Example 3.6.

Let $\mathbb{F}=\mathbb{F}_{2}$ and $n=8$. Let, moreover,
$\mathcal{D}=\mathcal{T}_{8}=(1,2,3,4,5,6,7,8)$ and write
$\mathbb{F}^{\mathcal{D}}$ for the space representing the upper triangular
$8\times 8$ matrices over $\mathbb{F}=\mathbb{F}_{2}$. Furthermore, consider
the degree $8$ extension field
$\mathbb{L}=\mathbb{F}_{2^{8}}=\mathbb{F}_{2}(\gamma)$, where
$\gamma^{8}+\gamma^{4}+\gamma^{3}+\gamma^{2}+1=0$. Let
$\sigma:\mathbb{F}_{2^{8}}\rightarrow\mathbb{F}_{2^{8}}$ be the $2$-Frobenius
automorphism which is defined as $\sigma(\alpha)=\alpha^{2}$ and let
$\bar{\sigma}=\sigma-\operatorname{id}$. As in Example 3.5 let
$\mathcal{F}=(\mathcal{F}_{0},\ldots,\mathcal{F}_{8})$ be the flag in the
$\mathbb{F}_{2}$-vector space $\mathbb{F}_{2^{8}}$ given by

$\mathcal{F}_{i}=\ker(\bar{\sigma}^{i}).$

Then an $\mathcal{F}$-compatible $\mathbb{F}_{2}$-basis of
$\mathbb{F}_{2^{8}}$ is given by
$\mathcal{B}=(1,\gamma^{170},\gamma^{136},\gamma^{204},\gamma^{222},\gamma^{38},\gamma^{143},\gamma^{5})$.
By Theorem 3.4, the algebra $\mathbb{F}_{2}^{\mathcal{D}}$ is isomorphic to

$\mathbb{F}_{2^{8}}[\sigma;\mathcal{D}]=\left\\{\sum_{i=1}^{8}\lambda_{i}\bar{\sigma}^{i-1}\,:\,\lambda_{i}\in\mathcal{F}_{i}\right\\},$

where the isomorphism depends on the $\mathbb{F}_{2}$-basis $\mathcal{B}$.
Let, for instance,
$f=\operatorname{id}+\gamma^{68}\bar{\sigma}^{2}\in\mathbb{F}_{2^{8}}[\sigma;\mathcal{D}]$:
a straightforward computation reveals that the matrix representation of $f$
with respect to $\mathcal{B}$ is

$\left(\begin{array}[]{ccccccccc}1\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&1\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0&0&1\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0&0&0&1\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0&0&0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0&0&0&0&0\pagecolor{blue!10}\end{array}\right)\in\mathbb{F}_{2}^{\mathcal{D}}.$

######  Example 3.7.

Let $\mathcal{D}=(4,4,4,4,8,8,8,8)$ be the $2$-monotone Ferrers diagram of
order $8$ from Example 2.21. With the same notation used in Example 3.6, the
space $\mathbb{F}_{2}^{\mathcal{D}}$ is isomorphic to

$\mathbb{F}_{2^{8}}[\sigma;\mathcal{D}]=\left\\{\sum_{i=1}^{8}\lambda_{i}\bar{\sigma}^{i-1}\,:\,\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\in\mathcal{F}_{4},\lambda_{5},\lambda_{6},\lambda_{7},\lambda_{8}\in\mathcal{F}_{8},\right\\},$

where the isomorphism is induced by the $\mathbb{F}_{2}$-basis
$\mathcal{B}=(1,\gamma^{170},\gamma^{136},\gamma^{204},\gamma^{222},\gamma^{38},\gamma^{143},\gamma^{5})$.

##  4 Ferrers diagram rank-metric codes

In this section we prove our main results: Theorem 4.2, Theorem 4.10, and
Theorem 4.22. We deal with the first two via the construction of MFD codes
from Theorem 3.4, relying on (4). We prove the third theorem with the aid of
Theorem 4.10 and Proposition 4.23.

###  4.1 Monotone and convex Ferrers diagrams

In this section we give a constructive proof of Conjecture 2.11 for
$p$-monotone Ferrers diagrams and their adjoints over fields of characteristic
$p$. To this end, we work under the same assumptions and use the same notation
as in Section 3. For an integer $1\leq d\leq n$ and a Ferrers diagram
$\mathcal{D}=(c_{1},\ldots,c_{n})$ of order $n=p^{m}$, we write additionally
$\mathbb{L}[\sigma;\mathcal{D}]_{n-d}$ for the $\mathbb{F}$-subspace

$\mathbb{L}[\sigma;\mathcal{D}]_{n-d}=\mathbb{L}[\sigma]_{n-d}\cap\mathbb{L}[\sigma;\mathcal{D}]=\bigoplus_{i=1}^{n-d}\mathcal{F}_{c_{i}}\bar{\sigma}^{i-1}=\left\\{\sum_{i=1}^{n-d}\lambda_{i}\bar{\sigma}^{i-1}\,:\,\lambda_{i}\in\mathcal{F}_{c_{i}}\right\\}$

of polynomials in $\mathbb{L}[\sigma;\mathcal{D}]$ of degree at most $n-d$.
The following result is stated implicitly assuming the choice of a basis
$\mathcal{B}$ of $\mathbb{L}$ over $\mathbb{F}$.

######  Theorem 4.1.

Let $\mathbb{F}$ be a finite field, of characteristic $p$ and order $q$, and
let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a $p$-monotone Ferrers diagram of
order $n=p^{m}$. Let $\mathbb{L}$ be an extension of $\mathbb{F}$ of degree
$n$ and let $\sigma$ be the $q$-Frobenius automorphism of $\mathbb{L}$. Let
$d$ be an integer with $1\leq d\leq n$ and write
$\bar{\sigma}=\sigma-\operatorname{id}$. Then the image under (3) of
$\mathbb{L}[\sigma;\mathcal{D}]_{n-d}$ in $\mathbb{F}^{n\times n}$ is an MFD
code.

