# Octonions and the Two Unique Tight Projective 5-Designs

Benjamin Nasmith Department of Physics and Space Science, Royal Military
College of Canada, Kingston, ON<EMAIL_ADDRESS>
###### Abstract.

There are precisely two unique tight projective 5-designs, one constructed
from the short vectors of the Leech lattice and the other corresponding to a
generalized hexagon structure in the octonion projective plane. This paper
describes a new connection between these two projective 5-designs—a common
construction using octonions. A set of well chosen octonion involutionary
matrices acts on a three-dimensional octonion vector space to produce the
first 5-design and these same matrices act on the octonion projective plane to
produce the second 5-design. This result draws on the octonion construction of
the Leech lattice due to Robert Wilson and provides a new link between the
generalized hexagon GH(2,8) and the Leech lattice.

##  1\. Introduction

A spherical $t$-design is a finite subset of points on the unit sphere in a
real vector space with the following special property: the average value of
any polynomial of degree at most $t$ over the sphere is equal to the average
value of the polynomial evaluated at the points of the $t$-design. A
projective $t$-design generalizes this concept from spheres in
$\mathbb{R}^{d}$ to projective spaces. Tight $t$-designs (whether spherical or
projective) meet an absolute bound. While $t$-designs are common, tight
$t$-designs are rare. It turns out that there are precisely two tight
projective $5$-designs—one constructed from the short vectors of the Leech
lattice and the other constructed from the lines of the unique generalized
hexagon $\mathrm{GH}(2,8)$. Some authors have conjectured that these two
objects are closely related [Hog82]. This paper provides a common construction
of both using well-chosen octonion involutionary matrices. These matrices can
act on both an octonion vector space $\mathbb{O}^{3}$ and on the octonion
projective plane $\mathbb{OP}^{2}$ to produce the two unique tight projective
$5$-designs. This common construction provides a new connection between the
generalized hexagon $\mathrm{GH}(2,8)$ and the Leech lattice. Projective
$t$-designs are interesting objects in part because of their connections to
certain real, complex, and quaternionic reflection groups, as well as certain
of sporadic simple groups. The octonionic construction of the two tight
projective $5$-designs in this paper makes clearer the role of the octonion
algebra in the structure of the sporadic simple groups associated to the Leech
lattice.

##  2\. The Two Unique Tight Projective 5-Designs

This section reviews tight projective $t$-designs and an important uniqueness
theorem. We denote a projective space by $\mathbb{FP}^{d-1}$ where
$\mathbb{F}$ is one of $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, or
$\mathbb{O}$ (with $d=2,3$ in the case of $\mathbb{F}=\mathbb{O}$). As
described in [Hog82, 234], there are two ways to model a projective space. The
first model applies for $\mathbb{F}\neq\mathbb{O}$. In the first model, a
point $[x]$ in $\mathbb{FP}^{d-1}$ (with $x\in\mathbb{F}^{d}$) is represented
by the set,

$\displaystyle[x]=\\{\lambda x\mid
0\neq\lambda\in\mathbb{F}\\},\quad\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}.$

The second model applies whenever the first model does, and also when
$\mathbb{F}=\mathbb{O}$. In the second model, a point $X$ in
$\mathbb{FP}^{d-1}$ is an idempotent $d\times d$ Hermitian matrix over
$\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$. The Euclidean inner
product is defined in terms of the matrix product as $\langle
X,Y\rangle=\frac{1}{2}\mathrm{Tr}(XY+YX)$. This inner product is a real-valued
positive definite symmetric bilinear form, and we can define an angle between
any two idempotent matrices $X,Y$ via $\cos^{2}\theta=\langle X,Y\rangle$.
Taking $x$ to be a row vector in $\mathbb{F}^{d}$, with
$\mathbb{F}\neq\mathbb{O}$, we can convert from the first model to the second
via the map $[x]\mapsto x^{\dagger}x/(xx^{\dagger})=X$, where $x^{\dagger}$ is
the complex conjugate transpose of $x$.

To define projective $t$-designs, we first need the define the polynomials
$Q_{k}(x)$ of degree $k$ [CD07, 54.33]:

$\displaystyle
Q_{k}(x)=\sum_{r=0}^{k}(-1)^{k-r}\binom{k}{r}(N+2k-1)\frac{(N)_{k+r-1}}{(m)_{r}k!}x^{r},$

where $m=\frac{1}{2}[\mathbb{F}:\mathbb{R}]$, $N=md$, and
$(a)_{i}=a(a+1)(a+2)\cdots(a+i-1)$. A projective $t$-design is a subset
$T\subset\mathbb{FP}^{d-1}$ that satisfies,

$\displaystyle\sum_{a\in T}\sum_{b\in T}Q_{k}(\langle a,b\rangle)=0,\quad
k=1,2,\ldots,t.$

We denote by $A$ the angle set of possible inner products $\langle
a,b\rangle=\cos^{2}\theta_{a,b}$ for $a\neq b$ in $T$. We can constrain the
size of $|T|$ based on properties of $A$. When the constraint depends on $|A|$
only, it is called an absolute bound. When it depends on the values in $A$, it
is called a special bound. A spherical or projective $t$-design is called
tight when it reaches the absolute bound [CD07, 54.17]. In general, a
projective $t$-design satisfies $t\leq 2l+\varepsilon$ and $|T|\geq
Q_{0}(1)+Q_{1}(1)+\ldots+Q_{l}(1)$ where we have [Hog89, 88],

$\displaystyle l=|A\setminus\\{0\\}|,$
$\displaystyle\varepsilon=\left\\{\begin{matrix}1,&\text{if }0\in|A|\\\
0,&\text{otherwise}\end{matrix}\right.$

A projective $t$-design is tight when it satisfies $t=2l+\varepsilon$. An
equivalent condition is that $|T|=Q_{0}(1)+Q_{1}(1)+\ldots+Q_{l}(1)$ for a
tight projective $t$-design.

######  Theorem 2.1 ([Hog89]).

A tight projective $t$-design always satisfies $t=2,3,5$ and for $t=5$ there
are only two examples. The only two tight projective $5$-designs are the
unique system of $98280$ points in $\mathbb{RP}^{23}$ defined by the short
vectors of the Leech lattice and the unique system of $819$ points in
$\mathbb{OP}^{2}$ corresponding to the lines of the generalized hexagon
$\mathrm{GH}(2,8)$.

###### Proof.

This theorem is the main result of [Hog89], which draws on [BS81] for the
uniqueness of the tight $5$-design in $\mathbb{RP}^{23}$ and on [CT85] for the
uniqueness of the tight $5$-design in $\mathbb{OP}^{2}$ via the uniqueness of
the generalized hexagon $\mathrm{GH}(2,8)$. ∎

In addition to the tight projective $5$-design in $\mathbb{RP}^{23}$, there
are other important projective $5$-designs constructed from the Leech lattice.
These include the designs in $\mathbb{CP}^{11}$ and $\mathbb{HP}^{2}$ that
exhibit the symmetries of sporadic simple groups $\mathrm{Suz}$ and
$\mathrm{HJ}$ respectively, shown in Fig. 1. The remarks in [Hog82] include a
conjecture that the tight projective $5$-design in $\mathbb{OP}^{2}$, which
exhibits ${}^{3}D_{4}(2)$ symmetries, is also constructed in terms of the
Leech lattice, which exhibits $\mathrm{Co}_{1}$ symmetries. The justification
offered is that $196560=819\cdot 240$, the number of points in the $5$-design
multiplied by the number of roots in the $E_{8}$ lattice equals the number of
short vectors in the Leech lattice. Against this possible connection, one
referee of [Hog82] highlights certain facts about the representation theory of
the groups $\mathrm{Co}_{1}$ and ${}^{3}D_{4}(2)$. However, the uniqueness of
these two tight projective $5$-designs, established in [Hog89], provides some
further motivation to find a common construction of both.

$\mathbb{F}$ | $d$ | $v$ | $A$ | $t$ | Isometries
---|---|---|---|---|---
$\mathbb{R}$ | $24$ | $98280$ | $\left\\{0,\frac{1}{16},\frac{1}{4}\right\\}$ | $5$ | $\mathrm{Co}_{1}$
$\mathbb{C}$ | $12$ | $32760$ | $\left\\{0,\frac{1}{12},\frac{1}{4},\frac{1}{3}\right\\}$ | $5$ | $\mathrm{Suz}$
$\mathbb{H}$ | $3$ | $315$ | $\left\\{0,\frac{1}{4},\frac{1}{2},\frac{1}{8}(3\pm\sqrt{5})\right\\}$ | $5$ | $\mathrm{HJ}$
$\mathbb{O}$ | $3$ | $819$ | $\left\\{0,\frac{1}{4},\frac{1}{2}\right\\}$ | $5$ | ${}^{3}D_{4}(2)$
Figure 1. Some projective $5$-designs related to the Leech lattice.

