[table]capposition=above

Low rank groups of Lie type acting point and line-primitively on finite
generalised quadrangles

Vishnuram Arumugam, John Bamberg and Michael Giudici

###### Abstract

Suppose we have a finite thick generalised quadrangle whose automorphism group
$G$ acts primitively on both the set of points and the set of lines. Then $G$
must be almost simple. In this paper, we show that $\textrm{soc}(G)$ cannot be
isomorphic to
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(2^{2m+1}}\right)$
or
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(3^{2m+1}}\right)$
where $m$ is a positive integer.

## 1 Introduction

A generalised polygon is a type of incidence structure introduced by Jacques
Tits (1959) [19] to realise groups of Lie type as symmetries (more precisely,
automorphism groups) of geometric objects. Let $n$ be a positive integer. A
generalised $n$-gon is an incidence structure whose incidence graph is a
bipartite graph with diameter $n$ and girth $2n$. We say that a generalised
$n$-gon has order $(s,t)$ if every line has $s+1$ points incident with it and
every point is incident with $t+1$ lines. Furthermore, a generalised $n$-gon
is said to be thick if it has order $(s,t)$ where $s,t>1$. Feit and Higman
showed in [7] that thick generalised $n$-gons exist if and only if
$n\in\\{2,3,4,6,8\\}$. The examples of (thick) generalised polygons that arise
from groups of Lie type via the construction of Tits are called _classical_.
Since then, many non-classical examples of projective planes (generalised
$3$-gons) and generalised quadrangles ($4$-gons) have been found [20, Section
3.7]. In the case of generalised hexagons ($6$-gons) and generalised octagons
($8$-gons), the only known examples are the classical ones.

Many attempts have been made to construct new examples. These attempts involve
introducing various symmetry conditions on the automorphism groups and
analysing the possible groups acting on generalised polygons with these
symmetry conditions. Buekenhout and Van Maldeghem (1994) [5] showed that if a
group acts distance-transitively on a generalised $n$-gon ($n\geqslant 4)$,
then in fact, it must act point-primitively on that generalised $n$-gon. We
summarise the results on generalised hexagons and octagons in Table 1 and the
results on generalised quadrangles in Table 2. Here, we let $\Gamma$ be a
generalised $n$-gon (generalised hexagon or octagon in Table 1 and generalised
quadrangle in Table 2) and $G\leqslant\textsf{Aut}(\Gamma)$. For a point
$\alpha$ in $\Gamma$, we write $G_{\alpha}$ to denote the point-stabiliser of
$\alpha$ in $G$. Also, $q$ is assumed to be a prime power. Finally, the column
of assumptions refers to the action of $G$ on $\Gamma$.

Assumptions |  Conclusion |  Reference
---|---|---
Point-primitive, line-primitive and flag-transitive |  $G$ is almost simple of Lie type | [16]
Point-primitive |  $G$ is almost simple of Lie type | [3]
Point-primitive and $\textrm{soc}(G)\cong\textsf{PSL}_{n}({q})$ for $n\geqslant 2$ |  $G_{\alpha}$ acts irreducibly on $V=\mathbb{F}_{q}^{n}$ | [9]
Point-primitive and $\textrm{soc}(G)\cong\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(2^{2m+1}}\right)$, $\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(3^{2m+1}}\right)$ or $\prescript{2}{}{\textrm{F}_{4}}\mathopen{}\mathclose{{}\left(2^{2m+1}}\right)$ |  $\textrm{soc}(G)\cong\prescript{2}{}{\textrm{F}_{4}}\mathopen{}\mathclose{{}\left(2^{2m+1}}\right)$ and $\Gamma$ is the classical generalised octagon or its dual | [14]
Table 1: Summary of results on generalised hexagons and octagons. Assumptions |  Conclusion |  Reference
---|---|---
Point-primitive and line-primitive |  $G$ is almost simple | [2]
Point-primitive line-primitive and flag-transitive |  $G$ is almost simple of Lie type | [2]
Point-primitive, flag-transitive and $\textrm{soc}(G)\cong A_{n}$ with $n\geqslant 5$ |  $G\leqslant S_{6}$ and $\Gamma$ is the unique generalised quadrangle of order $(2,2)$ | [2]
Point-primitive |  $\textrm{soc}(G)$ is not isomorphic to a sporadic group | [1]
Point-primitive and $\textrm{soc}(G)\cong\textsf{PSL}_{2}({q})$ for $q\geqslant 4$ |  $q=9$ and $\Gamma$ is the symplectic quadrangle $W(2)$ | [8]
Point-primitive and line-primitive |  $\textrm{soc}(G)$ is not isomorphic to $\textsf{PSU}_{3}({q})$ for $q\geqslant 3$ | [13]
Table 2: Summary of results on generalised quadrangles.

The main theorem in this paper (Theorem 1.1) is motivated by the work of
Morgan and Popiel [14] on generalised hexagons and octagons, where they showed
that if $G$ acts point-primitively on a generalised hexagon or an octagon
$\Gamma$ with socle
$\textrm{soc}(G)\cong\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(2^{2m+1}}\right)$,
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(3^{2m+1}}\right)$
or
$\prescript{2}{}{\textrm{F}_{4}}\mathopen{}\mathclose{{}\left(2^{2m+1}}\right)$,
then
$\textrm{soc}(G)\cong\prescript{2}{}{\textrm{F}_{4}}\mathopen{}\mathclose{{}\left(2^{2m+1}}\right)$
and $\Gamma$ is the classical generalised octagon or its dual.

######  Theorem 1.1.

Let $m$ be a positive integer. An almost simple group with socle isomorphic to
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(2^{2m+1}}\right)$
or
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(3^{2m+1}}\right)$
cannot act primitively on both the set of points and the set of lines of a
generalised quadrangle.

The case where
$\textrm{soc}(G)\cong\prescript{2}{}{\textrm{F}_{4}}\mathopen{}\mathclose{{}\left(2^{2m+1}}\right)$
is not included in this paper as it still requires further analysis and the
techniques needed to classify generalised quadrangles in this scenario may be
different.

## 2 Preliminaries

We develop some preliminary definitions and results from incidence geometry
and group theory.

### 2.1 Incidence Geometry

An incidence geometry is a triple $(\mathcal{P},\mathcal{L},\mathcal{I})$
where $\mathcal{P}$ is called the set of points, $\mathcal{L}$ is the set of
lines disjoint from $\mathcal{P}$ and
$\mathcal{I}\subseteq\mathcal{P}\times\mathcal{I}$ is the incidence relation.
We say that a point $\alpha\in\mathcal{P}$ is incident with a line
$L\in\mathcal{L}$ if $(\alpha,L)\in\mathcal{I}$. This pair $(\alpha,L)$ is
called a flag. We say two points $\alpha,\beta\in\mathcal{P}$ are collinear if
there exists a line $L\in\mathcal{L}$ such that $(\alpha,L)\in\mathcal{I}$ and
$(\beta,L)\in\mathcal{I}$. Finally, the dual of an incidence geometry
$(\mathcal{P},\mathcal{L},\mathcal{I})$ is the incidence geometry
$(\mathcal{P}^{\prime},\mathcal{L}^{\prime},\mathcal{I}^{\prime})$ where the
set of points $\mathcal{P}^{\prime}=\mathcal{L}$, the set of lines
$\mathcal{L}^{\prime}=\mathcal{P}$ and the incident pair
$(L,\alpha)\in\mathcal{I}^{\prime}$ precisely when $(\alpha,L)\in\mathcal{I}$.

We may also view an incidence structure as a graph. More precisely, given an
incidence structure $\Gamma=(\mathcal{P},\mathcal{L},\mathcal{I})$, we
construct a graph called the _incidence graph_ of $\Gamma$ in the following
way: we take the vertex set to be $\mathcal{P}\cup\mathcal{L}$ and we join an
edge between $\alpha\in\mathcal{P}$ and $L\in\mathcal{L}$ if
$(\alpha,L)\in\mathcal{I}$. Note that the disjointness of $\mathcal{P}$ and
$\mathcal{L}$ ensures that the incidence graph does not contain loops.
Finally, an automorphism of $\Gamma$ is a bijective map
$\theta:\mathcal{P}\cup\mathcal{L}\rightarrow\mathcal{P}\cup\mathcal{L}$ that
sends points to points and lines to lines as well as preserving the incidence
relation. The group of automorphisms of $\Gamma$ is denoted by
$\textsf{Aut}(\Gamma)$. While we have defined generalised polygons via this
incidence graph, we will often think of them as incidence geometries. For more
information on incidence structures and generalised polygons, we refer to [11,
15, 20].

From here on, a generalised quadrangle is assumed to be finite, _i.e._ , it
has a finite set of points and set of lines. Furthermore, a generalised
quadrangle with order $(s,t)$ has $(s+1)(st+1)$ points and $(t+1)(st+1)$ lines
[15, (1.2.1)].

######  Lemma 2.1 ([8, Corollary 2.3]).

Let $G$ be a group acting on a generalised quadrangle $\Gamma$ and suppose
$g\in G$. Let $\Gamma_{g}=(\mathcal{P}_{g},\mathcal{L}_{g},\mathcal{I}_{g})$
be the fixed substructure of $g$. If $|\mathcal{P}_{g}|\geqslant 2$,
$|\mathcal{L}_{g}|\geqslant 2$ and $\Gamma_{g}$ admits an automorphism group
$H$ that is transitive on both points and lines, then $\Gamma_{g}$ is a
generalised quadrangle of order $(s^{\prime},t^{\prime})$ for some positive
integers $s^{\prime}$ and $t^{\prime}$.

The next number-theoretic lemma concerns the solutions of $s$ and $t$ given a
certain number of points and lines.

######  Lemma 2.2.

Let $a,b,s,t$ be positive integers and $p$ be a prime. If

$(s+1)(st+1)=p^{a}\ \textrm{and}\ (t+1)(st+1)=p^{b},$ (1)

then $p=2$. Furthermore, we have: $(s=1\ \textrm{and}\ b=a/2+1)$ or $(t=1\
\textrm{and}\ a=b/2+1)$.

  * Proof.

First, let us suppose that $s,t\geqslant 2$. Since $s+1$ and $t+1$ are powers
of $p$ with $s+1,t+1\geqslant 3$, we can write $s+1=p^{c}$ and $t+1=p^{d}$ for
some integers $c,d\geqslant 2$. Then

$\displaystyle st+1={}$ $\displaystyle(p^{c}-1)(p^{d}-1)+1$ $\displaystyle={}$
$\displaystyle p^{c+d}-p^{c}-p^{d}+2$ $\displaystyle\equiv{}$ $\displaystyle
2\pmod{p}.$

Since $st+1$ divides $p^{a}$, we must have that $p=2$. Hence,

$st+1=2(2^{c+d-1}-2^{c-1}-2^{d-1}+1).$

Now, because $c,d\geqslant 2$, we find that $c+d-1\geqslant 3$, $c-1\geqslant
1$ and $d-1\geqslant 1$. Therefore, $2^{c+d-1}-2^{c-1}-2^{d-1}$ contains a
factor of $2$. Hence,

$st+1=2(2(2^{c+d-2}-2^{c-2}-2^{d-2})+1).$

Again, because $st+1$ is a power of $2$, we must have
$2^{c+d-2}-2^{c-2}-2^{d-2}=0$. This means that $2^{c+d}-2^{c}-2^{d}=0$.
Suppose $c\geqslant d$ and write $c=d+k$ for some non-negative integer $k$.
Then we find that $2^{2d+k}-2^{d+k}-2^{d}=0$. Simplifying further, we obtain
$2^{d+k}=2^{k}+1$. If $k=0$, then $2^{d}=2$, and so, $d=1$, which contradicts
our assumption. Now, if $k\geqslant 1$, then there are no solutions since
$2^{d+k}$ is even and $2^{k}+1$ is odd. The case where $d\geqslant c$ is
analogous. Therefore, there are no solutions for $s$ and $t$ if we assume that
$s,t\geqslant 2$.