###### Proof.

Thanks to Theorem 3.4, the algebra $\mathbb{L}[\sigma;\mathcal{D}]$ maps
isomorphically onto $\mathbb{F}^{\mathcal{D}}$ via (3). Moreover, it follows
from (4) that every element of $\mathbb{L}[\sigma;\mathcal{D}]_{n-d}$ is
mapped to a matrix of rank at least $d$. Computing the $\mathbb{F}$-dimension
of $\mathbb{L}[\sigma;\mathcal{D}]_{n-d}$ we derive from Lemma 3.1 and Remark
2.26 that

$\dim_{\mathbb{F}}(\mathbb{L}[\sigma;\mathcal{D}]_{n-d})=\sum_{i=1}^{n-d}\dim_{\mathbb{F}}\mathcal{F}_{c_{i}}=\sum_{i=1}^{n-d}c_{i}=\nu_{\min}(\mathcal{D},d).$

This concludes the proof. ∎

We derive the following result as an immediate corollary of Theorem 4.1 and
Lemma 2.24.

######  Theorem 4.2.

Let $d,m$ be positive integers and let $p$ be a prime number. Write $n=p^{m}$
and let $1\leq d\leq n$. Then the following hold:

  1. (1)

Conjecture 2.11 holds true for $p$-monotone Ferrers diagrams of order $n$ over
all finite fields of characteristic $p$.

  2. (2)

Conjecture 2.11 holds true for $p$-convex Ferrers diagrams of order $n$ over
all finite fields of characteristic $p$.

We illustrate Theorem 4.2 with a concrete example.

######  Example 4.3.

Let $\mathbb{F}=\mathbb{F}_{5}$, $n=5$ and $d=4$. Let
$\mathcal{D}=(1,3,4,5,5)$ be the monotone Ferrers diagram from Example 3.5. In
this case $\nu_{\min}(\mathcal{D},4)=4$ and Theorem 4.2, ensures the existence
of a $[\mathcal{D},4,4]_{\mathbb{F}_{5}}$ MFD code. A concrete example is
given by:

$\mathbb{F}_{5^{5}}[\sigma;\mathcal{D}]_{1}=\mathcal{F}_{1}\operatorname{id}\oplus\mathcal{F}_{3}\bar{\sigma}=\langle\operatorname{id},\bar{\sigma},\gamma^{2968}\bar{\sigma},\gamma^{1531}\bar{\sigma}\rangle_{\mathbb{F}_{5}}.$

Using the $\mathbb{F}_{5}$-basis
$\mathcal{B}=(1,\gamma^{2968},\gamma^{1531},\gamma^{1556},\gamma^{1566})$ of
$\mathbb{F}_{5^{5}}$ given in Example 3.5, we obtain that the
$[\mathcal{D},4,4]_{\mathbb{F}_{5}}$ MFD code in
$\mathbb{F}_{5}^{\mathcal{D}}$ is the $\mathbb{F}_{5}$-subspace generated by
the following matrices:

$\left(\begin{array}[]{ccccc}1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0\pagecolor{blue!10}&1\pagecolor{blue!10}\end{array}\right),\,\left(\begin{array}[]{ccccc}0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}\end{array}\right),\,\left(\begin{array}[]{ccccc}0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&4\pagecolor{blue!10}\\\
0&1\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0\pagecolor{blue!10}&2\pagecolor{blue!10}&2\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0\pagecolor{blue!10}&3\pagecolor{blue!10}&3\pagecolor{blue!10}\\\
0&0&0&0\pagecolor{blue!10}&4\pagecolor{blue!10}\end{array}\right),\,\left(\begin{array}[]{ccccc}0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&1\pagecolor{blue!10}&2\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&3\pagecolor{blue!10}&1\pagecolor{blue!10}&3\pagecolor{blue!10}\\\
0&0&0&1\pagecolor{blue!10}&2\pagecolor{blue!10}\\\ \end{array}\right).$

###  4.2 Strictly monotone and initially convex Ferrers diagrams

Among monotone Ferrers diagrams, a distinguished subfamily is given by
strictly monotone ones, whose adjoints are the initially convex Ferrers
diagrams.

######  Definition 4.4.

A Ferrers diagram $\mathcal{D}=(c_{1},\ldots,c_{n})$ is called strictly
monotone if, whenever $c_{i}>0$, one has $c_{i+1}>c_{i}$. A Ferrers diagram
$\mathcal{D}=(c_{1},\ldots,c_{n})$ is called initially convex if, for every
$i\in\\{1,\ldots,n-1\\}$, one has $c_{1}\leq 1$ and $c_{i+1}-c_{i}\leq 1$.

######  Definition 4.5.

Let $p$ be a prime number. A Ferrers diagram
$\mathcal{D}=(c_{1},\ldots,c_{n})$ is called strictly $p$-monotone if its
$p$-contraction is strictly monotone. A Ferrers diagram
$\mathcal{D}=(c_{1},\ldots,c_{n})$ is called initially $p$-convex if its
$p$-contraction is initially convex.

######  Remark 4.6.

Let $p$ be a prime number and let $\mathcal{D}$ be a strictly monotone Ferrers
diagram. Since $\mathcal{D}$ is non-empty, the $p$-height of $\mathcal{D}$ is
equal to $0$ and so $\mathcal{D}^{(p)}=\mathcal{D}$. In particular, strictly
monotone Ferrers diagram are strictly $p$-monotone for every choice of a prime
$p$.

The following result is straightforward.

######  Lemma 4.7.

Let $p$ be a prime number. A Ferrers diagram $\mathcal{D}$ is strictly
$p$-monotone if and only if $\mathcal{D}^{\top}$ is initially $p$-convex.

The following results ensures, up to iteration, that the property of being
strictly $p$-monotone is preserved via top-right embeddings of Ferrers
diagrams into larger square grids.

######  Lemma 4.8.

Let $p$ be a prime number and let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a
strictly $p$-monotone Ferrers diagram of order $n$ of $p$-height $h$. Then

$\mathcal{D}^{\prime}=(\underbrace{\,0\ldots,0\,}_{p^{h}\textup{
times}},c_{1},\ldots,c_{n})$

is a strictly $p$-monotone Ferrers diagram of order $n+p^{h}$ and $p$-height
$h$.