This paper will identify a common construction for the two unique projective
$5$-designs via the action of the same set of well-chosen octonion
involutionary matrices acting on (a) vectors in
$\mathbb{O}^{3}\cong\mathbb{R}^{24}$ and (b) idempotents in the octonion
projective plane $\mathbb{OP}^{2}$. The same involutionary matrices, acting on
different objects, yield the two unique tight projective $5$-designs. This
also establishes a previously elusive connection between the Leech lattice and
the $\mathrm{GH}(2,8)$ $5$-design on the octonion projective plane.

##  3\. Octonions and the Albert Algebra

This section reviews the octonion algebra, the Albert algebra, and the
octonion projective plane. The octonion algebra $\mathbb{O}$ is a real algebra
with a basis indexed by the projective line over $\mathbb{F}_{7}$, namely
$\\{i_{t}\mid t\in\mathrm{PL}(7)=\\{\infty\\}\cup\mathbb{F}_{7}\\}$. The
identity element $i_{\infty}=1$ spans the center of $\mathbb{O}$ and this
center is the real subalgebra $\mathbb{R}\cong\mathbb{R}i_{\infty}$. The
remaining basis vectors are square-roots of $-1$, so that $i_{t}^{2}=-1$ for
all $t\in\mathbb{F}_{7}$. Finally, remaining octonion products are given by,

$\displaystyle i_{t}$ $\displaystyle=i_{t+1}i_{t+3}=-i_{t+3}i_{t+1}$
$\displaystyle=i_{t+2}i_{t+6}=-i_{t+6}i_{t+2}$
$\displaystyle=i_{t+4}i_{t+5}=-i_{t+5}i_{t+4}\quad t\in\mathbb{F}_{7}.$

Figure 2. An Octonion Multiplication Wheel.

Fig. 2 provides a memory-aid to assist with octonion multiplication, which is
invariant under the maps $i_{t}\mapsto i_{t+1}$ and $i_{t}\mapsto i_{2t}$. The
map $i_{t}\mapsto i_{t+1}$ corresponds to rotating the diagram labels and the
map $i_{t}\mapsto i_{2t}$ corresponds to mapping the red arrow to the green
arrow to the blue arrow in a cycle.

Many details about the unique properties of the octonion algebra are available
in [Bae02] and [CS03]. It is important to remember that the octonion algebra
is not associative, but that it contains important associative subalgebras
such as the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, and
Hamilton’s quaternion algebra $\mathbb{H}$.

######  Lemma 3.1.

The subalgebra of $\mathbb{O}$ generated by a single octonion is commutative
(isomorphic to $\mathbb{R}$ or $\mathbb{C}$) and the subalgebra generated by
any two octonions is associative (isomorphic to $\mathbb{R}$, $\mathbb{C}$, or
$\mathbb{H}$).

###### Proof.

For details see [CS03, 76]. ∎

Just like the complex numbers and the quaternions, the octonion algebra
$\mathbb{O}$ is equipped with a conjugation map $X\mapsto\overline{X}$ sending
$i_{\infty}\mapsto i_{\infty}$ and $i_{t}\mapsto-i_{t}$ for
$t\in\mathbb{F}_{7}$. A Hermitian octonion matrix is a matrix with octonion
entries equal to its own octonion conjugate transpose, i.e.
$\mathcal{X}=\mathcal{X}^{\dagger}$ where $\dagger$ denotes conjugate
transpose. The Albert algebra $\mathrm{Herm}(3,\mathbb{O})$ is defined as the
algebra of $3\times 3$ Hermitian octonion matrices,

$\displaystyle(d,e,f\mid D,E,F)=\left(\begin{array}[]{ccc}d&F&\overline{E}\\\
\overline{F}&e&D\\\ E&\overline{D}&f\end{array}\right),\quad
d,e,f\in\mathbb{R},~{}D,E,F\in\mathbb{O},$

with a product $\circ$ defined in terms of standard matrix multiplication as,

$\displaystyle\mathcal{X}\circ\mathcal{Y}=\frac{1}{2}(\mathcal{X}\mathcal{Y}+\mathcal{Y}\mathcal{X}),\quad\mathcal{X},\mathcal{Y}\in\mathrm{Herm}(3,\mathbb{O}).$

The trace is defined as the sum of the diagonal entries, and
$\langle\mathcal{X},\mathcal{Y}\rangle=\mathrm{Tr}(\mathcal{X}\circ\mathcal{Y})$
is a real positive-definite symmetric bilinear form on the Albert algebra. The
Albert algebra is a real $27$-dimensional commutative but nonassociative
algebra with the exceptional Lie group $F_{4}$ for its automorphism group.

The octonion projective plane $\mathbb{OP}^{2}$ is the manifold of primitive
idempotents in the Albert algebra $\mathrm{Herm}(3,\mathbb{O})$. A primitive
idempotent is a matrix $\mathcal{X}$ in the Albert algebra that satisfies
$\mathcal{X}\circ\mathcal{X}=\mathcal{X}$ and $\mathrm{Tr}(\mathcal{X})=1$.
Equivalently, a primitive idempotent is an idempotent that cannot be written
as the sum of two other nonzero orthogonal idempotents. To illustrate the
axioms of the projective plane, we need to introduce a second product on
$\mathrm{Herm}(3,\mathbb{O})$. That is, we can define a symmetric cross
product [SV00, 122],

$\displaystyle\mathcal{X}\times\mathcal{Y}$
$\displaystyle=\mathcal{X}\circ\mathcal{Y}+\frac{I}{2}\left(\operatorname{Tr}(\mathcal{X})\operatorname{Tr}(\mathcal{Y})-\operatorname{Tr}(\mathcal{X}\circ\mathcal{Y})\right)-\frac{1}{2}\left(\operatorname{Tr}(\mathcal{X})\mathcal{Y}+\operatorname{Tr}(\mathcal{Y})\mathcal{X}\right)$

This cross product can be used to identify primitive idempotents.

######  Lemma 3.2.

[SV00, 141] Let $\mathcal{X}\in\mathrm{Herm}(3,\mathbb{O})$ with
$\operatorname{Tr}(\mathcal{X})\neq 0$. Then $\mathcal{X}\times\mathcal{X}=0$
if and only if $\mathcal{X}$ is a scalar multiple of a primitive idempotent.

Now we can describe the projective plane geometry using the cross product.
First, $\mathcal{X}$ is a point in the octonion projective plane if and only
if $\mathrm{Tr}(\mathcal{X})\neq 0$ and $\mathcal{X}\times\mathcal{X}=0$. Two
points $\mathcal{X}$ and $\mathcal{Y}$ are equivalent points if they are real
scalar multiples. We can always represent a point by normalizing so that
$\mathrm{Tr}(\lambda\mathcal{X})=1$, as we do when we select a primitive
idempotent to represent the point. Two points $\mathcal{X}$ and $\mathcal{Y}$
are joined by the line consisting of all the points orthogonal to
$\mathcal{X}\times\mathcal{Y}$. When two lines are represented by points
$\mathcal{M}$ and $\mathcal{N}$, they intersect in the common point
$\mathcal{M}\times\mathcal{N}$.

######  Lemma 3.3.

Let $\mathcal{X}=(d,e,f\mid D,E,F)$ be a primitive idempotent representing a
point in the octonion projective plane $\mathbb{OP}^{2}$. Then $D,E,F$ belong
to the same quaternion subalgebra of $\mathbb{O}$ and
$\mathcal{X}=x^{\dagger}x$ for some row vector $x$ in $\mathbb{H}^{3}$.

###### Proof.

For details see [DM15, 119-125]. ∎

More details about the Albert algebra and octonion projective plane are
available in [SV00], [Bae02], and [DM15].

##  4\. An Octonion Leech Lattice

This section reviews Wilson’s construction of the Leech lattice using octonion
vectors in $\mathbb{O}^{3}$, rather than $\mathbb{R}^{24}$ [Wil09a], [Wil09b].
We adopt his notation and specific Leech lattice construction in what follows.