Finally, let us suppose that $s=1$ or $t=1$. Without loss of generality, say
$t=1$. Then $t+1=2$ divides $p^{b}$ and so, $p=2$. Substituting for $t$ and
$p$ in (1), we obtain

$(s+1)^{2}=2^{a}\ \textrm{and}\ 2(s+1)=2^{b}.$

Hence, $s+1=2^{a/2}=2^{b-1}$, from which, we obtain $b=a/2+1$. If we consider
the case where $s=1$, then by the same argument, we obtain $a=b/2+1$. ∎

### 2.2 Group Theory

#### 2.2.1 Notation

Let $n$ be a positive integer and $q$ be a prime power. We denote the cyclic
group of order $n$ by $C_{n}$, the dihedral group of order $2n$ by $D_{n}$,
the elementary abelian group of order $q$ by $E_{q}$. Given two groups $H$ and
$K$, we write $H.K$ to mean an extension of $H$ by $K$, _i.e._ , a group $G$
with a normal subgroup $M$ where $M\cong H$ and $G/M\cong K$. When the
extension is a split extension, we write $H:K$ (or $H\rtimes K$).

#### 2.2.2 Elementary Results

In our study of groups of Lie type, we will come across subgroups that are
semidirect products of two groups with coprime order. A classical result that
will be useful in studying these subgroups is the Schur-Zassenhaus Theorem,
which can be found in [10, Section 3B].

######  Theorem 2.3 (Schur-Zassenhaus Theorem).

Let $G$ be a group with a normal subgroup $N$ such that $|G|$ and $|G/N|$ are
coprime. Then there exists a subgroup $K\leqslant G$ such that $G=N\rtimes K$,
_i.e._ , $G=NK$ and $N\cap K=\vmathbb{1}$. We call $K$ a _complement_ of $N$
in $G$. Moreover, all complements of $N$ in $G$ are conjugate to $K$.

######  Corollary 2.4.

Let $G=N\rtimes K$ where $\gcd(|N|,|K|)=1$. Suppose $M\leqslant G$ with
$|M|=|K|$. Then $M=K^{x}$ for some $x\in G$.

  * Proof.

Observe that $N\cap M=\vmathbb{1}$ since they have coprime orders. Thus,
$|NM|=|N||M|/|N\cap M|=|N||K|=|G|$. Hence, $M$ is a complement of $N$ in $G$.
By the Schur-Zassenhaus Theorem, all complements of $N$ are conjugate and so,
$M=K^{x}$ for some $x\in G$. ∎

######  Corollary 2.5.

Let $G=N\rtimes K$ where $\gcd(|N|,|K|)=1$ and suppose $g\in G$ such that
$|g|$ divides $|K|$. Then $g\in K^{h}$ for some $h\in G$. In particular, the
number of conjugacy classes of elements with order $|g|$ in $G$ is at most the
number of conjugacy classes of elements with order $|g|$ in $K$.

  * Proof.

First, observe that $g$ acts on $N$ by conjugation. Since $|g|$ divides $|K|$,
it follows that $g\notin N$. Hence, we have the subgroup
$M\vcentcolon=N\rtimes\mathopen{}\mathclose{{}\left\langle
g}\right\rangle\leqslant G$. Note that
$KM=K(N\rtimes\mathopen{}\mathclose{{}\left\langle
g}\right\rangle)=KN\mathopen{}\mathclose{{}\left\langle g}\right\rangle=G$ and
$|G|=|K||M|/|K\cap M|$. Thus,

$|K\cap
M|=\frac{|K||M|}{|G|}=\frac{|K||N||\mathopen{}\mathclose{{}\left\langle
g}\right\rangle|}{|G|}=\frac{|K||N||g|}{|K||N|}=|g|.$

By Corollary 2.4, we deduce that $K\cap M$ is a complement of $N$ in $M$. By
the Schur-Zassenhaus Theorem, all complements of $N$ in $M$ are conjugate.
Therefore, $K\cap M=\mathopen{}\mathclose{{}\left\langle g}\right\rangle^{x}$
for some $x\in M$. Thus, $\mathopen{}\mathclose{{}\left\langle
g}\right\rangle\leqslant K^{x^{-1}}$, whence, $g\in K^{h}$, where $h=x^{-1}$.
The number of conjugacy classes of elements with order $|g|$ is at most the
number in $K$ because any element of $G$ with order $|g|$ is conjugate to an
element of $K$. ∎

#### 2.2.3 Permutation Groups

Our investigation involves groups of Lie type acting on generalised
quadrangles. The following lemma provides a formula for calculating the number
of fixed points of an element with respect to a group action.

######  Lemma 2.6 (Formula for the Number of Fixed Points [12, Lemma 2.5]).

Let $G$ be a finite group acting transitively on a set $\Omega$. Let
$\alpha\in\Omega$ and $g\in G$. Then the number of fixed points of $g$,
denoted $\pi(g)$, is given by

$\pi(g)=\frac{|\Omega||g^{G}\cap G_{\alpha}|}{|g^{G}|}.$ (2)

######  Lemma 2.7 ([8, Lemma 3.2]).

Let $G$ act transitively on a set $\Omega$ and $g$ be a non-identity element
in $G$. Consider a point $\alpha\in\Omega_{g}$, where $\Omega_{g}$ is the set
of fixed points of $g$. If
$|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right):\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)\cap
G_{\alpha}|=|\Omega_{g}|$, then
$\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)$ acts transitively on
$|\Omega_{g}|$.

The hypothesis of Lemma 2.7 is satisfied when the conjugacy class of $g$ in
$G$ does not split into multiple conjugacy classes in $G_{\alpha}$. The
following lemma is an immediate consequence of [6, Lemma 1.2 (iii)].

######  Lemma 2.8.

Let $G$ act transitively on a set $\Omega$ and $g$ be a non-identity element
in $G$. Consider a point $\alpha\in\Omega_{g}$, where $\Omega_{g}$ is the set
of fixed points of $g$. If $g^{G}\cap G_{\alpha}=g^{G_{\alpha}}$ then
$|\Omega_{g}|=|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right):\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)\cap
G_{\alpha}|$.

## 3 Suzuki Groups

### 3.1 Structure of Suzuki Groups

In this subsection, we provide some background information on the family of
Suzuki groups. Throughout this section, we denote a Suzuki group by
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$,
where $q=2^{2m+1}$ for some positive integer $m$. We start with the following
result which can be found in [18, Theorem 9].

######  Theorem 3.1 (Subgroups of a Suzuki Group).

Let
$G=\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$.
Then $G$ contains the following subgroups:

  1. (i)

$M\cong E_{q}.E_{q}.C_{q-1}$;

  2. (ii)

$B_{0}=\textbf{N}_{G}(A_{0})\cong D_{q-1}$, where $A_{0}\cong C_{q-1}$;

  3. (iii)

$A_{1}\cong C_{q+\sqrt{2q}+1}$, $A_{2}\cong C_{q-\sqrt{2q}+1}$;

  4. (iv)

$B_{i}=\textbf{N}_{G}(A_{i})\cong C_{q\pm\sqrt{2q}+1}:C_{4}$ for
$i\in\\{1,2\\}$; and

  5. (v)

$N_{0}\cong\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$,
where $q=q_{0}^{r}$.

Moreover, any subgroup of $G$ is conjugate to a subgroup of one of the above.

The next lemma shows that all the involutions in a Suzuki group are conjugate.

######  Lemma 3.2 (Conjugacy Class of Involutions).

Let $G=\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
and $H$ be a maximal subgroup of $G$. Then $G$ and $H$ both have exactly one
conjugacy class of involutions.

  * Proof.

It was shown in [18, Proposition 7] that all the involutions are conjugate in
$G$. Furthermore, if $H\cong E_{q}.E_{q}.C_{q-1}$, then a cyclic subgroup of
$H$ of order $q-1$ permutes the set of involutions in $H$ transitively [18,
Proposition 8]. Therefore, we only need to consider the cases:111Note that the
subfield case:
$H\cong\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$,
we have that $H$ is a Suzuki group and so, it follows from [18, Proposition
7]. $H\cong D_{q-1}=C_{q-1}:C_{2}$ and $H\cong C_{q\pm\sqrt{2q}+1}:C_{4}$.
Since $q$ is a power of $2$, we have that $\gcd(q-1,2)=1$ and
$\gcd(q\pm\sqrt{2q}+1,4)=1$. Hence, by Corollary 2.5, we conclude that all the
involutions are conjugate in $G$. ∎

The following lemma provides information about the centraliser of an
involution in $G$.

######  Lemma 3.3 (Centraliser of an Involution).

Let $G=\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
with maximal subgroups $H,K\leqslant G$ where $H\cong D_{q-1}$ and $K\cong
C_{q\pm\sqrt{2q}+1}:C_{4}$. Let $g\in G$, $h\in H$ and $k\in K$ be
involutions. Then

$|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)|=q^{2},\
|\textbf{C}_{H}\mathopen{}\mathclose{{}\left(h}\right)|=2,\ \textrm{and}\
|\textbf{C}_{K}\mathopen{}\mathclose{{}\left(k}\right)|=4.$

  * Proof.

Consider
$G=\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$.
Note that $\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)\leqslant Q$
where $Q$ is a Sylow $2$-subgroup of order $q^{2}$ [18, Proposition 1].
Moreover, $g\in\textbf{Z}\mathopen{}\mathclose{{}\left(Q}\right)$ [18,
Proposition 7]. Therefore,
$|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)|=|Q|=q^{2}$.

Next, we find
$\textbf{C}_{H}\mathopen{}\mathclose{{}\left(h}\right)=\textbf{C}_{G}\mathopen{}\mathclose{{}\left(h}\right)\cap
H$. Observe that
$|\textbf{C}_{H}\mathopen{}\mathclose{{}\left(h}\right)|=|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(h}\right)\cap
H|\leqslant 2$ since
$|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(h}\right)|=q^{2}$ and
$|H|=2(q-1)$. Note that
$h\in\textbf{C}_{H}\mathopen{}\mathclose{{}\left(h}\right)$. Hence,
$\textbf{C}_{H}\mathopen{}\mathclose{{}\left(h}\right)\cong C_{2}$. Similarly,
we have
$|\textbf{C}_{K}\mathopen{}\mathclose{{}\left(k}\right)|=|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(k}\right)\cap
K|\leqslant 4$ since
$|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(k}\right)|=q^{2}$ and
$|K|=4(q\pm\sqrt{2q}+1)$. Let us now write
$K=\mathopen{}\mathclose{{}\left\langle
x}\right\rangle:\mathopen{}\mathclose{{}\left\langle y}\right\rangle$, where
$|x|=q\pm\sqrt{2q}+1$ and $|y|=4$. Using Corollary 2.5, we find that
$k=(y^{2})^{z}$ for some $z\in K$, whence,
$y^{z}\in\textbf{C}_{K}\mathopen{}\mathclose{{}\left(k}\right)$. Since
$|y|=4$, we have that
$\textbf{C}_{K}\mathopen{}\mathclose{{}\left(k}\right)\cong C_{4}$. ∎

######  Lemma 3.4 (Number of Fixed Points of an Involution).

Let $G=\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
act primitively on a set $\Omega$ and take a point $\alpha\in\Omega$. Suppose
$g\in G_{\alpha}$ is an involution. Then

$|\Omega_{g}|=\begin{cases}q^{2}/2=2^{4m+1}&\textrm{if $G_{\alpha}\cong
D_{q-1}$},\\\ q^{2}/4=2^{4m}&\textrm{if $G_{\alpha}\cong
C_{q\pm\sqrt{2q}+1}:C_{4}$},\\\ (q/q_{0})^{2}=2^{2n_{0}(r_{0}-1)}&\textrm{if
$G_{\alpha}\cong\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$
where $q_{0}=2^{n_{0}}>2$ and $q=q_{0}^{r_{0}}$ for some prime
$r_{0}$.}\end{cases}$

Furthermore, $\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)$ acts
transitively on $\Omega_{g}$, the set of points fixed by $g$.