Observe that Lemma 4.8 is not valid if we only assume the Ferrers diagram to
be monotone.

######  Theorem 4.9.

Let $p$ be a prime number and let $d,n$ be positive integers with $1\leq d\leq
n$. Then the following hold:

  1. (1)

Conjecture 2.11 holds true for strictly $p$-monotone Ferrers diagrams over any
finite field of characteristic $p$.

  2. (2)

Conjecture 2.11 holds true for initially $p$-convex Ferrers diagrams over any
finite field of characteristic $p$.

###### Proof.

We only prove the result for strictly $p$-monotone Ferrers diagrams: the one
for initially $p$-convex Ferrers diagrams follows then from Lemma 4.7. Let
$\mathcal{D}=(c_{1},\ldots,c_{n})$ be a strictly $p$-monotone Ferrers diagram
of $p$-height $h$. Let $\mathbb{F}$ be a finite field of characteristic $p$
and $m=\min\\{i\,:\,p^{i}\geq n\\}$. Since $p^{h}$ divides $p^{m}-n$,
iterating Lemma 4.8 precisely $(p^{m}-n)/p^{h}$ times, we embed $\mathcal{D}$
into a strictly $p$-monotone Ferrers diagram
$\mathcal{D}^{\prime}=(c_{1}^{\prime},\ldots,c_{p^{m}}^{\prime})$ of order
$p^{m}$, where

$c_{i}^{\prime}=\begin{cases}0&\mbox{ if }i\leq p^{m}-n,\\\ c_{i-p^{m}}&\mbox{
if }i>p^{m}-n.\end{cases}$

Theorem 4.2 ensures the existence of a
$[\mathcal{D}^{\prime},\nu_{\min}(\mathcal{D}^{\prime},d),d]_{\mathbb{F}}$ MFD
code, which is clearly isometric to a
$[\mathcal{D},\nu_{\min}(\mathcal{D},d),d]_{\mathbb{F}}$ MFD code. ∎

We derive the following as a direct consequence of Theorem 4.9 and Remark 4.6.

######  Theorem 4.10.

Let $d,n$ be positive integers with $1\leq d\leq n$. Then the following hold:

  1. (1)

Conjecture 2.11 holds true for strictly monotone Ferrers diagrams over any
finite field.

  2. (2)

Conjecture 2.11 holds true for initially convex Ferrers diagrams over any
finite field.

We remark that the last result holds over any finite field, in contrast with
Theorem 4.2, which requires the characteristic $p$ of the field to be the
unique prime divisor of $n$.

######  Corollary 4.11.

Conjecture 2.11 holds true for upper triangular matrices over any finite
field.

######  Corollary 4.12.

If $p$ is a prime number and $\mathcal{D}$ is a Ferrers diagram whose
$p$-contraction is upper triangular, then 2.11 holds true over any finite
field of characteristic $p$.

We observe that Corollary 4.12 partially answers an open problem from [5,
Section VIII] asking whether MFD codes exist for Ferrers diagrams of the form
$(2,2,4,4,\ldots,2r,2r)$.

######  Example 4.13.

In this example we illustrate how to construct an MFD code of upper triangular
matrices in the smallest open case over $\mathbb{F}_{2}$. For this, let $n=6$
and $d=4$ and consider the strictly monotone Ferrers diagram
$\mathcal{D}=(1,2,3,4,5,6)$. In this case, we have
$\nu_{\min}(\mathcal{D},4)=6$, and now we show how to construct a
$[\mathcal{D},6,4]_{\mathbb{F}_{2}}$ MFD code.

The smallest power of $2$ which is at least $n$ is $n^{\prime}=8$, and we can
extend $\mathcal{D}$ to the Ferrers diagram
$\mathcal{D}^{\prime}=(0,0,1,2,3,4,5,6)$ of order $8$, which is still strictly
monotone, as a consequence of Lemma 4.8. Thus, we consider the degree
$n^{\prime}=8$ extension field of $\mathbb{F}_{2}$ given by
$\mathbb{F}_{2^{8}}=\mathbb{F}_{2}(\gamma)$, where
$\gamma^{8}+\gamma^{4}+\gamma^{3}+\gamma^{2}+1=0$. As in Example 3.6, we let
$\sigma$ be the $2$-Frobenius automorphism of $\mathbb{F}_{2^{8}}$, we set
$\bar{\sigma}=\sigma-\operatorname{id}$, and take the full flag
$\mathcal{F}=(\mathcal{F}_{0},\ldots,\mathcal{F}_{8})$, where
$\mathcal{F}_{i}=\ker(\bar{\sigma}^{i})$. The basis
$\mathcal{B}=(1,\gamma^{170},\gamma^{136},\gamma^{204},\gamma^{222},\gamma^{38},\gamma^{143},\gamma^{5})$
is $\mathcal{F}$-compatible, and Theorem 3.4 yields that
$\mathbb{F}_{2}^{\mathcal{D}}\cong\mathbb{F}_{2}^{\mathcal{D}^{\prime}}$ is
isomorphic to

$\mathbb{F}_{2^{8}}[\sigma;\mathcal{D}^{\prime}]=\bigoplus_{i=3}^{8}\mathcal{F}_{i-2}\bar{\sigma}^{i-1}=\left\\{\sum_{i=3}^{8}\lambda_{i}\bar{\sigma}^{i-1}\,:\,\lambda_{i}\in\mathcal{F}_{i-2}\right\\},$

where the isomorphism is defined using the $\mathbb{F}_{2}$-basis
$\mathcal{B}$. The $[\mathcal{D}^{\prime},6,4]_{\mathbb{F}_{2}}$ MFD code
representation in $\mathbb{F}_{2^{8}}[\sigma;\mathcal{D}^{\prime}]$ is then
given by

$\mathbb{F}_{2^{8}}[\sigma;\mathcal{D}^{\prime}]_{4}=\langle\bar{\sigma}^{2},\bar{\sigma}^{3},\gamma^{170}\bar{\sigma}^{3},\bar{\sigma}^{4},\gamma^{170}\bar{\sigma}^{4},\gamma^{136}\bar{\sigma}^{4}\rangle_{\mathbb{F}_{2}}.$