First we need to construct subsets of $\mathbb{O}$ isometric to a (possibly
scaled) $E_{8}$ lattice. Let $B$ denote the $E_{8}$ lattice with the following
natural choice of $240$ roots, for all $t\in\mathbb{F}_{7}$:

$\displaystyle\pm 1,$ $\displaystyle\pm i_{t},$ $\displaystyle\frac{1}{2}(\pm
1\pm i_{t}\pm i_{t+1}\pm i_{t+3}),$ $\displaystyle\frac{1}{2}(\pm i_{t+2}\pm
i_{t+4}\pm i_{t+5}\pm i_{t+6}).$

We call this $E_{8}$ lattice Kirmse’s octonions. The $E_{8}$ lattice defined
by these roots is not closed under octonion multiplication. However, we can
construct $7$ distinct octonion $E_{8}$ lattices that do close under
multiplication as follows:

$\displaystyle A_{t}=\frac{1}{2}(1-i_{t})B(1-i_{t}),\quad t\in\mathbb{F}_{7}.$

The $A_{t}$ octonion rings are each known as Coxeter-Dickson integral
octonions. Finally, we can recover the standard coordinates for the $E_{8}$
lattice in both the left-handed $L$ and right-handed $R$ form as follows:

$\displaystyle L=(1+i_{t})A_{t}=B(1-i_{t}),$ $\displaystyle
R=A_{t}(1+i_{t})=(1-i_{t})B.$

The intersection $L\cap R$ is a standard copy of the $D_{8}$ lattice, namely
the span of all $112$ $D_{8}$ roots $\pm i_{r}\pm i_{t}$ for
$r,t\in\mathrm{PL}(7)$. The remaining roots in $L$ are the $128$ vectors of
the form $\frac{1}{2}(\pm 1\pm i_{0}\pm i_{1}\cdots\pm i_{6})$ with an odd
number of minus signs. The remaining roots in $R$ are the $128$ vectors of
this form with an even number of minus signs. So the vector
$s=\frac{1}{2}(-1+i_{0}+i_{1}+i_{2}+i_{3}+i_{4}+i_{5}+i_{6})$ is in $L$ while
the vector $\overline{s}$ is in $R$.

The Moufang laws that octonions satisfy yield the following simple relations
between $L$, $R$, and $B$:

$\displaystyle LR=2B,$ $\displaystyle BL=L,$ $\displaystyle RB=R.$

Wilson uses these relations to construct the following octonionic definition
of the Leech lattice as the row vectors $(x,y,z)$ in $\mathbb{O}^{3}$ that
satisfy,

  1. (1)

$x,y,z\in L$

  2. (2)

$x+y,y+z,x+z\in L\overline{s}$

  3. (3)

$x+y+z\in Ls$.

In this construction the Leech lattice norm is given by,

$\displaystyle
N(x,y,z)=\frac{1}{2}(x,y,z)(x,y,z)^{\dagger}=\frac{1}{2}(x\overline{x}+y\overline{y}+z\overline{z}).$

We will call this set of octonion vectors in $\mathbb{O}^{3}$ Wilson’s Leech
lattice. The short vectors (of norm $4$) of Wilson’s Leech lattice are listed
in Fig. 3, for $\lambda$ any root in $L$, for all $j,k$ in $\\{\pm i_{t}\mid
t\in\\{\infty\\}\cup\mathbb{F}_{7}\\}$, and all permutations of the
coordinates.

Shape | Count | Total
---|---|---
$(2\lambda,0,0)$ | $3\times 240$ | $720$
$(\lambda\overline{s},(\lambda\overline{s})j,0)$ | $3\times 240\times 16$ | $11520$
$((\lambda s)j,\lambda k,(\lambda j)k)$ | $3\times 240\times 16\times 16$ | $184320$
|  | $196560$
Figure 3. The octonion Leech lattice short vectors

Wilson shows how to construct $2\cdot\mathrm{Co}_{1}$, the automorphism group
of the Leech lattice, using $3\times 3$ octonion matrices acting on the right
of the Leech lattice row vectors $(x,y,z)$. In particular, Wilson shows that
the following actions preserve the Leech lattice. First, if we write,

$\displaystyle R_{x}=\left(\begin{array}[]{ccc}x&0&0\\\ 0&x&0\\\
0&0&x\end{array}\right),$

then the following maps preserve the Leech lattice,

$\displaystyle(x,y,z)\mapsto\frac{1}{2}\left((x,y,z)R_{1-i_{0}}\right)R_{1+i_{t}}.$

In fact, these nested matrix actions generate a $2\cdot A_{8}$ action on the
Leech lattice. If we adjoin right multiplication by the following matrices,

$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&i_{t}&0\\\
0&0&i_{t}\end{array}\right),$

Then we obtain the action of the maximal subgroup $2^{3+12}(A_{8}\times
S_{3})$ of the Leech lattice automorphism group $2\cdot\mathrm{Co}_{1}$. To
obtain the full $2\cdot\mathrm{Co}_{1}$ action, we need to adjoin right
multiplication by,

$\displaystyle-\frac{1}{2}\left(\begin{array}[]{ccc}0&\overline{s}&\overline{s}\\\
s&-1&1\\\ s&1&-1\end{array}\right).$

Wilson uses subsets of these actions, and closely related ones, to exhibit the
groups of the Suzuki chain of $\mathrm{Co}_{1}$ subgroups acting on this
octonion Leech lattice:

$\displaystyle
S_{3}<S_{4}<\mathrm{PSL}_{2}(7)<\mathrm{PSU}_{3}(3)<\mathrm{HJ}<G_{2}(4)<3\cdot\mathrm{Suz}.$

In this paper we identify a number of different octonion involutionary
matrices that generate the Conway group $\mathrm{Co}_{1}$ and use them to link
the two unique projective $5$-designs.

##  5\. Constructing the Unique Real Projective Tight 5-Design

We now construct the first tight projective $5$-design using vectors in
$\mathbb{O}^{3}$ and provide octonion involutionary matrices that generate the
actions of Suzuki chain subgroups on this $5$-design. The unique tight
projective $5$-design in $\mathbb{R}^{24}\cong\mathbb{O}^{3}$ is defined by
the $98280$ pairs of opposite Leech lattice short vectors, projected onto
$\mathbb{RP}^{23}$. In what follows we will always identify opposite octonion
Leech lattice vectors. The action of the Leech lattice automorphisms on
opposite pairs of short vectors is the simple group $\mathrm{Co}_{1}$ acting
on $98280$ points, the smallest permutation representation.

Let’s begin with the Suzuki chain subgroup $S_{3}<\mathrm{Co}_{1}$. We already
know that any permutation of the three components of the octonion triples
$(x,y,z)$ in Wilson’s Leech lattice is a Leech lattice automorphism, and
therefore an element of $\mathrm{Co}_{1}$. We can model this group action
using reflection matrices. First consider the vectors in Wilson’s Leech
lattice of Fig. 3 with the form
$(\lambda\overline{s},(\lambda\overline{s})j,0)$ and $\lambda=\pm s$ and
$j=\mp i_{\infty}=\mp 1$. There are six vectors of this form, e.g.
$(2,-2,0),(-2,2,0)$ and all permutations of the coordinates. These form a
scaled $\mathtt{A_{2}}$ system of roots. We can construct a Weyl reflection
matrix from a row vector $r$ as,

$\displaystyle\mathcal{W}(r)=I-2r^{\dagger}r/(rr^{\dagger}).$

To reflect any $(x,y,z)\in\mathtt{A_{2}}$ by any $r$ in $\mathtt{A_{2}}$ we
compute,

$\displaystyle(x,y,z)\mapsto(x,y,z)\mathcal{W}(r).$

In fact, for any $(x,y,z)$ in Wilson’s Leech lattice and $r$ in
$\mathtt{A_{2}}$ this map returns another member of Wilson’s Leech lattice.
That is, for $r$ in $\mathtt{A_{2}}$, right multiplication by $\mathcal{W}(r)$
is a Leech lattice automorphism and an element of $\mathrm{Co}_{1}$. This is
not true for any $r$ in general, but for certain well chosen vectors $r$,
right multiplication by $\mathcal{W}(r)$ is in fact a Leech lattice
automorphism. Restricting to $r$ in $\mathtt{A_{2}}$ for the moment, these
maps generate a $S_{3}=W(\mathtt{A_{2}})<\mathrm{Co}_{1}$ subgroup of Leech
lattice automorphisms, the first and smallest in the Suzuki chain. By using
other careful choices of reflection vector $r$, we can generalize this
procedure to recover the action of all the remaining Suzuki chain subgroups
acting on the Leech lattice short vector opposite pairs.