  * Proof.

Since $G_{\alpha}$ contains a single class of involutions (Lemma 3.2), we
conclude that $g^{G}\cap G_{\alpha}=g^{G_{\alpha}}$. Thus, applying Lemma 2.8,
we obtain
$|\Omega_{g}|=|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right):\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)\cap
G_{\alpha}|$. Using Lemma 3.3, we obtain the desired formulae for
$|\Omega_{g}|$. Next, by Lemma 2.7, we conclude that
$\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)$ acts transitively on
$\Omega_{g}$. ∎

### 3.2 Suzuki Groups Acting on Generalised Quadrangles

We now prove Theorem 1.1 in the case where the socle is isomorphic to
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$.

  * Proof.

Suppose that an almost simple group with socle
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$ acts
primitively on the point-set and the line-set of a generalised quadrangle
$\Gamma=(\mathcal{P},\mathcal{L},\mathcal{I})$ with order $(s,t)$ where
$s,t\geqslant 2$. Then $|\mathcal{P}|=(s+1)(st+1)$. Note that $s+1\geqslant 3$
and $st+1\geqslant 3$. Therefore, their product cannot be a prime. The
argument in [14, Section 3] shows that the socle of this almost simple group
acts primitively on both $\mathcal{P}$ and $\mathcal{L}$. Therefore, it
suffices to consider the Suzuki group
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
acting primitively on both $\mathcal{P}$ and $\mathcal{L}$. Let
$G=\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$.
For a point $\alpha\in\mathcal{P}$ and a line $L\in\mathcal{L}$, their
respective stabilisers $G_{\alpha}$ and $G_{L}$ are maximal subgroups of $G$.
By Theorem 3.1, a maximal subgroup of
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$ is
isomorphic to one of the following:

    1. (i)

(Parabolic) $E_{q}.E_{q}.C_{q-1}$;

    2. (ii)

(Dihedral) $D_{q-1}$;

    3. (iii)

(Frobenius) $C_{q\pm\sqrt{2q}+1}:C_{4}$; or

    4. (iv)

(Subfield)
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$,
where $q_{0}=2^{n_{0}}>2$ and $q=q_{0}^{r_{0}}$ for some prime $r_{0}$.

Moreover, there is only one conjugacy class of each type of maximal subgroup,
see also [4, Table 8.16].

Note that if $G_{\alpha}\cong E_{q}.E_{q}.C_{q-1}$, then by [18], the action
of $G$ is $2$-transitive. Thus, a pair of collinear points can be mapped to a
pair of non-collinear points, which is a contradiction. Therefore, we only
need to investigate the cases: $D_{q-1}$, $C_{q\pm\sqrt{2q}+1}:C_{4}$ and
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$
for the point and line-stabiliser and show that there are no generalised
quadrangles in those scenarios. To this end, suppose $G_{\alpha}$ is
isomorphic to $D_{q-1}$, $C_{q\pm\sqrt{2q}+1}:C_{4}$ or
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$
and $G_{L}$ is isomorphic to $D_{q-1}$, $C_{q\pm\sqrt{2q}+1}:C_{4}$ or
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q_{1}}\right)$,
where $q_{i}=2^{n_{i}}>2$ and $q_{i}^{r_{i}}=q$ for primes $r_{i}$ and
$i=1,2$. Observe that both $G_{\alpha}$ and $G_{L}$ contain an involution, say
$g\in G_{\alpha}$ and $h\in G_{L}$. Since all the involutions in $G$ are
conjugate (Lemma 3.2), there exists a $k\in G$ such that $g=h^{k}$. Thus,
$g\in G_{L}^{k}$. Since $G_{L}^{k}=G_{L^{\prime}}$ where
$L^{\prime}=L^{k}\in\mathcal{L}$, we can take $L$ to be $L^{\prime}$ and
assume that we have an involution $g\in G_{\alpha}\cap G_{L}$. Let
$\mathcal{P}_{g}$ and $\mathcal{L}_{g}$ be the set of points and the set of
lines fixed by $g$, respectively. By Lemma 3.4, we find that the number of
fixed points and lines are

$|\mathcal{P}_{g}|=2^{a}\ \textrm{and}\ |\mathcal{L}_{g}|=2^{b},$

where $a\in A\vcentcolon=\\{4m,4m+1,2n_{0}(r_{0}-1)\\}$ and $b\in
B\vcentcolon=\\{4m,4m+1,2n_{1}(r_{1}-1)\\}$. By Lemma 2.1, the fixed
substructure, $(\mathcal{P}_{g},\mathcal{L}_{g},\mathcal{I}_{g})$ is a
generalised quadrangle of order $(s^{\prime},t^{\prime})$ for some positive
integers $s^{\prime}$ and $t^{\prime}$. Therefore,

$|\mathcal{P}_{g}|=(s^{\prime}+1)(s^{\prime}t^{\prime}+1)=2^{a}\ \textrm{and}\
|\mathcal{L}_{g}|=(t^{\prime}+1)(s^{\prime}t^{\prime}+1)=2^{b}.$

By Lemma 2.2, we have that $b=a/2+1$ or dually, $a=b/2+1$. Let us suppose that
$b=a/2+1$. Thus, $a$ is even and so, $a=4m$ or $2n_{0}(r_{0}-1)$. If $a=4m$,
then $b-1=a/2=2m$. Hence, $b=2m+1$, which is odd. However, the only odd
element in $B$ is $4m+1$ and so, we have a contradiction. Next, if
$a=2n_{0}(r_{0}-1)$, then $b=a/2+1=n_{0}(r_{0}-1)+1$. However, since $r_{0}$
is an odd prime, it follows that $b$ is odd. Therefore, $b=4m+1$.
Consequently, we find $4m+1=b=n_{0}r_{0}-n_{0}+1<n_{0}r_{0}=2m+1$, which is a
contradiction. The case where $a=b/2+1$ is analogous. Therefore, the Suzuki
group $\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
cannot act primitively on both the set of points and the set of lines of a
generalised quadrangle. Consequently, an almost simple group with socle
isomorphic to
$\prescript{2}{}{\textrm{B}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
cannot act primitively on both the set of points and the set of lines of a
generalised quadrangle. ∎

## 4 Ree Groups

We provide some background information on the family of small Ree groups.
Throughout this section, we denote a small Ree group by
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$,
where $q=3^{2m+1}$ for some positive integer $m$.

### 4.1 Structure of Ree Groups

We have the following theorem regarding the maximal subgroups of
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$ (see
[4, Table 9.43]).

######  Theorem 4.1 (Maximal Subgroups of a Small Ree Group).

Let $G=\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
and $H$ be a maximal subgroup of $G$. Then is isomorphic to one of the
following:

  1. (i)

$E_{q}.E_{q}.E_{q}.C_{q-1}$;

  2. (ii)

$C_{2}\times\textsf{PSL}_{2}({q})$;

  3. (iii)

$(E_{4}\times D_{(q+1)/4}):C_{3}$;

  4. (iv)

$C_{q\pm\sqrt{3q}+1}:C_{6}$; and

  5. (v)

$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$,
where $q_{0}=3^{n_{0}}>3$ and $q=q_{0}^{r_{0}}$ for some prime $r_{0}$.

Moreover, there is only one conjugacy class of each type of maximal subgroup.

Analogous to the Suzuki groups, elements of order $3$ play a crucial role in
studying the action of a small Ree group on a generalised quadrangle. First,
we recall a definition from group theory. For a group $G$ and $g\in G$, we say
that $g$ is real in $G$ if $g$ is conjugate to its inverse in $G$, _i.e._ ,
there exists an element $h\in G$ such that $g^{-1}=g^{h}$. We may omit the “in
$G$” part and simply refer to $g$ as a real element.

The following lemma is useful for analysing the maximal subgroup isomorphic to
$(E_{4}\times D_{(q+1)/4}):C_{3}$.

######  Lemma 4.2.

Let $H=KR\cong(E_{4}\times D_{(q+1)/4}):C_{3}$ where $K\cong E_{4}\times
D_{(q+1)/4}$ and $R=\mathopen{}\mathclose{{}\left\langle y}\right\rangle\cong
C_{3}$. Then $y$ centralises an involution in $H$.

  * Proof.

Note that $\mathopen{}\mathclose{{}\left\langle y}\right\rangle$ acts by
conjugation on the set of involutions in $H$. We count the number of
involutions in $H$. Since $q+1=3^{2m+1}+1=3(3^{2})^{m}+1\equiv
3+1\pmod{8}\equiv 4\pmod{8}$, it follows that $(q+1)/4$ is odd and thus,
$D_{(q+1)/4}$ has $(q+1)/4$ involutions. Since an involution in $K$ is the
product of an element of $E_{4}$ and an involution in $D_{(q+1)/4}$, or is
just an involution in $E_{4}$, the number of involutions in $K$ is
$4(q+1)/4+3=q+4$. Now, all the involutions in $H$ are in $K$ because
$K\trianglelefteq H$ and $\gcd(|K|,|R|)=1$. Therefore, $H$ has $q+4$
involutions. Focusing on the action of $\mathopen{}\mathclose{{}\left\langle
y}\right\rangle$ on the set of involutions, suppose that
$\mathopen{}\mathclose{{}\left\langle y}\right\rangle$ has no fixed points,
_i.e._ , $y$ does not centralise any involution. Then the orbits of
$\mathopen{}\mathclose{{}\left\langle y}\right\rangle$ must have length
divisible by $3$. This implies that $q+4$ is divisible by $3$, which is a
contradiction as $q=3^{2m+1}$. Therefore, $y$ has a fixed point, _i.e._ , $y$
centralises an involution. ∎

######  Lemma 4.3 (Conjugacy Classes of Elements of Order Three).

Let $G=\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
and $H$ be a maximal subgroup of $G$. Then the following statements hold:

  1. (i)

In $G$, there is one conjugacy class of real elements of order $3$ and two
conjugacy classes of non-real elements of order $3$.

  2. (ii)

If $H\cong C_{2}\times\textsf{PSL}_{2}({q})$, $H\cong(E_{4}\times
D_{(q+1)/4}):C_{3}$, or $H\cong C_{q\pm\sqrt{3q}+1}:C_{6}$, then there are two
$H$-conjugacy classes of non-real elements of order $3$.