A straightforward computation shows that writing each element with respect to
the $\mathbb{F}_{2}$-basis $\mathcal{B}$ – and then taking the top-right
$6\times 6$ submatrices – we obtain the $[\mathcal{D},6,4]_{\mathbb{F}_{2}}$
MFD code in $\mathbb{F}_{2}^{\mathcal{D}}$ as the span of the following six
matrices:

$\left(\begin{array}[]{cccccccc}1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0&0&1\pagecolor{blue!10}\end{array}\right),\,\left(\begin{array}[]{cccccccc}0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&\pagecolor{blue!10}0&\pagecolor{blue!10}1&\pagecolor{blue!10}0&0\pagecolor{blue!10}\\\
0&0&0&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0&0\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0&0&0&0\pagecolor{blue!10}\end{array}\right),\,\left(\begin{array}[]{cccccccc}0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&1\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&1\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0&0&1\pagecolor{blue!10}\end{array}\right),$

$\left(\begin{array}[]{cccccccc}0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0&0&0\pagecolor{blue!10}\end{array}\right),\,\left(\begin{array}[]{cccccccc}0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0&0\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0&0&0\pagecolor{blue!10}\end{array}\right),\,\left(\begin{array}[]{cccccccc}0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}&1\pagecolor{blue!10}\\\
0&0&1\pagecolor{blue!10}&0\pagecolor{blue!10}&1\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&1\pagecolor{blue!10}&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0&0\pagecolor{blue!10}&0\pagecolor{blue!10}\\\
0&0&0&0&0&0\pagecolor{blue!10}\end{array}\right).$

Observe that, if we wanted to construct a $[\mathcal{D},6,4]_{\mathbb{F}}$ MFD
code over a finite field $\mathbb{F}$ of some other (positive) characteristic
we would have to redefine $n^{\prime}$. For instance, if we chose
$\mathbb{F}=\mathbb{F}_{3}$, then $\mathcal{D}$ would be mapped to
$\mathcal{D}^{\prime}=(0,0,0,1,2,3,4,5,6)$, which is of order $9=3^{2}$, and
consequently the representation of $\mathbb{F}_{3}^{\mathcal{D}^{\prime}}$
would become $\mathbb{F}_{3^{9}}[\sigma;\mathcal{D}^{\prime}]$.

######  Remark 4.14.

As a consequence of [3, Corollary 1], if for a pair $(\mathcal{D},d)$ there
exists a maximum Ferrers diagram code over some finite field $\mathbb{F}$,
then such a code can be lifted to an MFD code over $\mathbb{Q}$. In
particular, Theorem 4.2 and Theorem 4.10 yield the existence of MFD codes over
$\mathbb{Q}$ for $p$-monotone and strictly $p$-monotone Ferrers diagrams, and
their adjoints.

###  4.3 Extension to MDS-constructible pairs

We conclude the paper by extending Theorem 4.10 to all MDS-costructible pairs
$(\mathcal{D},d)$; cf. Proposition 4.15. We recall that, for MDS-costructible
pairs 2.11 is known to be true only over large finite fields; cf. Section 2.1.
Here, we will prove it for every finite field. We start by showing that
strictly monotone Ferrers diagrams and their adjoints are MDS-constructible
with respect to any sensible distance.

######  Proposition 4.15.

Let $\mathcal{D}$ be a strictly monotone or initially convex Ferrers diagram
of order $n$. Then, for every integer $2\leq d\leq n$, the pair
$(\mathcal{D},d)$ is MDS-constructible.

###### Proof.

We start by assuming that $\mathcal{D}$ is strictly monotone so, thanks to
Remark 2.26, we have that
$\nu_{\min}(\mathcal{D},d)=|\mathcal{D}\cap([n]\times[n-d+1])|$. Moreover, for
every $i\in[n]$, strict monotonicity implies $c_{i}\leq i$ and thus
$\mathcal{D}\subseteq\mathcal{T}_{n}$. With a slight abuse of notation
identifying $\mathcal{T}_{n-d+1}$ with its image in $[n^{2}]$, this means that

$\nu_{\min}(\mathcal{D},d)=|\mathcal{D}\cap\mathcal{T}_{n-d+1}|.$ (10)

Set $R=[n]\times([n]\setminus[n-d+1])$ and, for each $i\in[n-d+1]$, write
$r_{i}=|\mathcal{D}\cap\Delta_{i}^{n}\cap\mathcal{T}_{n-d+1}|$. Assuming that
$r_{i}>0$, Remark 2.17 yields that $(\Delta_{i}^{n}\cap
R)\subseteq\mathcal{D}$ and thus

$r_{i}=|\mathcal{D}\cap(\Delta_{i}^{n}\cap\mathcal{T}_{n-d+1})|=|\mathcal{D}\cap\Delta_{i}^{n}|-|\mathcal{D}\cap(\Delta_{i}^{n}\cap
R)|=|\mathcal{D}\cap\Delta_{i}^{n}|-d+1.$ (11)

Combining (10) and (11) we finally obtain

$\nu_{\min}(\mathcal{D},d)=|\mathcal{D}\cap\mathcal{T}_{n-d+1}|=\sum_{i=1}^{n-d+1}r_{i}=\sum_{i=1}^{n-d+1}\max\\{0,|\mathcal{D}\cap\Delta_{i}^{n}|-d+1\\},$

which shows that the pair $(\mathcal{D},d)$ is MDS-constructible. This also
concludes the proof as initially convex Ferrers diagrams are adjoints of
strictly monotone ones and MDS-constructibility is preserved under adjunction.
∎

Observe that Proposition 4.15 is not true in general if we consider the wider
classes of monotone or $p$-monotone Ferrers diagrams (and their adjoints), as
the next example shows.