To see which choices of $r$ are suitable for exhibiting Leech lattice
symmetries, note that there is a correspondence between idempotent matrices in
$\mathbb{OP}^{2}$ and involutive matrices in the Albert algebra
$\mathrm{Herm}(3,\mathbb{O})$. That is, for any idempotent matrix
$\mathcal{X}$ in $\mathbb{OP}^{2}$, the element $I-2\mathcal{X}$ squares to
$I$ (with $I$ the identity matrix):

$\displaystyle(I-2\mathcal{X})^{2}=I^{2}-4I\mathcal{X}+4\mathcal{X}^{2}=I.$

Suppose that $v=(x,y,z)$ is a vector in $\mathbb{O}^{3}$ with the components
contained in a common associative subalgebra of $\mathbb{O}$. Then
$\mathcal{X}=v^{\dagger}v/(vv^{\dagger})$ is an idempotent matrix in
$\mathbb{OP}^{2}$. So for any $v=(x,y,z)$ in $\mathbb{O}^{3}$ with $x,y,z$ in
a common quaternion subalgebra, we can define the reflection matrix of $v$ as,

$\displaystyle\mathcal{W}(v)={I}-2v^{\dagger}v/(vv^{\dagger}).$

This matrix has the involution property $\mathcal{W}(v)^{2}=I$.

Next, let $v_{1}$ and $v_{2}$ be vectors in $\mathbb{O}^{3}$ that admit the
construction of idempotents $v_{i}^{\dagger}v_{i}/(v_{i}v_{i}^{\dagger})$
(i.e., such that the components of $v_{i}$ belong to a common quaternion
subalgebra of $\mathbb{O}$) so that $\mathcal{W}(v_{1})$ and
$\mathcal{W}(v_{2})$ are well defined. These matrices act on row vectors of
the form $(x,y,z)$ through nesting—the composition of right matrix
multiplication maps—so that we never actually compute the matrix product
$\mathcal{W}(v_{1})\mathcal{W}(v_{2})$. That is, the successive application of
$\mathcal{W}(v_{1})$ and $\mathcal{W}(v_{2})$ has the form,

$\displaystyle(x,y,z)\mapsto\left[(x,y,z)\mathcal{W}(v_{1})\right]\mathcal{W}(v_{2}).$

Provided that $\mathcal{W}(v_{1})$ and $\mathcal{W}(v_{2})$ each permute Leech
lattice short vectors, the composition of these two maps will also permute
Leech lattice vectors. But in general, this composition is not represented by
the matrix product $\mathcal{W}(v_{1})\mathcal{W}(v_{2})$, due to the non-
associativity of the octonion algebra.

In what follows we will restrict ourselves to reflection matrices
$\mathcal{W}(v)$ of vectors $v=(x,y,z)$ with $x,y,z$ in a common complex
subalgebra of $\mathbb{O}$. There are $2\times 1260$ short vectors in Wilson’s
Leech lattice for which the three coordinates belong to a common complex
subalgebra of the octonions. We’ve already seen that the six $\mathtt{A_{2}}$
vectors identified above are among these, and that their reflection matrices
generate a $S_{3}=W(\mathtt{A_{2}})$ action on the Leech lattice short vector
pairs.

Next, consider the following vectors among the short vectors of Wilson’s Leech
lattice with all components in a common complex subalgebra of $\mathbb{O}$:

$\displaystyle V_{t}=\left\\{\sigma(\pm 2,\pm
2i_{t},0),~{}\sigma\in\mathrm{Sym}(3)\right\\},\quad
t\in\\{\infty\\}\cup\mathbb{F}_{7}.$

We obtain these by taking the short vectors of the form
$(\lambda\overline{s},(\lambda\overline{s})j,0)$ in Fig. 3 and setting
$\lambda=s$. We see that $V_{\infty}$ contains $6$ pairs of opposite Leech
lattice short vectors. In fact, $V_{\infty}$ is a scaled $\mathtt{A_{3}}$
system of roots. Each $V_{t}$ for $t\in\mathbb{F}_{7}$ contains $12$ pairs of
opposite Leech lattice short vectors.

Each $v$ in each $V_{t}$ defines an octonion involutionary matrix
$\mathcal{W}(v)$ corresponding to an involutionary automorphism of the Leech
lattice. In particular, the $V_{\infty}$ vectors yield involutions generating
an $S_{4}=W(\mathtt{A_{3}})<\mathrm{Co}_{1}$ action on the Leech lattice short
vector pairs.

######  Lemma 5.1.

Let $\mathcal{W}(v)=I-2v^{\dagger}v/(vv^{\dagger})$ for $v$ any vector in
$V_{t}$, for any $t\in\mathrm{PL}(7)$. Then the following map is a Leech
lattice automorphism:

$\displaystyle(x,y,z)\mapsto(x,y,z)\mathcal{W}(v).$

The action on opposite pairs is an automorphism of the real projective tight
$5$-design and an involution in $\mathrm{Co}_{1}$. The groups $G$ generated by
these involutions are described in terms of their order and centralizer in
$\mathrm{Co}_{1}$ in the following table (with $t\neq
t^{\prime}\in\mathbb{F}_{7}$):

Vectors defining $\mathcal{W}(v)$ | $|G|$ | $\mathrm{Cent}_{\mathrm{Co}_{1}}(G)$
---|---|---
$\mathtt{A_{2}}\subset V_{\infty}$ | $6=|S_{3}|$ | $A_{9}$
$\mathtt{A_{3}}=V_{\infty}$ | $24=|S_{4}|$ | $A_{8}$
$V_{t}$ | $96$ | $A_{6}$
$V_{t}\cup V_{t^{\prime}}$ | $384$ | $A_{5}$
$V_{t}\cup V_{t+1}\cup V_{t+3}$ | $1536$ | $A_{4}$
$V_{t+2}\cup V_{t+5}\cup V_{t+6}\cup V_{t+7}$ | $6144$ | $A_{3}$
$V_{\infty}\cup V_{0}\cup V_{1}\cup\cdots\cup V_{6}$ | $98304$ | $1$

###### Proof.

These facts are confirmed using GAP computations [noa20] acting on the pairs
of opposite short vectors in Wilson’s Leech lattice. ∎

We now provide a set of generators for a $\mathrm{PSL}_{2}(7)$ subgroup in
$\mathrm{Co}_{1}$. Notice that the octonion algebra $\mathbb{O}$ has the
following cyclic automorphism:

$\displaystyle\alpha:i_{t}\mapsto i_{t+1}.$

The action of $\alpha$ on the coefficients of Wilson’s Leech lattice short
vectors induces a permutation of the short vector pairs. This permutation is
an element of order $7$ of $\mathrm{Co}_{1}$. We may also denote this element
of order $7$ in $\mathrm{Co}_{1}$ by $\alpha$. The centralizer of $\alpha$ in
$\mathrm{Co}_{1}$ has the form $C_{7}\times\mathrm{PSL}_{2}(7)$. In fact,
$\alpha$ fixes exactly $21$ opposite pairs of short vectors in Wilson’s Leech
lattice, and these $21$ vectors all have components within the same complex
subalgebra $\mathbb{R}(s)$ of $\mathbb{O}$. In fact, $s$ has the form
$\frac{1}{2}(-1+\sqrt{-7})$ in $\mathbb{R}(s)\cong\mathbb{C}$. We denote by
$S$ the $2\times 21$ short vectors fixed by $\alpha$.

To obtain the vectors in $S$, we take all the short vectors in Fig. 3
corresponding to $\lambda=s$ and $j,k=\pm 1$. This yields six short vectors of
the form $(\lambda,0,0)$, $12$ short vectors of the form
$(\lambda\overline{s},(\lambda\overline{s})j,0)$, and $24$ short vectors of
the form $((\lambda s)j,\lambda k,(\lambda j)k)$.

######  Lemma 5.2.

For each short vector $v$ in $S$, i.e. the short vectors in Wilson’s Leech
lattice fixed under $i_{t}\mapsto i_{t+1}$, the map
$(x,y,z)\mapsto(x,y,z)\mathcal{W}(v)$ is a Leech lattice involutionary
automorphism. There are $21$ such involutions corresponding to fixed vectors.
The group generated by these $21$ involutions is the group
$\mathrm{PSL}_{2}(7)$ with centralizer $A_{7}$ in $\mathrm{Co}_{1}$. This
$\mathrm{PSL}_{2}(7)$ is also the centralizer of $A_{7}$ in $\mathrm{Co}_{1}$.