  * Proof.

First, we find that the centre of a Sylow $3$-subgroup of $G$ contains one
conjugacy class of elements with order $3$ with representative labelled $X$
[21, Chapter III, Paragraph 4]. Also, in [21, Chapter III, Paragraph 7], we
find that there are two conjugacy classes of elements of order $3$ with
representatives labelled $T$ and $T^{-1}$. This yields (i).

Now we focus on (ii). From [21, Chapter III, Paragraphs 1 and 3], we deduce
that an order $3$ element in $G$ is not real precisely when it centralises an
involution. Therefore, it suffices to show the following two points:

    1. (a)

An element of order $3$ in $H$ centralises an involution.

    2. (b)

There are at most two conjugacy classes of elements with order $3$ in $H$.

Indeed, once we have established that there are at most two conjugacy classes,
it follows that we have exactly two conjugacy classes since the elements are
not real.

Let us suppose $H\cong C_{2}\times\textsf{PSL}_{2}({q})$. Note that all the
order $3$ elements of $H$ are in $\textsf{PSL}_{2}({q})$. Therefore, the
involution in $C_{2}$ is centralised by the order $3$ elements of $H$. Also,
there are exactly two conjugacy classes of elements with order $3$ in
$\textsf{PSL}_{2}({q})$ (see [17, Chapter 3, (6.3) (iii)]) and hence, two
conjugacy classes in $H$.

Next, we consider $H=KR\cong(E_{4}\times D_{(q+1)/4}):C_{3}$ where $K\cong
E_{4}\times D_{(q+1)/4}$ and $R=\mathopen{}\mathclose{{}\left\langle
y}\right\rangle\cong C_{3}$. Observe that $\gcd(|E_{4}\times
D_{(q+1)/4}|,|C_{3}|)=\gcd(2(q+1),3)=1$. Hence, by Corollary 2.5, any element
of order $3$ must be conjugate in $H$ to $y$ or $y^{-1}$. So there are at most
two conjugacy classes of elements with order $3$. Furthermore, $y$ centralises
an involution in $H$ by Lemma 4.2.

Finally, let us suppose that $H=KR\cong C_{q\pm\sqrt{3q}+1}:C_{6}$ where
$K\cong C_{q\pm\sqrt{3q}+1}$ and $R=\mathopen{}\mathclose{{}\left\langle
y}\right\rangle\cong C_{6}$. We find that
$\gcd(|C_{q\pm\sqrt{3q}+1}|,|C_{6}|)=\gcd(q\pm\sqrt{3q}+1,6)=1$ since
$q\pm\sqrt{3q}+1\equiv 1\pm 1+1\pmod{2}\equiv 1\pmod{2}$ and
$q\pm\sqrt{3q}+1\equiv 1\pmod{3}$. There are two conjugacy classes of elements
with order $3$ in $R$, namely, $(y^{2})^{R}$ and $(y^{4})^{R}$. Using
Corollary 2.5 again, we find that the number of conjugacy classes of elements
with order $3$ in $H$ is at most $2$. Moreover, we note that $y^{2}$
centralises $y^{3}$, which is an involution. ∎

######  Lemma 4.4 (Centralisers of Elements of Order Three).

Let
$G=\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$,
$x\in G$ be a real element of order $3$ and $y\in G$ be a non-real element of
order $3$. Then

$|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(x}\right)|=q^{3}\ \textrm{and}\
|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(y}\right)|=2q^{2}.$

Consider maximal subgroups $H_{1}$, $H_{2}$ and $H_{3}$ of $G$ where
$H_{1}\cong C_{2}\times\textsf{PSL}_{2}({q})$, $H_{2}\cong(E_{4}\times
D_{(q+1)/4}):C_{3}$ and $H_{3}\cong C_{q\pm\sqrt{3q}+1}:C_{6}$. Suppose
$h_{i}\in H_{i}$ are elements of order $3$ for $i\in\\{1,2,3\\}$. Then

$|\textbf{C}_{H_{1}}\mathopen{}\mathclose{{}\left(h_{1}}\right)|=2q,\
|\textbf{C}_{H_{2}}\mathopen{}\mathclose{{}\left(h_{2}}\right)|=6\
\textrm{and}\
|\textbf{C}_{H_{3}}\mathopen{}\mathclose{{}\left(h_{3}}\right)|=6.$

  * Proof.

From [21, Chapter III, Paragraphs 2 and 3], we obtain
$|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(y}\right)|=2q^{2}$ and
$|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(x}\right)|=q^{3}$, respectively.

From [17, Chapter 3, (6.4) (i)], we find that
$|\textbf{C}_{\textsf{PGL}_{2}({q})}\mathopen{}\mathclose{{}\left(x}\right)|=q$
when $x$ is an element of order $3$. Since $\textsf{PSL}_{2}({q})$ has index
$2$ in $\textsf{PGL}_{2}({q})$ and
$C\vcentcolon=\textbf{C}_{\textsf{PGL}_{2}({q})}\mathopen{}\mathclose{{}\left(x}\right)$
has odd order, it follows that $C\leqslant\textsf{PSL}_{2}({q})$. Now, back to
$H_{1}$. Since the $C_{2}$ in $H_{1}$ also centralises $h_{1}$, we obtain
$|\textbf{C}_{H_{1}}\mathopen{}\mathclose{{}\left(h_{1}}\right)|=2q$.

Next, observe that
$|\textbf{C}_{H_{2}}\mathopen{}\mathclose{{}\left(h_{2}}\right)|=|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(h_{2}}\right)\cap
H_{2}|\leqslant 6$ since
$\gcd(|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(h_{2}}\right)|,|H_{2}|)=\gcd(2q^{2},4\cdot
2(q+1)/4\cdot 6)=\gcd(2q^{2},12(q+1))=6$. Furthermore, $h_{2}$ centralises an
involution in $H_{2}$ (Lemma 4.2). Therefore,
$|\textbf{C}_{H_{2}}\mathopen{}\mathclose{{}\left(h_{2}}\right)|=6$.

Finally, we consider $H_{3}$. By the argument as above, we see that
$\gcd(|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(h_{3}}\right)|,|H_{3}|)=\gcd(2q^{2},6(q\pm\sqrt{3q}+1))=6$,
whence,
$|\textbf{C}_{H_{3}}\mathopen{}\mathclose{{}\left(h_{3}}\right)|=|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(h_{3}}\right)\cap
H_{3}|\leqslant 6$. Since $h_{3}$ lies in some subgroup isomorphic to $C_{6}$
in $H_{3}$ (Corollary 2.5), it follows that $h_{3}$ is centralised by $6$
elements. Therefore,
$|\textbf{C}_{H_{3}}\mathopen{}\mathclose{{}\left(h_{3}}\right)|=6$. ∎

######  Lemma 4.5 (Number of Fixed Points of Order Three Elements).

Let $G=\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
act primitively on a set $\Omega$ and take a point $\alpha\in\Omega$. Suppose
we have a non-real element $g\in G_{\alpha}$ of order $3$. Then

$|\Omega_{g}|=\begin{cases}q&\textrm{if $G_{\alpha}\cong
C_{2}\times\textsf{PSL}_{2}({q})$},\\\ q^{2}/3&\textrm{if
$G_{\alpha}\cong(E_{4}\times D_{(q+1)/4}):C_{3}$ or $G_{\alpha}\cong
C_{q\pm\sqrt{3q}+1}:C_{6}$}\\\ (q/q_{0})^{2}&\textrm{if
$G_{\alpha}\cong\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$
where $q_{0}=3^{n_{0}}>3$ and $q=q_{0}^{r_{0}}$ for some prime
$r_{0}$}.\end{cases}$

Furthermore, $\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)$ acts
transitively on $\Omega_{g}$, the set of points fixed by $g$.

  * Proof.

By Lemma 4.3, $G_{\alpha}$ has precisely two conjugacy classes of order $3$
non-real elements. Since $G$ also has two conjugacy classes of these elements,
the conjugacy classes do not split, _i.e._ , $g^{G}\cap
G_{\alpha}=g^{G_{\alpha}}$. Thus, applying Lemma 2.8, we obtain
$|\Omega_{g}|=|\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right):\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)\cap
G_{\alpha}|$. Using Lemma 4.4, we obtain the desired formulae for
$|\Omega_{g}|$. Next, by Lemma 2.7, we conclude that
$\textbf{C}_{G}\mathopen{}\mathclose{{}\left(g}\right)$ acts transitively on
$\Omega_{g}$. ∎

### 4.2 Ree Groups Acting on Generalised Quadrangles

We now prove Theorem 1.1 in the case where the socle is isomorphic to
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$.

  * Proof.

Suppose that an almost simple group with socle
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$ acts
primitively on the point-set and the line-set of a generalised quadrangle
$\Gamma=(\mathcal{P},\mathcal{L},\mathcal{I})$ with order $(s,t)$ where
$s,t\geqslant 2$. Then $|\mathcal{P}|=(s+1)(st+1)$. Note that $s+1\geqslant 3$
and $st+1\geqslant 3$. Therefore, their product cannot be a prime. The
argument in [14, Section 4] shows that the socle of this almost simple group
acts primitively on both $\mathcal{P}$ and $\mathcal{L}$. Therefore, it
suffices to consider the Ree group
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
acting primitively on both $\mathcal{P}$ and $\mathcal{L}$. Let
$G=\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$.
For a point $\alpha\in\mathcal{P}$ and a line $L\in\mathcal{L}$, their
respective stabilisers $G_{\alpha}$ and $G_{L}$ are maximal subgroups of $G$.
By Theorem 4.1, a maximal subgroup of
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$ is
isomorphic to one of the following:

    1. (i)

$E_{q}.E_{q}.E_{q}.C_{q-1}$;

    2. (ii)

$C_{2}\times\textsf{PSL}_{2}({q})$;

    3. (iii)

$(E_{4}\times D_{(q+1)/4}):C_{3}$;

    4. (iv)

$C_{q\pm\sqrt{3q}+1}:C_{6}$; and

    5. (v)

$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$
where $q=q_{0}^{r}$ for some prime $r$ and $q_{0}>3$.

Moreover, there is only one conjugacy class of each type of maximal subgroup.