######  Example 4.16.

In this example, for given positive integers $n$ and $2\leq d\leq n$, write

$\nu_{\mathrm{MDS}}(\mathcal{D},d)=\sum_{i=1}^{n-d+1}\max\\{0,|\mathcal{D}\cap\Delta_{i}^{n}|-d+1\\}.$

In the following table, we collect the values of $\nu_{\min}(\mathcal{D},d)$
and $\nu_{\mathrm{MDS}}(\mathcal{D},d)$ for the first two monotone Ferrers
diagrams from Example 2.15, which we label in order $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$. Note that here $n=5$ and $\mathcal{D}_{1}$ is strictly
monotone, whilst $\mathcal{D}_{2}$ is not.

$d$ | $\nu_{\min}(\mathcal{D}_{1},d)$ | $\nu_{\mathrm{MDS}}(\mathcal{D}_{1},d)$ | $\nu_{\min}(\mathcal{D}_{2},d)$ | $\nu_{\mathrm{MDS}}(\mathcal{D}_{2},d)$
---|---|---|---|---
2 | 4 | 4 | 12 | 10
3 | 1 | 1 | 7 | 6
4 | 0 | 0 | 3 | 3
5 | 0 | 0 | 1 | 1

######  Definition 4.17.

Let $\mathcal{D}=(c_{1},\ldots,c_{n})$ be a Ferrers diagram of order $n$, let
$d\in\\{2,\ldots,n\\}$ and let $j\in\\{0,\ldots,d-1\\}$. The pair
$(\mathcal{D},d)$ is called $j$-Singleton if

$\nu_{\min}(\mathcal{D},d)=\sum_{i=1}^{n-j}\max\\{0,c_{i}-d+1+j\\}.$

Informally speaking, a pair $(\mathcal{D},d)$ is $j$-Singleton if
$\nu_{\min}(\mathcal{D},d)$ is obtained by counting the number of dots in
$\mathcal{D}$ after removing the last $j$ columns and the first $d-j-1$ rows.
Because of this and (6), every pair $(\mathcal{D},d)$, where $\mathcal{D}$ is
a Ferrers diagram of order $n$ and $2\leq d\leq n$, is $j$-Singleton for some
$j\in\\{0,\ldots,d-1\\}$. It is not difficult to construct examples of pairs
$(\mathcal{D},d)$ that are $j$-Singleton for more than one choice of $j$.

Until the end of the paper, we will use the following notation. For any
positive integers $n$ and $d$ with $2\leq d\leq n$ and every
$j\in\\{0,\ldots,d-1\\}$, define the quantity

$\displaystyle\nu_{j}(\mathcal{D},d)$
$\displaystyle=\sum_{i=1}^{n-j}\max\\{0,c_{i}-d+1+j\\},$

and also the following subsets of $[n^{2}]$:

$\displaystyle\mathcal{S}_{n,d,j}$
$\displaystyle=\\{(i,\ell)\,:\,i\in\\{d-j,\ldots,n\\},\ell\in[n-j]\\},$
$\displaystyle\mathcal{T}_{n,d,j}$
$\displaystyle=\mathcal{S}_{n,d,j}\cap\mathcal{T}_{n},$
$\displaystyle\mathcal{L}_{n,d,j}$
$\displaystyle=[n]^{2}\setminus\mathcal{S}_{n,d,j}.$

With the notation above, observe that, for any Ferrers diagram $\mathcal{D}$
of order $n$, one has

$\sum_{i=1}^{n-j}\max\\{0,c_{i}-d+1+j\\}=|\mathcal{D}\cap\mathcal{S}_{n,d,j}|=|\mathcal{D}|-|\mathcal{D}\cap\mathcal{L}_{n,d,j}|=\nu_{j}(\mathcal{D},d),$

and hence, the value $\nu_{\min}(\mathcal{D},d)$ in (6) can be written as

$\nu_{\min}(\mathcal{D},d)=\min_{0\leq j\leq
d-1}\nu_{j}(\mathcal{D},d)=\min_{0\leq j\leq
d-1}|\mathcal{D}\cap\mathcal{S}_{n,d,j}|.$

In particular, $(\mathcal{D},d)$ is $j$-Singleton if and only if
$\nu_{\min}(\mathcal{D},d)=\nu_{j}(\mathcal{D},d)$.

######  Example 4.18.

For $n=8$, $d=4$ and $j=1$, the figure below represents the three sets
$\mathcal{S}_{8,4,1}$, $\mathcal{T}_{8,4,1}$ and $\mathcal{L}_{8,4,1}$ as
follows. The black dots represent $\mathcal{S}_{8,4,1}$, the orange dots
represent $\mathcal{L}_{8,4,1}$ and the dots contained in the red triangular
area correspond to $\mathcal{T}_{8,4,1}$.

######  Lemma 4.19.

Let $\mathcal{D}$ be a Ferrers diagram of order $n$, let
$d\in\\{2,\ldots,n\\}$, and let $j\in\\{0,\ldots,d-1\\}$ be such that
$(\mathcal{D},d)$ is $j$-Singleton. Let
$\mathcal{D}^{\prime}\subseteq\mathcal{D}$ be a Ferrers diagram of order $n$
with the property that
$\mathcal{D}\cap\mathcal{L}_{n,d,j}=\mathcal{D}^{\prime}\cap\mathcal{L}_{n,d,j}$.
Then $(\mathcal{D}^{\prime},d)$ is $j$-Singleton.

###### Proof.

For any $i\in\\{0,\ldots,d-1\\}$ we have

$\displaystyle\nu_{\min}(\mathcal{D},d)\leq\nu_{i}(\mathcal{D},d)$
$\displaystyle=|\mathcal{D}|-|\mathcal{D}\cap\mathcal{L}_{n,d,i}|=|\mathcal{D}^{\prime}|+|\mathcal{D}\setminus\mathcal{D}^{\prime}|-|\mathcal{D}\cap\mathcal{L}_{n,d,i}|$
$\displaystyle\leq|\mathcal{D}^{\prime}|+|\mathcal{D}\setminus\mathcal{D}^{\prime}|-|\mathcal{D}^{\prime}\cap\mathcal{L}_{n,d,i}|=\nu_{i}(\mathcal{D}^{\prime},d)+|\mathcal{D}\setminus\mathcal{D}^{\prime}|.$
(12)