###### Proof.

These facts have been confirmed by computation using GAP [noa20]. ∎

We can now use the generators corresponding to the vectors in the sets $S$ and
$V_{t}$, with $t\in\mathbb{F}_{7}$, to generate the Suzuki chain subgroups of
$\mathrm{Co}_{1}$.

######  Lemma 5.3.

The following Suzuki chain subgroups can be generated via the right action,

$\displaystyle(x,y,z)\mapsto(x,y,z)\mathcal{W}(v),$

where $\mathcal{W}(v)$ are octonion involution matrices constructed from the
following vectors:

Vectors defining $\mathcal{W}(v)$ | $G$ | $\mathrm{Cent}_{\mathrm{Co}_{1}}(G)$
---|---|---
$\mathtt{A_{2}}\subset V_{\infty}$ | $S_{3}$ | $A_{9}$
$\mathtt{A_{3}}=V_{\infty}$ | $S_{4}$ | $A_{8}$
$S$ | $\mathrm{PSL}_{2}(7)$ | $A_{7}$
$S\cup V_{t}$ | $\mathrm{PSU}_{3}(3)$ | $A_{6}$
$S\cup V_{t}\cup V_{t^{\prime}}$ | $\mathrm{HJ}$ | $A_{5}$
$S\cup V_{t}\cup V_{t+1}\cup V_{t+3}$ | $G_{2}(4)$ | $A_{4}$
$S\cup V_{t+2}\cup V_{t+5}\cup V_{t+6}\cup V_{t+7}$ | $3\cdot\mathrm{Suz}$ | $A_{3}$
$S\cup V_{\infty}\cup V_{0}\cup V_{1}\cup\cdots\cup V_{6}$ | $\mathrm{Co}_{1}$ | $1$

###### Proof.

These facts have been confirmed by computation in GAP [noa20] and making use
of the automorphism $i_{t}\mapsto i_{t+1}$. ∎

The groups $G_{2}(4)$, $3\cdot\mathrm{Suz}$, and $\mathrm{Co}_{1}$ are each
transitive on the Leech lattice short vector pairs. This means that the
octonion reflection matrices corresponding to the vectors listed for each of
these three groups in Lemma 5.3 are sufficient to construct the unique tight
real projective $5$-design in $\mathbb{RP}^{23}$. Minimally, we can construct
the unique tight real projective $5$-design in $\mathbb{RP}^{23}$ using only a
$G_{2}(4)$ subgroup of Leech lattice automorphisms. That is, the vectors of
$S\cup V_{t}\cup V_{t+1}\cup V_{t+3}$ for $t\in\mathbb{F}_{7}$ define the
reflection matrices $\mathcal{W}(v)$ needed to construct the unique tight
projective $5$-design in $\mathbb{RP}^{23}$. The design itself is independent
of our choice of $t\in\mathbb{F}_{7}$.

######  Remark 5.4.

We can make better sense of the choices of vectors in Lemmas 5.1 and 5.3 by
returning to Fig. 2 and replacing the labels $i_{0},i_{1},\ldots,i_{6}$ with
$V_{0},V_{1},\ldots,V_{6}$. In particular, the generators of
$\mathrm{PSU}_{3}(3)$ correspond to any point on the edge of the diagram,
those of $\mathrm{HJ}$ to any pair of points or an arc, those of $G_{2}(4)$ to
a triangle constructed from an arc and its product (i.e. $1,3,7$), those of
$3\cdot\mathrm{Suz}$ to any pair of arcs, and those of $\mathrm{Co}_{1}$ to
all seven points.

##  6\. Action on the Octonion Projective Plane

In this section we use the same set of generating vectors $v$ in $S$ and
$V_{t}$, for $t\in\mathbb{F}_{7}$, to both (a) define points in the octonion
projective plane of the form $\mathcal{U}(v)=v^{\dagger}v/(vv^{\dagger})$ and
(b) act on these points in $\mathbb{OP}^{2}$ via the action,

$\displaystyle\mathcal{U}(v)\mapsto\mathcal{W}(v^{\prime})^{\dagger}\mathcal{U}(v)\mathcal{W}(v^{\prime}).$

By using the generating vectors $v$ of various different subgroups in the
Suzuki chain, described in Lemma 5.3, we can construct a number of interesting
projective $t$-designs in $\mathbb{OP}^{2}$. Most notably, the generating
vectors of both $\mathrm{HJ}$ and $G_{2}(4)$ both yield the same projective
$t$-design—the unique octonionic projective tight $5$-design with $819$ points
in $\mathbb{OP}^{2}$.

The first task is to confirm that the action
$\mathcal{U}(v)\mapsto\mathcal{W}(v^{\prime})^{\dagger}\mathcal{U}(v)\mathcal{W}(v^{\prime})$
is well-defined given the non-associativity of the octonion algebra.

######  Lemma 6.1 ([DM15]).

Suppose that the components of $\mathcal{W}(v^{\prime})$ belong to the same
complex subalgebra of $\mathbb{O}$. Then we have,

$\displaystyle\mathcal{W}(v^{\prime})^{\dagger}(\mathcal{U}(v)\mathcal{W}(v^{\prime}))=(\mathcal{W}(v^{\prime})^{\dagger}\mathcal{U}(v))\mathcal{W}(v^{\prime}),$

for all $\mathcal{U}(v)$ in the octonion projective plane $\mathbb{OP}^{2}$.

###### Proof.

The Hermitian octonion matrix $\mathcal{U}(v)$ has the form $(d,e,f\mid
D,E,F)$, being an element in the Albert algebra $\mathrm{Herm}(3,\mathbb{O})$.
This means that we can write this matrix as a sum of terms of the form
$(d,0,0\mid 0,0,0)$, $(0,e,0\mid 0,0,0),\ldots,(0,0,0\mid 0,0,F)$. Let
$\mathcal{X}$ be one of these six matrices. The coordinates of $\mathcal{X}$
and of $\mathcal{W}(v^{\prime})$ belong to a common quaternion subalgebra of
$\mathbb{O}$, since $\mathcal{X}$ only contains one independent octonionic
coordinate and this coordinate combines with the common complex subalgebra of
$\mathbb{O}$ given by the coordinates of $\mathcal{W}(v^{\prime})$ to defined
at the most a quaternion subalgebra of $\mathbb{O}$. This means that we have
in each case,

$\displaystyle\mathcal{W}(v^{\prime})^{\dagger}(\mathcal{X}\mathcal{W}(v^{\prime}))=(\mathcal{W}(v^{\prime})^{\dagger}\mathcal{X})\mathcal{W}(v^{\prime}).$

Since each term associates, the sum also associates. ∎

We see then that the action,

$\displaystyle\mathcal{U}(v)\mapsto\mathcal{W}(v^{\prime})^{\dagger}\mathcal{U}(v)\mathcal{W}(v^{\prime}),$

is well defined for our choice of $v$ in $S$ or $V_{t}$ for some
$t\in\mathbb{F}_{7}$. It turns out that this action defines an involutionary
automorphism of the Albert algebra $\mathrm{Herm}(3,\mathbb{O})$, contained in
the Lie group $F_{4}$. As before, we compose two involutions corresponding to
$v_{1}$ and $v_{2}$ by nesting:

$\displaystyle\mathcal{U}\mapsto\mathcal{W}(v_{2})^{\dagger}(\mathcal{W}(v_{1})^{\dagger}\mathcal{U}~{}\mathcal{W}(v_{1}))\mathcal{W}(v_{2}).$

We do not need to evaluate $\mathcal{W}(v_{1})\mathcal{W}(v_{2})$ and in
general this matrix will not represent the composition of the involutions
given by the two factors, due to the non-associativity of $\mathbb{O}$.

We now have a method to construct subgroups of $F_{4}$ acting on the octonion
projective plane $\mathbb{OP}^{2}$ generated by involutions corresponding to a
set of suitable octonion involutionary matrices $\mathcal{W}(v)$. The orbit of
$\mathcal{U}(v)\in\mathbb{OP}^{2}$ under the action of these involutions can
form a projective $t$-design.

######  Lemma 6.2.

The following projective $t$-designs can be generated via the action,

$\displaystyle\mathcal{U}(v)\mapsto\mathcal{W}(v^{\prime})^{\dagger}\mathcal{U}(v)\mathcal{W}(v^{\prime}),$

for $v,v^{\prime}$ in the following subsets of $S\cup V_{0}\cup
V_{1}\cup\cdots\cup V_{6}$.