If $G_{\alpha}\cong E_{q}.E_{q}.E_{q}.C_{q-1}$, then the action of $G$ is
$2$-transitive [21, Theorem (v)]. Thus, a pair of collinear points can be
mapped to a pair of non-collinear points, which is a contradiction. Therefore,
we only need to investigate the cases: $C_{2}\times\textsf{PSL}_{2}({q})$,
$(E_{4}\times D_{(q+1)/4}):C_{3}$, $C_{q\pm\sqrt{3q}+1}:C_{6}$ and
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$
for the point and line-stabiliser and show that there are no generalised
quadrangles in these scenarios. To this end, suppose $G_{\alpha}$ is
isomorphic to $C_{2}\times\textsf{PSL}_{2}({q})$, $(E_{4}\times
D_{(q+1)/4}):C_{3}$, $C_{q\pm\sqrt{2q}+1}:C_{4}$ or
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q_{0}}\right)$
and $G_{L}$ is isomorphic to $C_{2}\times\textsf{PSL}_{2}({q})$, $(E_{4}\times
D_{(q+1)/4}):C_{3}$, $C_{q\pm\sqrt{2q}+1}:C_{4}$ or
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q_{1}}\right)$,
where $q_{i}=3^{n_{i}}>3$ and $q_{i}^{r_{i}}=q$ for primes $r_{i}$ and
$i=1,2$. By Lemma 4.3, both $G_{\alpha}$ and $G_{L}$ contain a non-real
element of order $3$, say $g\in G_{\alpha}$ and $h\in G_{L}$. Moreover, there
are precisely two classes of non-real elements of order $3$ in $G$, and so $g$
is conjugate to $h$ or $h^{-1}$. Without loss of generality, suppose $g$ is
conjugate to $h$, and so, we can write $g=h^{k}$ for some $k\in G$. Hence,
$g\in G_{L}^{k}$. Since $G_{L}^{k}=G_{L^{\prime}}$ where
$L^{\prime}=L^{k}\in\mathcal{L}$, we can take $L$ to be $L^{\prime}$ and
assume that we have an order $3$ non-real element $g\in G_{\alpha}\cap G_{L}$.
Let $\mathcal{P}_{g}$ and $\mathcal{L}_{g}$ be the set of points and the set
of lines fixed by $g$, respectively. By Lemma 4.5, we find that the number of
fixed points and lines are

$|\mathcal{P}_{g}|=3^{a}\ \textrm{and}\ |\mathcal{L}_{g}|=3^{b},$

where $a\in A\vcentcolon=\\{2m+1,4m+1,2n_{0}(r_{0}-1)\\}$ and $b\in
B\vcentcolon=\\{2m+1,4m+1,2n_{1}(r_{1}-1)\\}$. By Lemma 2.1, the fixed
substructure, $(\mathcal{P}_{g},\mathcal{L}_{g},\mathcal{I}_{g})$ is a
generalised quadrangle of order $(s^{\prime},t^{\prime})$ for some positive
integers $s^{\prime}$ and $t^{\prime}$. Therefore,

$|\mathcal{P}_{g}|=(s^{\prime}+1)(s^{\prime}t^{\prime}+1)=3^{a}\ \textrm{and}\
|\mathcal{L}_{g}|=(t^{\prime}+1)(s^{\prime}t^{\prime}+1)=3^{b}.$

However, there are no solutions for $s^{\prime}$ and $t^{\prime}$ by Lemma 2.2
and so, we have a contradiction. Therefore, the small Ree group
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
cannot act primitively on both the set of points and the set of lines of a
generalised quadrangle. Consequently, an almost simple group with socle
isomorphic to
$\prescript{2}{}{\textrm{G}_{2}}\mathopen{}\mathclose{{}\left(q}\right)$
cannot act primitively on both the set of points and the set of lines of a
generalised quadrangle. ∎

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# Unlocking the Potential of Local CSI in
Cell-Free Networks with Channel Aging and Fronthaul Delays ††thanks: L.
Miretti and S. Stańczak acknowledge the financial support by the Federal
Ministry of Education and Research of Germany in the programme of “Souverän.
Digital. Vernetzt.” Joint project 6G-RIC, project identification number:
16KISK020K and 16KISK030. ††thanks: © 2024 IEEE. Personal use of this material
is permitted. Permission from IEEE must be obtained for all other uses, in any
current or future media, including reprinting/republishing this material for
advertising or promotional purposes, creating new collective works, for resale
or redistribution to servers or lists, or reuse of any copyrighted component
of this work in other works.

Lorenzo Miretti and Sławomir Stańczak  _Technische Universität Berlin_ and
_Fraunhofer Institute for Telecommunications Heinrich-Hertz-Institut_ ,
Berlin, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It is generally believed that downlink cell-free networks perform best under
centralized implementations where the local channel state information (CSI)
acquired by the access-points (AP) is forwarded to one or more central
processing units (CPU) for the computation of the joint precoders based on
global CSI. However, mostly due to limited fronthaul capabilities, this
procedure incurs some delay that may lead to partially outdated precoding
decisions and hence performance degradation. In some scenarios, this may even
lead to worse performance than distributed implementations where the precoders
are locally computed by the APs based on partial yet timely local CSI. To
address this issue, this study considers the problem of robust precoding
design merging the benefits of timely local CSI and delayed global CSI. As
main result, we provide a novel distributed precoding design based on the
recently proposed team minimum mean-square error method. As a byproduct, we
also obtain novel insights related to the AP-CPU functional split problem. Our
main conclusion, corroborated by simulations, is that the opportunity of
performing some local precoding computations at the APs should not be
neglected, even in centralized implementations.

##  I Introduction

Cell-free massive MIMO is one of the most promising candidate technologies for
enhancing the performance of future generation wireless networks [1]. Most of
the related current research effort focuses on the development of practical
methods and architectures for turning the known theoretical gains of
coordinated multi-point concepts into commercially attractive solutions [2, 3,
4, 5, 6, 7, 8]. Of particular relevance is the debate on the type of joint
precoding and combining implementation, and on its impact on the functional
split problem in cloud radio access network (C-RAN) architectures [9].

More specifically, taken aside promising yet exotic schemes based, e.g., on
sequential processing over serial fronthauls [10, 11] or iterative
bidirectional over-the-air processing [12], this debate is essentially
centered around the comparison between fully distributed and centralized
implementations. In fully distributed implementations, the access points (AP)
locally compute their precoders and combiners based on local channel state
information (CSI) only [2]. In contrast, in centralized implementations, these
functions are moved to one or more central processing units (CPU) endowed with
global CSI [1]. Hence, due to their ability to form joint precoders and
combiners based on a broader view of the channel state, centralized
implementations are often considered superior, especially in terms of spectral
efficiency. However, the theoretical superiority of centralized
implementations is typically shown under ideal assumptions such as those
related to fronthaul capabilities, which can be challenged in many practical
deployments.

Against this background, this paper studies centralized downlink cell-free
networks with fronthaul delays, which is a key impairment in real-world
settings. In particular, we focus on scenarios where, due to fronthaul
limitations and mobility, the delay incurred by the CPUs in collecting global
CSI, computing the precoders, and forwarding the result is non-negligible with
respect to channel aging [13, 14, 15]. Intuitively, in these scenarios,
centralized implementations experience performance degradation and may be even
outperformed by fully distributed implementations with precoders formed using
partial yet more timely local CSI. To adress this issue, we formulate a
distributed precoding design problem that aims to jointly exploit timely local
CSI and delayed global CSI. To the best of our knowledge, this is the first
time that a similar problem is addressed in the literature. Then, we derive an
optimal solution based on a novel application of the recent team theoretical
framework [11]. In addition, we discuss the structure of the optimal solution
and related practical implementation aspects in C-RAN architectures.
Interestingly, our theoretical and numerical results show that carefully
designed implementations that delegate the computation of at least a portion
of the precoders to the APs may significantly outperform both centralized and
fully distributed implementations, even under pedestrian mobility and
relatively small delays.

Notation: We denote by $\mathbbmss{R}_{++}$ the set of positive reals. The
Euclidean and Frobenius norms are denoted by $\|\cdot\|$ and
$\|\cdot\|_{\mathsf{F}}$, respectively. Let $(\Omega,\Sigma,\mathbb{P})$ be a
probability space. We denote by $\mathcal{H}^{K}$ the set of random vectors,
i.e., $K$-tuples of $\Sigma$-measurable functions $\Omega\to\mathbbmss{C}$,
satisfying $(\forall\mathbbm{x}\in\mathcal{H}^{K})$
$\mathsf{E}[\|\mathbbm{x}\|^{2}]<\infty$. Given a random variable
$X\in\mathcal{H}$, we denote by $\mathsf{E}[X]$ and $\mathsf{V}(X)$ its
expected value and variance, respectively.

##  II Problem statement

###  II-A Basic definitions and assumptions

We consider the downlink of a cell-free wireless network [1] composed of $L$
APs indexed by $\mathcal{L}:=\\{1,\ldots,L\\}$, each of them equipped with $N$
antennas, and $K$ single-antenna user equipments (UEs) indexed by
$\mathcal{K}:=\\{1,\ldots,K\\}$. By assuming a standard flat-fading channel
model for each time-frequency resource element, we denote an arbitrary
realization of the $(NL\times K)$-dimensional global channel matrix by

$\mathbbm{H}:=\begin{bmatrix}\mathbbm{h}_{1}&\ldots&\mathbbm{h}_{K}\end{bmatrix}=\begin{bmatrix}\mathbbm{H}_{1}\\\
\vdots\\\
\mathbbm{H}_{L}\end{bmatrix}=\begin{bmatrix}\mathbbm{h}_{1,1}&\ldots&\mathbbm{h}_{1,K}\\\
\vdots&\ddots&\vdots\\\
\mathbbm{h}_{L,1}&\ldots&\mathbbm{h}_{L,K}\end{bmatrix},$

where $\mathbbm{h}_{l,k}\in\mathcal{H}^{N}$ is a random vector modeling the
fading state between AP $l\in\mathcal{L}$ and UE $k\in\mathcal{K}$. As
customary in the channel aging literature [13, 14], we assume that the random
channel realizations $\mathbbm{H}$ evolve over time according to a stationary
and ergodic discrete-time random process
$\\{\mathbbm{H}[t]\\}_{t\in\mathbbmss{Z}}$, without any further specific
assumption on the time correlation. In addition, we assume that the portions
$\\{\mathbbm{h}_{l,k}[t]\\}_{t\in\mathbbmss{Z}}$ of
$\\{\mathbbm{H}[t]\\}_{t\in\mathbbmss{Z}}$ corresponding to different AP-UE
pairs are mutually independent random processes.

Similarly, by focusing on simple multi-user cooperative transmission
techniques based on linear precoding and on treating interference as noise
[1], we denote an arbitrary realization of the $(NL\times K)$-dimensional
joint precoding matrix by

$\mathbbm{T}=\begin{bmatrix}\mathbbm{t}_{1}&\ldots&\mathbbm{t}_{K}\end{bmatrix}=\begin{bmatrix}\mathbbm{T}_{1}\\\
\vdots\\\
\mathbbm{T}_{L}\end{bmatrix}=\begin{bmatrix}\mathbbm{t}_{1,1}&\ldots&\mathbbm{t}_{1,K}\\\
\vdots&\ddots&\vdots\\\
\mathbbm{t}_{L,1}&\ldots&\mathbbm{t}_{L,K}\end{bmatrix},$

where $\mathbbm{t}_{l,k}\in\mathcal{H}^{N}$ is a linear precoding vector
applied by AP $l\in\mathcal{L}$ to the coded and modulated data stream for UE
$k\in\mathcal{K}$. The joint precoding matrix $\mathbbm{T}$ evolves over time
according to a random process $\\{\mathbbm{T}[t]\\}_{t\in\mathbbmss{Z}}$,
where $\mathbbm{T}[t]$ is adapted to the random channel realization
$\mathbbm{H}[t]$ based on the available channel state information (CSI) at
time $t\in\mathbbmss{Z}$. In particular, we focus on the case where the
submatrices $\mathbbm{T}_{l}[t]$ of $\mathbbm{T}[t]$ corresponding to the
precoding matrices applied by each AP $l\in\mathcal{L}$ may depend on
different CSI. This aspect is treated in more details next.