On the other hand, by hypothesis $(\mathcal{D},d)$ is $j$-Singleton, and thus
we have

$\nu_{\min}(\mathcal{D},d)=\nu_{j}(\mathcal{D},d)=|\mathcal{D}\cap\mathcal{S}_{n,d,j}|=|\mathcal{D}|-|\mathcal{D}\cap\mathcal{L}_{n,d,j}|=\nu_{j}(\mathcal{D}^{\prime},d)+|\mathcal{D}\setminus\mathcal{D}^{\prime}|.$
(13)

Combining (4.3) with (13), we get
$\nu_{\min}(\mathcal{D}^{\prime},d)=\nu_{j}(\mathcal{D}^{\prime},d)$. ∎

######  Lemma 4.20.

Let $\mathcal{D}$ be a Ferrers diagram of order $n$ and let
$d\in\\{2,\ldots,n\\}$, $j\in\\{0,\ldots,d-1\\}$. Assume that
$(\mathcal{D},d)$ is MDS-constructible and $j$-Singleton. Then the following
hold:

  1. (1)

One has
$\mathcal{D}\cap\mathcal{S}_{n,d,j}=\mathcal{D}\cap\mathcal{T}_{n,d,j}$.

  2. (2)

One has $\\{i\in[n-d+1]\,:\,|\mathcal{D}\cap\Delta_{i}^{n}|\geq
d\\}=\\{i\in[n-d+1]\,:\,\mathcal{D}\cap\Delta_{i}^{n}\cap\mathcal{S}_{n,d,j}\neq\emptyset\\}$.

  3. (3)

If $i\in[n-d+1]$ and
$\mathcal{D}\cap\Delta_{i}^{n}\cap\mathcal{S}_{n,d,j}\neq\emptyset$, then
$\mathcal{D}\cap\Delta_{i}^{n}\supseteq\Delta_{i}^{n}\cap\mathcal{L}_{n,d,j}$.

###### Proof.

Define the subsets $X$ and $Y$ of $[n-d+1]$ as

$X=\\{i\in[n-d+1]\,:\,|\mathcal{D}\cap\Delta_{i}^{n}|\geq d\\}\ \ \textup{ and
}\ \
Y=\\{i\in[n-d+1]\,:\,\mathcal{D}\cap\Delta_{i}^{n}\cap\mathcal{S}_{n,d,j}\neq\emptyset\\}$

and note that $X\subseteq Y$. We will first prove (1) and then (2) and (3)
together.

(1) Since $(\mathcal{D},d)$ is $j$-Singleton, we have that

$\nu_{\min}(\mathcal{D},d)=|\mathcal{D}\cap\mathcal{S}_{n,d,j}|.$ (14)

On the other hand, $(\mathcal{D},d)$ is MDS-constructible and thus we have

$\displaystyle\nu_{\min}(\mathcal{D},d)$ $\displaystyle=\sum_{i\in
X}(|\mathcal{D}\cap\Delta_{i}^{n}|-d+1)=\sum_{i\in
X}(|\mathcal{D}\cap\Delta_{i}^{n}|-|\Delta_{i}^{n}\cap\mathcal{L}_{n,d,j}|)$
$\displaystyle\leq\sum_{i\in
X}(|\mathcal{D}\cap\Delta_{i}^{n}|-|\mathcal{D}\cap(\Delta_{i}^{n}\cap\mathcal{L}_{n,d,j})|)=\sum_{i\in
X}(|\mathcal{D}\cap(\Delta_{i}^{n}\cap\mathcal{S}_{n,d,j})|)$ (15)
$\displaystyle\leq\sum_{i\in
Y}(|\mathcal{D}\cap(\Delta_{i}^{n}\cap\mathcal{S}_{n,d,j})|)=|\mathcal{D}\cap\mathcal{T}_{n,d,j}|.$

Combining (14) and (4.3), we derive that
$\mathcal{D}\cap\mathcal{S}_{n,d,j}=\mathcal{D}\cap\mathcal{T}_{n,d,j}$.

(2)-(3) Thanks to the proof of (1), we know that the inequalities in (4.3) are
in fact all equalities and so, for each $i\in X$ we have
$\mathcal{D}\cap(\Delta_{i}^{n}\cap\mathcal{L}_{n,d,j})=\Delta_{i}^{n}\cap\mathcal{L}_{n,d,j}$.
Furthermore, by the definition of $Y$ and using the fact that the second to
last inequality in (4.3) is an equality, we deduce that $Y=X$ and the proof is
complete. ∎

######  Example 4.21.

Let $\mathcal{D}=(0,2,2,3,3,5,6,8)$ be a Ferrers diagram of order $8$ and let
$d=4$. It is easy to verify that $(\mathcal{D},4)$ is MDS-constructible and
$j$-Singleton for $j=1$. We can see that
$\mathcal{D}\cap\mathcal{S}_{8,4,1}=\mathcal{D}\cap\mathcal{T}_{8,4,1}$.
Moreover, for every index $2\leq i\leq 5$ we have that
$\mathcal{D}\cap\Delta_{i}^{8}\cap\mathcal{T}_{8,4,1}\neq 0$ and indeed, for
each such index, it holds that
$\mathcal{D}\cap\Delta_{i}^{8}\supseteq\Delta_{i}^{8}\cap\mathcal{L}_{8,4,1}$.

The following is the main result of this section, which ensures in particular
that Conjecture 2.11 holds true for every MDS-constructible pair
$(\mathcal{D},d)$.

######  Theorem 4.22.

Let $\mathcal{D}$ be a Ferrers diagram of order $n$ and let $2\leq d\leq n$ be
an integer. If $(\mathcal{D},d)$ is MDS-constructible, then there exists a
$[\mathcal{D},\nu_{\min}(\mathcal{D},d),d]_{\mathbb{F}}$ MFD code over any
finite field $\mathbb{F}$.

Before giving a proof of Theorem 4.22, we recall a useful criterion to
construct MFD codes on a given Ferrers diagram from MFD codes on related
Ferrers diagrams. This strategy was first adopted in the proof of [5, Theorem
7] and the explicit statement can be found in [2, Remarks II.12 & II.14].