  * •

$v,v^{\prime}\in V_{\infty}\subset S$ generates a $S_{4}$ action on a $6$
point projective $1$-design in $\mathbb{RP}^{2}$.

  * •

$v,v^{\prime}\in S$ generates a $\mathrm{PSL}_{2}(7)$ action on a $21$ point
projective $2$-design in $\mathbb{CP}^{2}$.

  * •

$v,v^{\prime}\in S\cup V_{t}$ for any $t\in\mathbb{F}_{7}$ generates a
$\mathrm{PSU}_{3}(3)$ action on a $63$ point projective $3$-design in
$\mathbb{HP}^{2}$.

  * •

$v,v^{\prime}\in S\cup V_{t}\cup V_{t^{\prime}}$ with $t\neq
t^{\prime}\in\mathbb{F}_{7}$ generates a ${}^{3}D_{4}(2)$ action on a $819$
point projective $5$-design in $\mathbb{OP}^{2}$.

  * •

$v,v^{\prime}\in S\cup V_{t}\cup V_{t+1}\cup V_{t+3}$ with
$t\in\mathbb{F}_{7}$ generates a ${}^{3}D_{4}(2)$ action on a $819$ point
projective $5$-design in $\mathbb{OP}^{2}$.

###### Proof.

These facts have been confirmed by computation in GAP [noa20] and making use
of the automorphism $i_{t}\mapsto i_{t+1}$. ∎

The $5$-design in $\mathbb{OP}^{2}$ described in Lemma 6.2 is the unique tight
octonion projective $5$-design. The construction via generators corresponding
to the vectors in $S\cup V_{t}\cup V_{t+1}\cup V_{t+3}$ yields seven
intersecting $5$-designs, one for each value of $t\in\mathbb{F}_{7}$. The
intersection of these $5$-designs is the $2$-design generated by the vectors
in $S$ alone.

##  7\. Conclusion

We have seen that the vectors in $v\in S\cup V_{t}\cup V_{t+1}\cup V_{t+3}$
with $t\in\mathbb{F}_{7}$ define octonion reflection matrices $\mathcal{W}(v)$
that exhibit a $G_{2}(4)$ action on octonion row vectors in $\mathbb{O}^{3}$
and a corresponding ${}^{3}D_{4}(2)$ action on points in the octonion
projective plane $\mathbb{OP}^{2}$. In the first case, the $G_{2}(4)$ action
yields a single orbit of $98280$ pairs of opposite vectors in
$\mathbb{O}^{3}\cong\mathbb{R}^{24}$, which form a tight projective $5$-design
in $\mathbb{RP}^{23}$. The corresponding ${}^{3}D_{4}(2)$ action yields a
single orbit of $819$ points in the octonion projective plane
$\mathbb{OP}^{2}$, which form another tight projective $5$-design. These are
the only two tight projective $5$-designs and we can now see that they share a
common construction.

## References

  * [Bae02] John Baez, _The octonions_ , Bulletin of the American Mathematical Society 39 (2002), no. 2, 145–205.
  * [BS81] Eiichi Bannai and N. J. A. Sloane, _Uniqueness of Certain Spherical Codes_ , Canadian Journal of Mathematics 33 (1981), no. 2, 437–449 (en), Publisher: Cambridge University Press.
  * [CD07] Charles J. Colbourn and Jeffrey H. Dinitz, _Handbook of Combinatorial Designs_ , CRC Press, 2007 (en).
  * [CS03] John H. Conway and Derek A. Smith, _On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry_ , CRC Press, 2003.
  * [CT85] A. M. Cohen and J. Tits, _On Generalized Hexagons and a Near Octagon whose Lines have Three Points_ , European Journal of Combinatorics 6 (1985), no. 1, 13–27 (en).
  * [DM15] Tevian Dray and Corinne A. Manogue, _The geometry of the octonions_ , World Scientific, 2015.
  * [Hog82] Stuart G. Hoggar, _t-Designs in projective spaces_ , European Journal of Combinatorics 3 (1982), no. 3, 233–254.
  * [Hog89] by same author, _Tight 4 and 5-designs in projective spaces_ , Graphs and Combinatorics 5 (1989), no. 1, 87–94.
  * [noa20] _GAP – Groups, Algorithms, and Programming, Version 4.11.0_ , 2020\.
  * [SV00] Tonny A. Springer and Ferdinand D. Veldkamp, _Octonions, Jordan Algebras and Exceptional Groups_ , Springer Monographs in Mathematics, Springer-Verlag, Berlin Heidelberg, 2000 (en).
  * [Wil09a] Robert Wilson, _The Finite Simple Groups_ , Graduate Texts in Mathematics, Springer-Verlag, London, 2009 (en).
  * [Wil09b] Robert A. Wilson, _Octonions and the Leech lattice_ , Journal of Algebra 322 (2009), no. 6, 2186–2190 (en).


# A Federated Channel Modeling System using Generative Neural Networks
††thanks: This work was partially supported by the European Union under the
Italian National Recovery and Resilience Plan (PNRR) of NextGenerationEU,
partnership on "Telecommunications of the Future" (PE00000001 - program
"RESTART") and by the HORIZON-CL4-2021-SPACE-01 project "5G+ evoluTion to
mutioRbitAl multibaNd neTwORks" (TRANTOR) No. 101081983

Saira Bano1,2, Pietro Cassarà2, Nicola Tonellotto1, Alberto Gotta2
1Department of Information Engineering, University of Pisa, Pisa, Italy
2Information Science and Technology Institute "A. Faedo", National Research
Council, Pisa, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The paper proposes a data-driven approach to air-to-ground channel estimation
in a millimeter-wave wireless network on an unmanned aerial vehicle. Unlike
traditional centralized learning methods that are specific to certain
geographical areas and inappropriate for others, we propose a generalized
model that uses Federated Learning (FL) for channel estimation and can predict
the air-to-ground path loss between a low-altitude platform and a terrestrial
terminal. To this end, our proposed FL-based Generative Adversarial Network
(FL-GAN) is designed to function as a generative data model that can learn
different types of data distributions and generate realistic patterns from the
same distributions without requiring prior data analysis before the training
phase. To evaluate the effectiveness of the proposed model, we evaluate its
performance using Kullback-Leibler divergence (KL), and Wasserstein distance
between the synthetic data distribution generated by the model and the actual
data distribution. We also compare the proposed technique with other
generative models, such as FL-Variational Autoencoder (FL-VAE) and stand-alone
VAE and GAN models. The results of the study show that the synthetic data
generated by FL-GAN has the highest similarity in distribution with the real
data. This shows the effectiveness of the proposed approach in generating
data-driven channel models that can be used in different regions.

###### Index Terms:

Federated learning, Unmanned aerial vehicles, Channel modeling, Generative
neural networks

##  I Introduction

Non-terrestrial networks (NTNs), such as near-earth satellite constellations
(LEO), high-altitude platforms, and unmanned aerial vehicles (UAVs) have
traditionally been used for disaster management and remote sensing [1].
However, they are now being seen as promising technologies for providing
ubiquitous connectivity in the future generation of the Internet [2]. Such
radio access networks (RAN), operating in the millimeter wave (mmWave) range,
are very promising, providing global coverage and high capacity for reliable
and efficient communications services [3]. The 3rd Generation Partnership
Project (3GPP), has also recognized the potential of mmWave technology to
support satellite communications.

Accurate statistical channel models are essential to characterize the mmWave
link and to determine the underlying channel parameters to improve the
transmission performance of wireless communication systems. Extensive research
has been conducted to develop effective methods for accurate channel modeling,
such as the mathematical propagation model proposed in [4] for estimating
ground-to-air path loss between wireless devices and low-altitude platforms
using mmWave frequency bands. Furthermore, deterministic channel models, such
as ray-tracing techniques, as well as stochastic channel models are commonly
used and require extensive technical knowledge and expertise for analyzing
measurement data to estimate a comprehensive set of different channel
parameters [5]. However, building statistical channel models to determine the
underlying channel parameters that accurately capture the delay, direction,
and path gains of individual links is difficult, especially in the mmWave
domain.