###  II-B Delayed CSI sharing

Canonical cell-free network models based on time-division duplex operations
assume each AP $l\in\mathcal{L}$ to acquire local measurements of the downlink
local channel $\mathbbm{H}_{l}=[\mathbbm{h}_{l,1}\;\ldots\;\mathbbm{h}_{l,K}]$
by means of uplink pilot signals. In centralized implementations [1], these
local measurements are then typically forwarded by the APs to one or more
central processors for the computation of the joint precoding matrix based on
measurements of the global channel $\mathbbm{H}$. However, this process
inevitably incurs some delay and may lead to outdated precoding decisions in
many practical scenarios, making distributed implementations based on timely
local measurements [2] a competitive alternative. This observation is also
corroborated by our simulations in Section IV for relatively small delays.

In this work, we study the impact of delayed CSI sharing on performance and
robust precoding design by considering the following simplified model. We
assume that each AP $l\in\mathcal{L}$ can form its precoding matrix based on
perfect instantaneous knowledge of the local channel $\mathbbm{H}_{l}[t]$, and
perfect $d$-step delayed knowledge of the global channel $\mathbbm{H}[t]$.
More precisely, we assume that the precoders $\mathbbm{T}_{l}[t]$ of AP
$l\in\mathcal{L}$ at time $t\in\mathbbmss{Z}$ are constrained to be functions
of the CSI

$S_{l}[t]:=(\mathbbm{H}_{l}[t],Z[t]),\quad Z[t]:=\mathbbm{H}[t-d],\quad
d\in\mathbbmss{N}.$ (1)

As we will see, the key feature of the above model is that it allows us to
design robust precoders that combine the benefits of (delayed) centralized
interference management with the opportunity of performing timely local
refinements.

######  Remark 1.

To avoid technical digressions, cumbersome notation, and to better focus on
the essence of the problem, in this study we do not consider aspects such as
channel estimation errors, user-centric network clustering, and the
opportunity of storing and exploiting CSI history such as $\mathbbm{H}[t-i]$
for $i>d$. However, these aspects can be easily incorporated in our model and
results following the approach in [16], and will be covered in details in an
extended version of this study.

######  Remark 2.

Our model and main derivations do not explicitly consider centralized
precoding computation, and assume a distributed system where all precoders are
locally computed by the APs after a preliminary CSI sharing step. However, we
remark that all computations involving $\mathbbm{H}[t-d]$ only can also be
implemented on a central processor. Hence, our model implicitly covers both
the aforementioned centralized and fully distributed implementations, which
correspond to extreme functions that discard either $\mathbbm{H}_{l}[t]$ or
$\mathbbm{H}[t-d]$, respectively. More interestingly, our model also covers
intermediate cases where the computation of the precoders is split among a
central processor operating on the basis of global delayed CSI, and the APs
operating on the basis of timely local CSI. Additional details are given in
Section III-C.

###  II-C Team MMSE precoding

Following the approach in [11], which introduced a novel non-heuristic method
for optimal distributed precoding design when the APs are endowed with
different CSI, we consider the following parametric Team MMSE precoding
problem:

$\underset{\mathbbm{T}\in\mathcal{T}}{\text{minimize}}~{}\mathsf{E}\left[\|\bm{P}^{\frac{1}{2}}\mathbbm{H}^{\mathsf{H}}\mathbbm{T}-\bm{I}_{K}\|_{\mathsf{F}}^{2}\right]+\sum_{l=1}^{L}\sigma_{l}\mathsf{E}\left[\|\mathbbm{T}_{l}\|_{\mathsf{F}}^{2}\right],$
(2)

where $\mathcal{T}\subseteq\mathcal{H}^{K\times LN}$ is a given information
constraint [11, 16] induced by the CSI structure (1),
$\bm{P}=\text{diag}(\bm{p})$ with
$\bm{p}=(p_{1},\ldots,p_{K})\in\mathbbmss{R}_{++}^{K}$, and
$\bm{\sigma}=(\sigma_{1},\ldots,\sigma_{L})\in\mathbbmss{R}_{++}^{L}$ are
given parameters.111Note that the Team MMSE precoding problem is equivalently
formulated in [11, 16] as $K$ separate problems for each of the $K$ precoding
vectors $(\mathbbm{t}_{k})_{k=1}^{K}$, coupled by the problem parameters
$(\bm{p},\bm{\sigma})$. The difference between [11] and [16] is that [11]
focuses on the case $\bm{\sigma}=\bm{1}$. Note that, to improve readability of
the paper, we omit the dependency on the time index $t$ since
$\\{\mathbbm{H}[t],S_{1}[t],\ldots,S_{L}[t]\\}_{t\in\mathbbmss{Z}}$ is a
stationary random process. However, we remark that the impact of the delay $d$
is still fully captured in (3) by means of the constraint set $\mathcal{T}$.

Informally, the role of the constraint set $\mathcal{T}$ is to enforce the
precoders of each AP $l\in\mathcal{L}$ to be functions of
$S_{l}=(\mathbbm{H}_{l},Z)$ only, where $S_{l}$ denotes a realization of (1)
at some arbitrary time $t\in\mathbbmss{Z}$. As proposed in [11, 16], we
formally define the constraint set $\mathcal{T}$ as follows: we let

$(\forall
k\in\mathcal{K})~{}\mathbbm{t}_{k}\in\mathcal{T}_{k}:=\mathcal{H}_{1}^{N}\times\ldots\times\mathcal{H}_{L}^{N},$

where $\mathcal{H}_{l}^{N}\subseteq\mathcal{H}^{N}$ denotes the set of
$N$-tuples of $\Sigma_{l}$-measurable functions $\Omega\to\mathbbmss{C}$
satisfying $(\forall\mathbbm{x}\in\mathcal{H}_{l}^{N})$
$\mathsf{E}[\|\mathbbm{x}\|^{2}]<\infty$, and where
$\Sigma_{l}\subseteq\Sigma$ is the sub-$\sigma$-algebra induced by the CSI
$S_{l}=(\mathbbm{H}_{l},Z)$ available at AP $l\in\mathcal{L}$. In the team
theoretical literature, $\Sigma_{l}$ is also called the _information subfield_
of AP $l$. Then, we let
$\mathcal{T}:=\mathcal{T}_{1}\times\ldots\times\mathcal{T}_{K}$. The
interested reader is referred to [17] for an introduction to the measure
theoretical notions used in the above definitions. However, we stress that
these notions are by no means required for understanding the key results of
this study.

######  Remark 3.

Problem (2) can be motivated under multiple points of view. For instance, the
solution to Problem (2) can be interpreted as the best distributed
approximation of regularized channel inversion (recovered for $d=0$), where
the parameters $(\bm{p},\bm{\sigma})$ can be tuned to balance UE priorities
and APs power consumption. Furthermore, solving (2) corresponds to minimizing
the individual MSE between the transmit and receive data-bearing symbols after
linear processing over a dual uplink channel with UE uplink powers $\bm{p}$
and AP noise powers $\bm{\sigma}$. Finally, an information theoretical
motivation is obtained by evaluating performance using the so-called
_hardening_ inner bound [1] on the ergodic capacity region. Specifically, by
leveraging the known uplink-downlink duality principle for fading channels
under a sum power constraint (see, e.g., [1]), [11] proves that, by choosing
$\bm{\sigma}=\bm{1}$ and $\bm{p}$ such that $\sum_{k=1}^{K}p_{l}=P$, the
solution to (2) is Pareto optimal, in the sense that it produces rate tuples
on the boundary of the considered inner bound under a sum power constraint $P$
(and unitary noise powers). Conversely, [11] shows that all boundary points
under a sum power constraint $P$ can be achieved by solutions to (2) for
$\bm{\sigma}=\bm{1}$ and for some $\bm{p}$ such that $\sum_{k=1}^{K}p_{l}=P$.
Furthermore, [16] proves that all boundary points under per-AP power
constraints can be achieved by solutions to (2) for some
$(\bm{p},\bm{\sigma})$.

The parameters $(\bm{p},\bm{\sigma})$ of Problem (2) can be tuned to maximize
some network utility function under some power and/or quality of service
constraints as in, e.g, [18, 16], or set heuristically as for the many
variants of the MMSE or regularized zero forcing precoding schemes [1].
Additional details on the tuning of these parameters are left for the extended
version of this study, since they mostly relate to resource allocation and
power control aspects that are not specific to the delayed CSI sharing model
(1). The rest of this study is devoted to solving the Team MMSE precoding
problem (2) under the considered delayed CSI sharing model (1), for arbitrary
problem parameters $(\bm{p},\bm{\sigma})$. Only our simulations in Section IV
will focus on a specific example of $(\bm{p},\bm{\sigma})$.

##  III Problem solution

###  III-A Optimal solution

Problem (2) is a functional (i.e., infinite dimensional) optimization problem
belonging to the class of team decision problems [17], which are notoriously
difficult, even when convex as in our case. The main difficulty lies in the
information constraint $\mathcal{T}$, which prevents the direct application of
standard methods and numerical routines for finite dimensional convex
problems. However, [11] showed that Problem (2) can be mapped to a minor
variation of the subclass of quadratic team problems [17], and, as a
consequence, that the following necessary and sufficient optimality conditions
hold.

######  Proposition 1.

For given $\bm{p}\in\mathbbmss{R}_{++}^{K}$ and
$\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, Problem (2) admits a unique solution,
which is also the unique $\mathbbm{T}\in\mathcal{T}$ satisfying

$(\forall
l\in\mathcal{L})~{}\mathbbm{T}_{l}=\mathbbm{F}_{l}\left(\bm{I}_{K}-\sum_{j\in\mathcal{L}\backslash\\{l\\}}\bm{P}^{\frac{1}{2}}\mathsf{E}\left[\mathbbm{H}_{j}^{\mathsf{H}}\mathbbm{T}_{j}\Big{|}S_{l}\right]\right),$
(3)

where
$\mathbbm{F}_{l}:=\left(\mathbbm{H}_{l}\bm{P}\mathbbm{H}_{l}^{\mathsf{H}}+\sigma_{l}\bm{I}_{N}\right)^{-1}\mathbbm{H}_{l}\bm{P}^{\frac{1}{2}}\in\mathcal{H}^{N\times
K}$.

###### Proof.

The proof follows readily by replacing $\bm{\sigma}=\bm{1}$ with an arbitrary
$\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$ from the proof of [11, Lemma 2].
Informally, (3) is obtained by minimizing the objective in (2) with respect to
$\mathbbm{T}_{l}$, and by fixing $\mathbbm{T}_{j}$ for $j\neq l$. This gives a
set of necessary optimality conditions, related to the game theoretical notion
of Nash equilibrium. The key step of the proof shows that these conditions are
also sufficient. ∎

Proposition 1 and its extensions covering channel estimation errors and user-
centric network clustering are used in [11, 16] to derive optimal distributed
precoders under local CSI models of the type $(\forall
l\in\mathcal{L})~{}S_{l}=\mathbbm{H}_{l}$ for fully distributed
implementations, or under sequential CSI sharing models of the type $(\forall
l\in\mathcal{L})~{}S_{l}=(\mathbbm{H}_{1},\ldots,\mathbbm{H}_{l})$ for
partially distributed implementations exploiting the properties of serial
fronthauls. In the following, we use Proposition 1 to derive the main result
of this study, i.e., the solution to Problem (2) under the delayed CSI sharing
model $(\forall l\in\mathcal{L})~{}S_{l}=(\mathbbm{H}_{l},Z)$ in (1).