######  Proposition 4.23.

Let $\mathcal{D}$ be a Ferrers diagram of order $n$ and let $2\leq d\leq n$ be
an integer.

  1. (1)

Let $\mathcal{D}^{\prime}\subseteq\mathcal{D}$ be a Ferrers diagram of order
$n$ such that $\nu_{\min}(\mathcal{D},d)=\nu_{\min}(\mathcal{D}^{\prime},d)$.
If $\mathcal{C}$ is a
$[\mathcal{D}^{\prime},\nu_{\min}(\mathcal{D}^{\prime},d),d]_{\mathbb{F}}$ MFD
code, then it is also a
$[\mathcal{D},\nu_{\min}(\mathcal{D},d),d]_{\mathbb{F}}$ MFD code.

  2. (2)

Let $\mathcal{D}^{\prime}\supseteq\mathcal{D}$ be a Ferrers diagram of order
$n$ such that
$\nu_{\min}(\mathcal{D},d)=\nu_{\min}(\mathcal{D}^{\prime},d)-|\mathcal{D}^{\prime}\setminus\mathcal{D}|$.
If $\mathcal{C}$ is a
$[\mathcal{D}^{\prime},\nu_{\min}(\mathcal{D}^{\prime},d),d]_{\mathbb{F}}$ MFD
code, then $\mathcal{C}\cap\mathbb{F}^{\mathcal{D}}$ is a
$[\mathcal{D},\nu_{\min}(\mathcal{D},d),d]_{\mathbb{F}}$ MFD code.

We give here the proof of Theorem 4.22 and we illustrate its steps explicitly
in Example 4.24.

###### Proof of Theorem 4.22.

Let $\mathbb{F}$ be a finite field, let $(\mathcal{D},d)$ be an MDS-
constructible pair and let $j\in\\{0,\ldots,d-1\\}$ be such that
$(\mathcal{D},d)$ is $j$-Singleton. Define

$Y=\\{i\in[n-d+1]\,:\,\mathcal{D}\cap\Delta_{i}^{n}\cap\mathcal{S}_{n,d,j}\neq\emptyset\\}$

and note that, as a consequence of (7) and Lemma 4.20(2), one has

$\displaystyle\nu_{\min}(\mathcal{D},d)=\sum_{i=1}^{n}\max\\{0,|\mathcal{D}\cap\Delta_{i}^{n}|-d+1\\}$
$\displaystyle=\sum_{i=1}^{n-d+1}\max\\{0,|\mathcal{D}\cap\Delta_{i}^{n}|-d+1\\}$
$\displaystyle=\sum_{i\in Y}\max\\{0,|\mathcal{D}\cap\Delta_{i}^{n}|-d+1\\}.$

If $Y=\emptyset$, then $\nu_{\min}(\mathcal{D},d)=0$ and there is nothing to
prove. Hence, assume $Y\neq\emptyset$. By the definition of $Y$, there exists
$\ell\in[n-d+1]$ such that $Y=\\{i\,:\,\ell\leq i\leq n-d+1\\}$. Fix such
$\ell$ and define

$\mathcal{D}^{\prime}=(\mathcal{D}\cap\mathcal{S}_{n,d,j})\cup\Big{(}\bigcup_{i=\ell}^{n}(\Delta_{i}^{n}\cap\mathcal{L}_{n,d,j})\Big{)}.$

By Lemma 4.20(3), for every $\ell\leq i\leq n-d+1$ we have that
$\Delta_{i}^{n}\cap\mathcal{L}_{n,d,j}\subseteq\mathcal{D}$ while the same is
satisfied for $i\geq n-d+2$ thanks to the fact that $(\mathcal{D},d)$ is MDS-
constructible. It follows in particular that
$\mathcal{D}^{\prime}\subseteq\mathcal{D}$. Moreover, it also holds that

$\mathcal{D}^{\prime}\subseteq\mathcal{D}^{\prime\prime}=\bigcup_{i=\ell}^{n}\Delta_{i}^{n}.$

The Ferrers diagram $\mathcal{D}^{\prime\prime}$ is a copy of
$\mathcal{T}_{n-\ell+1}$ (considered as embedded into $[n^{2}]$) and hence it
is strictly monotone and $(\mathcal{D}^{\prime\prime},d)$ is
$j^{\prime}$-Singleton for every $j^{\prime}$. In particular, it is
$j$-Singleton. Furthermore, $\mathcal{D}^{\prime}$ and
$\mathcal{D}^{\prime\prime}$ have the same intersection with
$\mathcal{L}_{n,d,j}$. Thus, Lemma 4.19 yields that $(\mathcal{D}^{\prime},d)$
is also $j$-Singleton. Since $\mathcal{D}^{\prime\prime}$ is strictly
monotone, by Theorem 4.10 there exists a
$[\mathcal{D}^{\prime\prime},\nu_{\min}(\mathcal{D}^{\prime\prime},d),d]_{\mathbb{F}}$
MFD code $\mathcal{C}^{\prime\prime}$, which we fix. Since
$(\mathcal{D}^{\prime},d)$ and $(\mathcal{D}^{\prime\prime},d)$ are both
$j$-Singleton and
$\mathcal{D}^{\prime}\cap\mathcal{L}_{n,d,j}=\mathcal{D}^{\prime\prime}\cap\mathcal{L}_{n,d,j}$,
we compute

$\displaystyle\nu_{\min}(\mathcal{D}^{\prime\prime},d)$
$\displaystyle=|\mathcal{D}^{\prime\prime}\cap\mathcal{S}_{n,d,j}|=|(\mathcal{D}^{\prime}\cap\mathcal{S}_{n,d,j})|+|(\mathcal{D}^{\prime\prime}\setminus\mathcal{D}^{\prime})\cap\mathcal{S}_{n,d,j}|$
$\displaystyle=\nu_{\min}(\mathcal{D}^{\prime},d)+|(\mathcal{D}^{\prime\prime}\setminus\mathcal{D}^{\prime})\cap\mathcal{S}_{n,d,j}|=\nu_{\min}(\mathcal{D}^{\prime},d)+|\mathcal{D}^{\prime\prime}\setminus\mathcal{D}^{\prime}|.$