Machine Learning (ML) techniques, such as Neural Networks (NNs), can be used
to develop statistical channel models that overcome the limitations of
conventional channel modeling systems [6]. However, these models result in
channel parameters that are site-specific and may not be generally applicable.
In this regard, generative NNs, which have proven to be very successful in
modeling images and text, provide a suitable approach to data-driven channel
modeling and can accurately represent complex environments. Initial research
has explored the use of generative NNs for site-specific wireless channels.
For example, in [7], the authors proposed generative networks to model channel
parameters and trained five different models for five different cities. In
contrast, our main goal is to develop a general model that can be used for all
participating cities, considering an acceptable model performance for each of
these different locations.

To this end, we propose a location-agnostic statistical channel propagation
model based on Federated Learning (FL) that focuses on predicting the path
loss component between a UAV and terrestrial nodes in mmWave communication
networks. FL is a paradigm developed by Google that aims to build ML models
with distributed datasets across multiple devices while maintaining privacy
[8]. Participating users communicate parameters or gradients to a central
server, which updates and distributes a global model without access to user
data [9, 10, 11]

@inproceedingsbano2022federated,. However, in this work, we used the FL
frameworks as distributed training engines to train our models on different
datasets and develop the generalized channel model using Variational
Autoencoder (VAE) and Conditional Generative Adversarial Network (CGAN)
architectures, i.e. FL-VAE and FL-GAN. In our study, we rely on the
statistical characteristics of the urban environment of the target area
collected through ray tracing simulations to train the models. The performance
of the proposed approach is determined using various statistical parameters.

The remainder of the paper is organised as follows. Section II discusses the
system model, while sections III and IV present the federated VAE and GAN
approaches for channel modeling, respectively. Section V shows the
experimental evaluation performed. Finally, Section VI draws conclusions.

##  II System Model

In this work, we focus on channel parameter modeling with the main focus on
the path loss component connecting UAVs to cellular base stations on the
ground, i.e. gNB. We propose a distributed training approach using FL for
channel model estimation with two generative NNs. For modelling purposes, we
assume that the UAVs act as transmitters and the ground base stations act as
receivers, but the roles can be reversed. To model the air-to-ground channel,
we assume two ground gNBs, one terrestrial and the other aerial, as in [12].
The aerial gNBs serve as dedicated stations (mounted on rooftops and tilted
upward), while the terrestrial gNBs are for ground users (mounted at street
level), as shown in Figure 1. In addition, we assume three link states between
the transmitter and receiver, including Line of Sight (LOS), Non-LOS (NLOS),
and no link (i.e., no paths are available). However, when modelling path loss
between UAVs and gNBs, we mainly focus on NLOS paths since for LOS, path loss
can be calculated using Friis’ law [13].

We adopt the channel parameters estimated with the raytracer package by [7] as
a benchmark dataset for our investigation. The raytracer simulations estimate
the channel parameters, including path losses, azimuth and elevation angles of
arrival and departure, and propagation delays. According to the dataset, there
is a total of 20 paths per link and six parameters per path, resulting in 120
parameters per link with a maximum path loss of 200 dB [7]. The dataset
consists of channel parameters for different cities estimated by using the
ray-tracer package. Using this dataset, we train the generative models for
each city in a decentralized manner. These standalone models can learn the
channel representation of a UAV’s local dataset in a given region but may have
biases and be applicable only in a limited spatial domain. Therefore, a
general model that is not tied to a specific environment is essential. To this
end, we use FL to aggregate these standalone models and obtain a global model.
We validate the generated model using CDF of path loss.

In the proposed approach, we use two generative NN models, both of which have
a two-stage structure, i.e., link and path models [12]. In the first stage, an
NN is used as a link model to determine the state of the link - whether it is
LOS, NLOS, or no link, according to 3GPP requirements [14]. To determine the
link state, the relative position of the UAV to the gNB and the type of gNB
are used as inputs. After the link state is determined, a generative model,
i.e., a path model, is used in the second stage to generate the path
parameters. This generative model is trained to match the distribution of the
training dataset. To perform the distributed training using FL, we trained the
link-state model for each city and stored it on the corresponding station to
use with the path model in FL. We then aggregate these generative models as
described in Section III and Section IV, respectively. Once the model is
trained, it can be used in simulations to statically determine channel
parameters considering the link status.

Figure 1: System Model

##  III Federated VAE

In this section, we describe our FL-VAE for channel modeling of the path loss
component. We first introduce the basic concepts of VAE to understand its
content (Section III-A) and then describe in detail the FL-VAE approach
(Section III-B) used for modeling the channel parameters.

###  III-A Variational Autoencoder (VAE)

VAE consists of encoder and decoder modules, where the encoder, defined as
$q_{\theta}(z|x)$, characterizes the distribution of the input variables $x$
according to the encoding in the latent space $z$ (encoded representation of
the input variables). On the other hand, the decoder, defined as
$p_{\phi}(x|z)$, characterizes the distribution of the decoded variables based
on the latent space, where $\theta$ and $\phi$ are the parameters of the
encoder and decoder NNs, respectively. The loss function of the VAE given in
[15] is as follows:

$\mathcal{L}(\phi,\theta)=-\mathbb{E}_{q_{\theta}(z|x_{i})}\bigl{[}\log
p_{\phi}(x_{i}|z)\bigr{]}+KL(q_{\theta}(z|x_{i})\|p(z))$ (1)

The first component of the expression represents the reconstruction loss
corresponding to the expected negative log likelihood of each data point. The
expected value is calculated based on the encoder’s distribution over the
representations, and this component is intended to provide an incentive for
the decoder to acquire the ability to reconstruct the data. The second term is
the KL divergence, which acts as a regulizer and measures the loss of
information when we use $q_{\theta}(z|x_{i})$ to represent $p(z)$, which is
the posterior distribution defined for the latent space, i.e., a Gaussian
distribution.

###  III-B FL-VAE

FL-VAE uses the same VAE architecture proposed in [12] and trains generative
(path) model using the FL framework developed in [16]. The goal of FL-VAE is
to capture the conditional distribution $p(x|u)$ of all participating cities
such that it tends to encode the local latent space of all cities into a
single latent space and form the generic global model for generating channel
parameters. VAEs can easily be trained in an FL framework since their encoder
and decoder components comprise of NNs. Let
$\mathcal{V}\coloneqq(\theta^{e},\theta^{d})$ be the VAE parameters, and
$\theta^{e}$ and $\theta^{d}$ be the weights of the encoder and decoder,
respectively. A centralized server initiates the training by communicating the
initial weights of VAE $\mathcal{V}^{t}$ to all agents in the participating
city stations. Each agent in a city initializes its own VAE model with these
weights and uses local training data and a pre-trained link model to obtain a
latent representation of its own data. Local updates of each city $k$ is given
by:

$\mathcal{V}^{t+1}_{k}\longleftarrow\mathcal{V}^{t}_{k}-\eta\nabla\mathcal{L}(\mathcal{V}^{t}_{k})$
(2)

Where $\eta$ is the learning rate. Each city agent uses equation (2) to
perform some local training epochs on local data and send the updates
$\mathcal{V}^{t+1}_{k}$ to the central server. The server finally amalgamates
the received updates with a weighted average approach given by:

$\mathcal{V}^{t+1}=\sum_{k=1}^{K}\frac{n_{k}}{n}\mathcal{V}^{t}_{k}$ (3)

$n_{k}$ are the number of training examples at each agent $k$ and $n$ is the
total number of training data of each city. The server continues training
until it obtains a global latent representation sufficient to represent all
training data.

##  IV Federated CGAN

In this section, we describe our FL-GAN approach to channel modeling. We first
describe the Generative Adversarial Network (GAN) (Section IV-A) and then the
FL-GAN (Section IV-B) used to model the channel parameters to form the
generalised or universal model.

###  IV-A Generative Adversarial Network (GAN)

The GAN is a popular concept first proposed in [17]. Its main purpose is to
generate synthetic data that closely resembles real data. GANs use an
unsupervised learning approach to detect patterns in the input data and
generate new samples with the same distribution as the original data. It
consists of two NNs: the generator (G) and the discriminator (D), which
compete in a "min-max two-player game." The G generates synthetic (fake) data
from the learned latent vector, while the D discriminates the synthetic data
from the real data. These models are trained until the G replicates the
original data so well that it becomes difficult for the D to distinguish
between the fake and the real data.