######  Proposition 2.

For given $\bm{p}\in\mathbbmss{R}_{++}^{K}$ and
$\bm{\sigma}\in\mathbbmss{R}_{++}^{L}$, the unique solution to Problem (2) is
given by

$(\forall l\in\mathcal{L})~{}\mathbbm{T}_{l}=\mathbbm{F}_{l}\mathbbm{C}_{l},$
(4)

where
$\mathbbm{F}_{l}:=\left(\mathbbm{H}_{l}\bm{P}\mathbbm{H}_{l}^{\mathsf{H}}+\sigma_{l}\bm{I}_{N}\right)^{-1}\mathbbm{H}_{l}\bm{P}^{\frac{1}{2}}\in\mathcal{H}^{N\times
K}$, and $\mathbbm{C}_{l}\in\mathcal{H}^{K\times K}$ is given by the unique
solution to the linear system of equations

$(\forall
l\in\mathcal{L})~{}\mathbbm{C}_{l}+\sum_{j\in\mathcal{L}\backslash\\{l\\}}\mathsf{E}[\bm{P}^{\frac{1}{2}}\mathbbm{H}_{j}^{\mathsf{H}}\mathbbm{F}_{j}|Z]\mathbbm{C}_{j}=\bm{I}_{K}.$
(5)

###### Proof.

(Sketch) The proof follows by verifying that (4) satisfies (3) via simple
algebraic manipulations. ∎

###  III-B Interpretation and computation of the optimal solution

Proposition 2 states the optimality of the two-stage precoding structure in
(4), where each stage depends on a distinct portion of the CSI
$S_{l}=(\mathbbm{H}_{l},Z)$. Specifically, the optimal precoder
$\mathbbm{T}_{l}$ at AP $l\in\mathcal{L}$ is composed by a first $N\times K$
local MMSE precoding stage $\mathbbm{F}_{l}$ [1], function of the timely local
CSI $\mathbbm{H}_{l}$, and a second $K\times K$ precoding stage
$\mathbbm{C}_{l}$, function of the delayed global CSI $Z$. To better
understand the dependency of $\mathbbm{C}_{l}$ on $Z$, we observe that the
linear system of equations (5) defining $\mathbbm{C}_{l}$ has random
coefficients
$\mathsf{E}[\bm{P}^{\frac{1}{2}}\mathbbm{H}_{l}^{\mathsf{H}}\mathbbm{F}_{l}|Z]$
which are functions of $Z$. In particular, each realization of the $L$
precoding stages $(\mathbbm{C}_{l})_{l=1}^{L}$ can be obtained by solving (5)
disjointly for each realization of $Z$. More precisely, a realization
$(\bm{C}_{l})_{l=1}^{L}$ of $(\mathbbm{C}_{l})_{l=1}^{L}$ for a given
realization $z$ of $Z$ is given by the solution to the finite dimensional
linear system of equations

$(\forall
l\in\mathcal{L})~{}\bm{C}_{l}+\sum_{j\in\mathcal{L}\backslash\\{l\\}}\mathsf{E}[\bm{P}^{\frac{1}{2}}\mathbbm{H}_{j}^{\mathsf{H}}\mathbbm{F}_{j}|Z=z]\bm{C}_{j}=\bm{I}_{K},$

which can be computed using standard techniques, provided that the
coefficients
$\mathsf{E}[\bm{P}^{\frac{1}{2}}\mathbbm{H}_{l}^{\mathsf{H}}\mathbbm{F}_{l}|Z=z]$
are known.

By reintroducing the time index $t$, we notice that these coefficients take
the form
$\mathsf{E}[\bm{P}^{\frac{1}{2}}\mathbbm{H}_{l}[t]^{\mathsf{H}}\mathbbm{F}_{l}[t]|Z[t]]=$

$\displaystyle\mathsf{E}[\bm{P}^{\frac{1}{2}}\mathbbm{H}_{l}[t]^{\mathsf{H}}\left(\mathbbm{H}_{l}[t]\bm{P}\mathbbm{H}_{l}[t]^{\mathsf{H}}+\sigma_{l}\bm{I}_{N}\right)^{-1}\mathbbm{H}_{l}[t]\bm{P}^{\frac{1}{2}}|\mathbbm{H}_{l}[t-d]],$

i.e., they are functions of $\mathbbm{H}_{l}[t-d]$ defined by the conditional
distribution of $\mathbbm{H}_{l}[t]$ given $\mathbbm{H}_{l}[t-d]$, which we
recall is independent of $t$ due to stationarity. Importantly, these
coefficients can be computed in parallel for each AP. Furthermore, we notice
that they can be interpreted as the optimal $d$-step MMSE predictors of the
$K\times K$ effective channels
$\bm{P}^{\frac{1}{2}}\mathbbm{H}_{l}[t]^{\mathsf{H}}\mathbbm{F}_{l}[t]$ after
local MMSE precoding. Unfortunately, closed-form expressions for the
coefficients
$\mathsf{E}[\bm{P}^{\frac{1}{2}}\mathbbm{H}_{l}^{\mathsf{H}}\mathbbm{F}_{l}|Z]$
may not be available in many practical cases. However, we remark that many
approximate numerical methods taken from the vast literature on
estimation/prediction theory could be potentially applied. Of particular
practical interest are data driven techniques that do not require explicit
knowledge of the conditional distribution of $\mathbbm{H}_{l}[t]$ given
$\mathbbm{H}_{l}[t-d]$. For simplicity, in this work, we evaluate numerically
each expectation using an empirical average over a sample set generated
according to the conditional distribution of $\mathbbm{H}_{l}[t]$ given
$\mathbbm{H}_{l}[t-d]$, assumed known. In addition, in Section III-D, we
discuss some suboptimal approximations. We leave the evaluation of more
advanced techniques as a promising future line of research.

###  III-C C-RAN functional split aspects

\begin{overpic}[width=144.54pt,tics=20]{distributed} \put(12.0,20.0){AP$\,1$}
\put(12.0,10.0){$\mathbbm{H}_{1}[t]$} \put(73.0,20.0){AP$\,2$}
\put(73.0,10.0){$\mathbbm{H}_{2}[t]$} \put(44.0,55.0){CPU}
\put(36.0,27.0){$\mathbbm{H}_{1}[t-d]$} \put(36.0,6.0){$\mathbbm{H}_{2}[t-d]$}
\put(26.0,37.0){$\mathbbm{u}[t]$} \put(64.0,37.0){$\mathbbm{u}[t]$}
\end{overpic} (a)

\begin{overpic}[width=144.54pt,tics=20]{centralized_1}
\put(12.0,20.0){AP$\,1$} \put(12.0,10.0){$\mathbbm{H}_{1}[t]$}
\put(73.0,20.0){AP$\,2$} \put(73.0,10.0){$\mathbbm{H}_{2}[t]$}
\put(44.0,55.0){CPU} \put(0.0,55.0){$\mathbbm{H}_{1}[t-d]$}
\put(70.0,55.0){$\mathbbm{H}_{2}[t-d]$}
\put(33.0,23.0){$\mathbbm{C}_{1}[t]\;\,\mathbbm{C}_{2}[t]$}
\put(28.0,37.0){$\mathbbm{u}[t]$} \put(62.0,37.0){$\mathbbm{u}[t]$}
\end{overpic} (b)

\begin{overpic}[width=144.54pt,tics=20]{centralized_2}
\put(11.0,20.0){AP$\,1$} \put(11.0,10.0){$\mathbbm{H}_{1}[t]$}
\put(73.0,20.0){AP$\,2$} \put(73.0,10.0){$\mathbbm{H}_{2}[t]$}
\put(44.0,57.0){CPU} \put(0.0,55.0){$\mathbbm{H}_{1}[t-d]$}
\put(70.0,55.0){$\mathbbm{H}_{2}[t-d]$}
\put(20.0,37.0){$\mathbbm{C}_{1}\mathbbm{u}[t]$}
\put(60.0,37.0){$\mathbbm{C}_{2}\mathbbm{u}[t]$} \end{overpic} (c)

Figure 1: Pictorial representation of possible implementations of the proposed
team MMSE solution (4) in C-RAN architectures: (a) distributed precoding with
CSI sharing; (b) locally refined centralized precoding (compress-before-
precoding); (c) locally refined centralized precoding (compress-after-
precoding).

In this section we discuss the implementation of the optimal two-stage
precoding structure identified in (4) by focusing on the functional split
problem in C-RAN architectures [9]. First, we recall that the considered
solution in (4) is derived under the delayed CSI sharing model (1), by
assuming that the precoders are computed locally at each AP after a
preliminary CSI sharing step. This model can be directly mapped to the
functional split depicted in Fig. 1a, where (from a physical layer
perspective) the CPU only forms and forwards the $K$-dimensional vector of
coded and modulated data streams $\mathbbm{u}[t]$. This is the closest
implementation to the original fully distributed cell-free massive MIMO
concept proposed in [2].

However, as anticipated in Remark 2, (4) can also be implemented by splitting
the computation of the two stages between the APs and the CPU, as depicted in
Fig. 1b and Fig. 1c. These implementations are closer to the centralized cell-
free massive MIMO concept described, e.g., in [1]. In both these functional
splits, the CPU forms the precoding stages $\mathbbm{C}_{l}[t]$ based on
delayed global CSI, and the APs form their local MMSE stages
$\mathbbm{F}_{l}[t]$ based on timely local CSI. The difference between these
two functional splits is that in Fig. 1b both precoding stages are applied to
the data streams by the APs, while in Fig. 1c the CPU computes and forwards
the $K$-dimensional intermediate signals $\mathbbm{C}_{l}[t]\mathbbm{u}[t]$
for each AP.222In both cases, the delay $d$ should be rather interpreted as
the round-trip delay incurred by the two-way information sharing procedure.

Choosing the best functional split is a notoriously challenging problem
encompassing many different system aspects. For instance, Fig. 1b and Fig. 1c
can be respectively interpreted as novel distributed versions of the compress-
before-precoding and compress-after-precoding functional splits compared in
[9] in terms of fronthaul rate requirements. Interestingly, the results of
this study may be used as a novel approach to extend the current literature on
C-RAN functional splits covering fronthaul and processing delay requirements.

We conclude this section by pointing out that, in the current form, none of
the implementations in Fig. 1 is scalable with respect to the number of UEs
$K$. This is mostly due to the fact that the information to be shared and
processed is proportional to $K$, as in [2]. However, we remark that this
issue can be significantly mitigated by omitting the sharing and processing of
information that does not contribute significantly to performance, for
instance because of large path loss between a certain UE-AP pair. A popular
way of implementing this idea is via the user-centric network clustering
paradigm [1]. The extension of our results to this paradigm can be done as in
[16], and it is left for the extended version of this study.

###  III-D Suboptimal solutions

####  III-D1 Local precoding

By discarding the potentially useful information $\mathbbm{H}[t-d]$ in (1),
i.e., by letting $(\forall l\in\mathcal{L})~{}S_{l}[t]=\mathbbm{H}_{l}[t]$,
the solution to Problem (2) is given by a variant of (4) with $(\forall
l\in\mathcal{L})~{}\mathbbm{C}_{l}=\bm{C}_{l}$, where $(\bm{C}_{l})_{l=1}^{L}$
are fixed (deterministic) precoding stages given by the solution to

$(\forall
l\in\mathcal{L})~{}\bm{C}_{l}+\sum_{j\in\mathcal{L}\backslash\\{l\\}}\mathsf{E}[\bm{P}^{\frac{1}{2}}\mathbbm{H}_{j}^{\mathsf{H}}\mathbbm{F}_{j}]\bm{C}_{j}=\bm{I}_{K}.$
(6)

This corresponds to the local team MMSE solution derived in [11, 16], which we
recall is an enhanced version of the known local MMSE scheme and its variants
[1].