Proposition 4.23(2) yields that
$\mathcal{C}^{\prime}=\mathcal{C}^{\prime\prime}\cap\mathbb{F}^{\mathcal{D}^{\prime}}$
is a
$[\mathcal{D}^{\prime},\nu_{\min}(\mathcal{D}^{\prime},d),d]_{\mathbb{F}}$ MFD
code. Finally, we have

$\nu_{\min}(\mathcal{D}^{\prime},d)=|\mathcal{D}^{\prime}\cap\mathcal{S}_{n,d,j}|=|\mathcal{D}\cap\mathcal{S}_{n,d,j}|=\nu_{\min}(\mathcal{D},d)$

and $\mathcal{D}^{\prime}\subseteq\mathcal{D}$. By Proposition 4.23(1), the
code $\mathcal{C}^{\prime}$ is thus also a
$[\mathcal{D},\nu_{\min}(\mathcal{D},d),d]_{\mathbb{F}}$ MFD code. ∎

######  Example 4.24.

Let $\mathcal{D}=(0,2,2,3,3,5,6,8)$ be the Ferrers Diagram from Example 4.21
and let $\mathbb{F}$ be an arbitrary finite field. In this case
$(\mathcal{D},4)$ is $1$-Singleton and
$\nu_{\min}(\mathcal{D},4)=\nu_{1}(\mathcal{D},4)=9$. In order to construct a
$[\mathcal{D},9,4]_{\mathbb{F}}$ MFD code, we follow the proof of Theorem 4.22
step by step. With the terminology introduced there and as illustrated below,
we have

$Y=\\{i\in[5]\,:\,\mathcal{D}\cap\Delta_{i}^{n}\cap\mathcal{S}_{n,d,j}\neq\emptyset\\}=\\{2,3,4,5\\},$

and

$\mathcal{D}^{\prime}=(\mathcal{D}\cap\mathcal{S}_{8,4,1})\cup\Big{(}\bigcup_{i=2}^{n}(\Delta_{i}^{n}\cap\mathcal{L}_{8,4,1})\Big{)}=(0,1,2,3,3,5,6,7).$

Furthermore, we have $\mathcal{D}^{\prime\prime}=(0,1,2,3,4,5,6,7)$, as given
in the third picture below.

It is easy to verify that
$\nu_{\min}(\mathcal{D}^{\prime},4)=\nu_{\min}(\mathcal{D},4)=9$ and
$\nu_{\min}(\mathcal{D}^{\prime\prime},4)=10=\nu_{\min}(\mathcal{D}^{\prime},4)+|\mathcal{D}^{\prime\prime}\setminus\mathcal{D}^{\prime}|$.
Theorem 4.10 ensures the existence of a
$[\mathcal{D}^{\prime\prime},10,4]_{\mathbb{F}}$ MFD code
$\mathcal{C}^{\prime\prime}$. Then, by Proposition 4.23(2), the code
$\mathcal{C}^{\prime}=\mathcal{C}^{\prime\prime}\cap\mathbb{F}^{\mathcal{D}}$
is a $[\mathcal{D}^{\prime},9,4]_{\mathbb{F}}$ MFD code and it is also a
$[\mathcal{D},9,4]_{\mathbb{F}}$ MFD code thanks to Proposition 4.23(1).

## References

  * [1] J. Antrobus.  The State of Lexicodes and Ferrers Diagram Rank-Metric Codes.  PhD thesis, University of Kentucky, 2019.  https://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1067&context=math_etds.
  * [2] J. Antrobus and H. Gluesing-Luerssen.  Maximal Ferrers diagram codes: constructions and genericity considerations.  IEEE Transactions on Information Theory, 65(10):6204–6223, 2019\.
  * [3] E. Ballico.  Linear subspaces of matrices associated to a Ferrers diagram and with a prescribed lower bound for their rank.  Linear Algebra and its Applications, 483:30–39, 2015.
  * [4] P. Delsarte.  Bilinear forms over a finite field, with applications to coding theory.  Journal of Combinatorial Theory, Series A, 25(3):226–241, 1978\.
  * [5] T. Etzion, E. Gorla, A. Ravagnani, and A. Wachter-Zeh.  Optimal Ferrers diagram rank-metric codes.  IEEE Transactions on Information Theory, 62(4):1616–1630, 2016\.
  * [6] T. Etzion and N. Silberstein.  Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams.  IEEE Transactions on Information Theory, 55(7):2909–2919, 2009\.
  * [7] E. M. Gabidulin.  Theory of codes with maximum rank distance.  Probl. Peredachi Inf., 21(1):3–16, 1985.
  * [8] E. Gorla and A. Ravagnani.  Subspace codes from Ferrers diagrams.  Journal of Algebra and Its Applications, 16(07):1750131, 2017.
  * [9] R. Gow and R. Quinlan.  Galois extensions and subspaces of alternating bilinear forms with special rank properties.  Linear algebra and its applications, 430(8-9):2212–2224, 2009.
  * [10] A. Gruica and A. Ravagnani.  Rook theory of the Etzion-Silberstein conjecture.  arXiv preprint arXiv:2209.05114, 2022.
  * [11] R. M. Guralnick.  Invertible preservers and algebraic groups.  Linear Algebra and its Applications, 212/213:249–257, 1994.
  * [12] S. Liu, Y. Chang, and T. Feng.  Constructions for optimal Ferrers diagram rank-metric codes.  IEEE Transactions on Information Theory, 65(7):4115–4130, 2019\.
  * [13] S. Liu, Y. Chang, and T. Feng.  Several classes of optimal Ferrers diagram rank-metric codes.  Linear Algebra and its Applications, 581:128–144, 2019.
  * [14] R. M. Roth.  Maximum-rank array codes and their application to crisscross error correction.  IEEE Transactions on Information Theory, 37(2):328–336, 1991.
  * [15] D. Silva, F. R. Kschischang, and R. Kötter.  A rank-metric approach to error control in random network coding.  IEEE Trans. Inform. Theory, 54(9):3951–3967, 2008.