####  III-D2 Centralized (delay-tolerant) precoding

On the other extreme, by discarding $(\mathbbm{H}_{l}[t])_{l=1}^{L}$ in (1),
i.e., by letting $(\forall l\in\mathcal{L})~{}S_{l}[t]=\mathbbm{H}[t-d]$, the
solution to Problem (2) becomes

$\mathbbm{T}[t]=\left(\hat{\mathbbm{H}}[t]\bm{P}\hat{\mathbbm{H}}[t]^{\mathsf{H}}+\bm{\Psi}+\bm{\Sigma}\right)^{-1}\hat{\mathbbm{H}}[t]\bm{P}^{\frac{1}{2}},$
(7)

where $\hat{\mathbbm{H}}[t]:=\mathsf{E}[\mathbbm{H}[t]|\mathbbm{H}[t-d]]$,
$\bm{\Psi}:=\mathsf{E}[(\hat{\mathbbm{H}}[t]-\mathbbm{H}[t])\bm{P}(\hat{\mathbbm{H}}[t]-\mathbbm{H}[t])^{\mathsf{H}}]$,
and
$\bm{\Sigma}:=\text{blkdiag}(\sigma_{1}\bm{I}_{N},\ldots,\sigma_{L}\bm{I}_{N})$.
This essentially corresponds to the known centralized MMSE scheme in [1] or to
the delay-tolerant zero-forcing scheme based on channel prediction in [15],
carefully optimized and adapted to our setup to ensure a fair comparison
against (4).

####  III-D3 Naïve distributed precoding

A simple baseline distributed precoding scheme that takes into account the
full information in (1) is given by letting each AP $l\in\mathcal{L}$ locally
compute a version of the centralized solution (7) based on its local estimate
of the global channel state, obtained by replacing the submatrix
$\hat{\mathbbm{H}}_{l}[t]:=\mathsf{E}[\mathbbm{H}_{l}[t]|\mathbbm{H}_{l}[t-d]]$
of $\hat{\mathbbm{H}}[t]$ with $\mathbbm{H}_{l}[t]$. This baseline approach
was termed naïve precoding in the early works on distributed precoding [19],
since it essentially corresponds to letting each AP believe that its
information on $\mathbbm{H}[t]$ is the same information at all APs.

####  III-D4 Structure-aware distributed precoding

As an alternative to the above baseline distributed precoding scheme that
takes into account the structure of the optimal solution, we propose a variant
of (4) based on approximating the coefficients of the linear system in (5) as
$\mathsf{E}[\bm{P}^{\frac{1}{2}}\mathbbm{H}_{l}[t]^{\mathsf{H}}\mathbbm{F}_{l}[t]|\mathbbm{H}_{l}[t-d]]\approx$

$\bm{P}^{\frac{1}{2}}\hat{\mathbbm{H}}_{l}[t]^{\mathsf{H}}\left(\hat{\mathbbm{H}}_{l}[t]\bm{P}\hat{\mathbbm{H}}_{l}[t]^{\mathsf{H}}+\bm{\Psi}_{l}+\sigma_{l}\bm{I}_{N}\right)^{-1}\hat{\mathbbm{H}}_{l}[t]\bm{P}^{\frac{1}{2}},$

where
$\bm{\Psi}_{l}:=\mathsf{E}[(\hat{\mathbbm{H}}_{l}[t]-\mathbbm{H}_{l}[t])\bm{P}(\hat{\mathbbm{H}}_{l}[t]-\mathbbm{H}_{l}[t])^{\mathsf{H}}]$.
This is similar to the centralized approach described above, but confined to
the computation of the precoding stages $(\mathbbm{C}_{l}[t])_{l=1}^{L}$.

##  IV Numerical simulations and conclusions

We consider a network composed by $K=50$ UEs are uniformly distributed within
a squared service area of size $0.5\times 0.5~{}\text{km}^{2}$, and $L=16$
regularly spaced APs with $N=4$ antennas each. By neglecting for simplicity
spatial correlation, we let $\mathbbm{h}_{l,k}$ be independently distributed
as
$\mathbbm{h}_{l,k}\sim\mathcal{CN}\left(\bm{0},\gamma_{l,k}\bm{I}_{N}\right)$,
where $\gamma_{l,k}>0$ denotes the channel gain between AP $l$ and UE $k$. We
follow the same 3GPP-like path-loss model adopted in [1] for a $2$ GHz carrier
frequency:

$\gamma_{l,k}=-36.7\log_{10}\left(D_{l,k}/1\;\text{m}\right)-30.5+Z_{l,k}-\sigma^{2}\quad\text{[dB]},$

where $D_{l,k}$ is the distance between AP $l$ and UE $k$ including a
difference in height of $10$ m, and $Z_{l,k}\sim\mathcal{N}(0,\rho^{2})$ [dB]
are shadow fading terms with deviation $\rho=4$. The shadow fading is
correlated as
$\mathsf{E}[Z_{l,k}Z_{j,i}]=\rho^{2}2^{-\frac{\delta_{k,i}}{9\text{ [m]}}}$
for all $l=j$ and zero otherwise, where $\delta_{k,i}$ is the distance between
UE $k$ and UE $i$. The noise power is
$\sigma^{2}=-174+10\log_{10}(B/1\;\text{Hz})+F$ [dBm], where $B=20$ MHz is the
bandwidth, and $F=7$ dB is the noise figure.

The time evolution of the channel is modeled as in [13] and many related
studies by assuming that each $\\{\mathbbm{h}_{l,k}[t]\\}_{t\in\mathbbmss{Z}}$
is a zero-mean stationary ergodic complex Gaussian process, where the joint
distribution of $(\mathbbm{h}_{l,k}[t],\mathbbm{h}_{l,k}[t-d])$ is fully
characterized by the autocovariance matrix
$\mathsf{E}[\mathbbm{h}_{l,k}[t]\mathbbm{h}_{l,k}[t-d]^{\mathsf{H}}]=r_{l,k}\gamma_{l,k}\bm{I}_{N}$
for a given autocorrelation coefficient $r_{l,k}\in[0,1]$. The autocorrelation
coefficient can be used to model the joint impact of the CSI sharing delay $d$
and UE mobility, for example by using Clarke’s model
$r_{l,k}=J_{0}(2\pi\nu_{l,k}Td)$ as in [13], where $\nu_{l,k}$ denotes the
Doppler spread for the $(l,k)$th AP-UE pair, and $T$ is the symbol time. We
focus on two representative scenarios obtained by letting $(\forall
l\in\mathcal{L})(\forall k\in\mathcal{K})~{}r_{l,k}=r\in\\{0.99,0.9\\}$,
which, according to Clarke’s model, can be mapped to pedestrian mobility
$(\nu_{l,k}\approx 10$ Hz) for all UEs and delay
$Td\in\\{1~{}\text{ms},10~{}\text{ms}\\}$.

###  IV-A Figure-of-merit

We evaluate performance in terms of downlink ergodic achievable rates
estimated by the hardening inner bound [1] for a given (network-wide) sum
power $P=5$ W. To facilitate the connection with the solution to Problem (2),
we leverage the known uplink-downlink duality principle for fading channels
(see, e.g., [1]) and compute the downlink rate of each UE $k\in\mathcal{K}$
for a given precoding scheme $\mathbbm{T}$ and for some downlink power
allocation policy using the equivalent expressions

$R_{k}(\mathbbm{T},\bm{p}):=\log_{2}(1+\mathrm{SINR}_{k}(\mathbbm{t}_{k},\bm{p})),$
$\mathrm{SINR}_{k}(\mathbbm{t}_{k},\bm{p}):=\leavevmode\resizebox{299.19885pt}{}{$\dfrac{p_{k}|\mathsf{E}[\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{k}]|^{2}}{p_{k}\mathsf{V}(\mathbbm{h}_{k}^{\mathsf{H}}\mathbbm{t}_{k})+\underset{j\neq
k}{\sum}p_{j}\mathsf{E}[|\mathbbm{h}_{j}^{\mathsf{H}}\mathbbm{t}_{k}|^{2}]+\mathsf{E}[\|\mathbbm{t}_{k}\|^{2}]}$},$

corresponding to the achievable rates over a virtual uplink channel for some
virtual uplink powers $\bm{p}\in\mathbbmss{R}_{+}^{K}$ satisfying
$\sum_{k=1}^{K}p_{k}=P$. We recall that the uplink-downlink duality principle
guarantees the existance of a downlink power allocation such that the same
rates are also achievable in the downlink using $\mathbbm{T}$. It can be shown
that the above uplink rates are maximized by the solution to Problem (2) with
problem parameters $\bm{p}$ equal to the uplink powers and
$\bm{\sigma}=\bm{1}$ [11, 18]. As an example, we choose a fractional power
allocation policy with exponent $-1$ [6], i.e., we let $(\forall
k\in\mathcal{K})~{}p_{k}\propto(\sum_{l=1}^{L}\gamma_{l,k})^{-1}$, which
approximates a max-min fair policy.

###  IV-B Results

(a)

(b)

Figure 2: Cumulative density function of downlink ergodic rates achieved by
the optimal team MMSE solution (4) and the suboptimal designs in Sect. III-D,
assuming pedestrian mobility and a CSI sharing delay of (a) $10$ ms ($r=0.9$)
and (b) $1$ ms ($r=0.99$).

Figure 2 shows the cumulative density function of the ergodic rates achieved
by different precoding schemes, approximated using $100$ independent user
drops and $100$ independent realizations of
$(\mathbbm{H}[t],\mathbbm{H}[t-d])$ per user drop. In particular, we compare
the performance of the optimal Team MMSE solution (3) against the suboptimal
solutions described in Section III-D. We remark that the schemes termed
centralized and local correspond, respectively, to the best known schemes for
the centralized and (fully) distributed cell-free massive MIMO implementations
described in [1]. The Team MMSE solution is computed via the numerical method
described in Section III-B.

A first important observation is that, even for pedestrian mobility, a CSI
sharing delay of $10$ ms ($r=0.9$, Figure 2a) can significantly degrade the
performance of centralized precoding to the point where it becomes noticeably
worse than the performance of local precoding. Moreover, a second important
observation is that that the proposed Team MMSE precoding scheme largely
outperforms both centralized and local precoding in all the considered
scenarios. Perhaps surprisingly, the gains are significant even for a
relatively small CSI sharing delay of $1$ ms ($r=0.99$, Figure 2b), a scenario
where centralized precoding is still quite effective and outperforms local
precoding. This suggests that the APs’ local interference management
capabilities based on timely local CSI should not be neglected in most
practical scenarios. Indeed, our results suggests that significant rate gains
can be achieved by a careful distributed precoding design, such as the
proposed Team MMSE scheme, that is able to merge the benefits of local and
centralized interference management.

Finally, Figure 2b shows that, for small CSI sharing delays, the
aforementioned rate gains can be achieved by simple approximations of the Team
MMSE scheme. In particular, in the considered scenario, we observe that the
proposed structure-aware distributed precoding scheme is able to exploit the
benefits of local and centralized interference management, at a significantly
lower computational cost than the Team MMSE scheme. However, Figure 2a shows
that this may not be the case for higher CSI sharing delays, since the Team
MMSE solution shows non-negligible performance gains.

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