To generate samples from a given target, the CGAN was introduced in [18]. A
CGAN learns the mapping from an observed sample $x$ and a random noise vector
$z$ to an output sample $y$, represented as $G:{x,z}\xrightarrow{}y$, where
$G$ is the generator function. Both networks in CGAN aim to solve a "min-max
loss" like GAN given by [18]:

$\mathcal{L}_{cGAN}(\mathcal{G},\mathcal{D})=\mathop{\mathbb{E}_{x,y}}\bigl{[}\log(\mathcal{D}(x,y))\bigr{]}+\\\
\mathop{\mathbb{E}_{x,z}}\bigl{[}1-\log(\mathcal{D}(x,\mathcal{G}(x,z)))\bigr{]}$
(4)

G and D compete according to equation (2), where D tries to maximize the
probability of assigning correct labels, and G tries to minimize that
probability. In the next section, we describe the distributed approach using
FL to train the CGAN.

###  IV-B FL-GAN

We use the FL technique to train CGAN in a distributed manner. The training
process is initiated by a central server, which communicates the initial
parameters of generator and discriminator i.e., $\theta^{G}$ and $\theta^{D}$
to the agents in the cities. Each city agent initializes its own CGAN instance
with the received parameters and trains it using local data and associated
link state models. The updated parameters are then reported back to the
server, which aggregates the updates from all cities as follows:

$\theta^{G}=\sum_{k=1}^{K}\frac{n_{k}}{n}\theta_{k}^{G}\quad;\quad\theta^{D}=\sum_{k=1}^{K}\frac{n_{k}}{n}\theta_{k}^{D}$
(5)

$\theta^{G}$ and $\theta^{D}$ in equation (5) are the aggregate parameter
estimates of G and D, respectively. The server goes through this process until
it develops a global CGAN that can generate synthetic samples from the
distribution that captures the local data distributions. After training, each
local city unit can generate the path parameters with $\theta^{G}$.

##  V Simulation Results

In this section, we describe the performed experiments to assess the
efficiency and effectiveness of the proposed FL approach.

###  V-A Dataset and settings

In this work, we use raytracer data provided by [7]. The dataset consists of
channel parameters from five different cities, each with different landscapes
and structures. However, for this work, we use the channel parameters of three
cities (Beijing, London, and Boston). In the raytracer simulation, the
transmitting UAVs are positioned at different horizontal locations in each
environment, with four possible heights: 30, 60, 90 and 120 m, to create the
whole city dataset. A total of 36k links were created for Beijing, 25.8k for
London, and 23k for Boston. All simulations were performed at a frequency of
28 GHz.

For our learning models, we used two generative NNs and trained them in a
distributed manner using FL to make FL-VAE and FL-GAN model. The main goal is
to develop a distributed model using FL framework that can be used universally
for estimating channel parameters. In this context, we compare the generative
models trained in a distributed manner and analyse which model is better at
capturing channel characteristics of different latent spaces. We compared the
results of these distributed models with the basic stand-alone models trained
for each city using different statistical parameters, i.e., KL divergence and
Wessterstein distance. The architecture and hyperparameters used to train
these models are shown in Table I and Table II respectively.

As mentioned earlier, in all cases our generative models consist of two
cascaded models, the first of which is the link predictor and the second is
the path generator. We first train the link predictor for each city separately
and then use these pre-trained link models for simulation.

###  V-B Results

In this work, we propose a promising solution for extending the channel model
to large-scale application scenarios by using a cooperative modeling approach
with multiple distributed channel datasets. We first describe the results
obtained in both centralized and distributed approaches. To ensure a fair
comparison, we train all models with the same number of epochs and
hyperparameters. In particular, we train the stand-alone models for 500 epochs
and for the FL-VAE and FL-GAN models, we perform 100 rounds of local training,
where each city trains its respective model for 5 epochs on its local data
within each FL round.

####  V-B1 Stand-alone Models

Our goal is to measure the extent to which the data generated by the
generative models (VAE and GAN networks) are comparable to the test data. To
this end, we compare the CDF of the path losses of the generated and test
data. Both trained generative models are able to capture the dual-slope nature
of CDF, which is a crucial component for the effectiveness of our proposed
framework. However, due to space constraints, we only show in Table III the
distance between the distribution of the standalone models (VAE and GAN) and
the distributed models, i.e., (FL-VAE) and (FL-GAN).

####  V-B2 FL-VAE

To evaluate the performance of our proposed decentralized model, we created
CDF plots for the path losses of both the test data and the path losses
generated by the FL-VAE model for each city. This allowed us to evaluate the
generalizability of our federated global model, particularly in terms of its
ability to accurately capture the channel characteristics of all participating
cities. The results in Figures 2a, 2b, and 2c show that our federated model
performs better compared to the individual models of each city. In addition,
the FL-VAE approach helps address potential privacy and security issues
related to data sharing between different cities. These measures ensure that
individual city data sets are not shared outside of the city, thus maintaining
privacy and security.

####  V-B3 FL-GAN

Now we use the CGAN instead of the VAE to generate the channel parameters and
compare its performance with the results we obtained with the FL-VAE and
standalone GAN models. Our results show that the generative network learns the
distribution of the channel modelling data very well and generates samples
that exactly reflect the same distribution of the training dataset. It is also
clear from Figures 3a, 3b, and 3c that FL-GAN produces better results for the
path loss component of the channel parameters compared to FL-VAE. The results
show that the channel parameters reconstructed using the FL-GAN approach are
closest to the original test data and outperform the VAE-based methods. This
can be attributed to the fact that it is difficult for VAEs to encode
heterogeneous datasets from different cities into a common latent space, while
GANs are better at learning diverse data. The FL-GAN approach is therefore
better suited to deal with the challenges of heterogeneous data and produce
synthetic data that accurately represents the actual data distribution.

###  V-C Performance Metrics and Evaluation Results

This section presents the evaluation metrics used to assess the performance of
the proposed distributed techniques FL-VAE and FL-GAN in generating synthetic
data compared to the standalone models trained separately for each city. Table
III summarizes the KL divergence and Wasserstein distance results obtained by
comparing the test data distribution with the synthetic data distribution
generated by the VAE, GAN, FL-VAE, and FL-GAN networks. These metrics are used
to measure the distance between the test data distribution and the synthetic
data generated by each model, and provide information about the accuracy and
quality of the generated data. These evaluation metrics used in Table III show
that the distribution of synthetic data generated by the FL-GAN network is
much closer to the true distribution compared to the other methods, i.e., the
standalone networks and FL-VAE. This highlights the superiority of the
proposed approach FL-GAN in accurately modeling the data and generating
synthetic data that is very similar to the real data.

As shown in Table III, the KL-divergence between the test data distribution
and the synthetic data distribution of the standalone GAN -model is much
higher than that of the other alternatives. FL-GAN achieves the lowest KL
-divergence among the alternatives, which is due to the fact that GANs
generally require more training time than VAEs, but can generate better
samples. We also evaluate our method using Wasserstein distance, which
considers metric space. Table III shows that FL-GAN significantly outperforms
all other methods and achieves satisfactory performance in developing a global
model for channel estimation parameters.

Table I: Model summary of the link model, path model (VAE and CGAN)

Model | Number of | Hidden | Number of | Number of
---|---|---|---|---
Inputs | Units | Outputs | Parameters
Link Model | $5$ | $[25,10]$ | $3$ | $1,653$
VAE (Enc) | $125$ | $[200,80]$ | $40$ | $44,520$
VAE (Dec) | $25$ | $[80,200]$ | $240$ | $40,720$
GAN (Disc) | $125$ | $[1120,560,280]$ | $40$ | $1,055,761$
GAN (Gen) | $25$ | $[280,560,1120]$ | $240$ | $1,094,360$

Table II: Hyperparameter settings for link model, Federated VAE and CGAN
models

Item | Link Model | Generative Model (VAE) | Generative Model (GAN)
---|---|---|---
Communication Rounds | $N/A$ | $100$ | $100$
Epochs | $30$ | $5$ | $5$
Batch Size | $100$ | $100$ | $100$
Learning Rate | $10^{-3}$ | $10^{-4}$ | $10^{-4}$
Optimizer | Adam | Adam | Adam

(a) Beijing

(b) Boston

(c) London

Figure 2: Federated Model (FL-VAE)

(a) Beijing

(b) Boston

(c) London

Figure 3: Federated Model (FL-GAN) Table III: Distance between real and
generated data distribution for channel modeling parameters, the bold letters
represents the smaller distance

City | Method | KL-Divergence | Wasserstein Distance
---|---|---|---
Beijing | VAE | $1.91$ | $13.92$
GAN | $3.08$ | $13.09$
FL-VAE | $1.63$ | $13.55$
FL-GAN | 1.51 | 12.47
Boston | VAE | $2.35$ | $12.48$
GAN | $1.66$ | $11.63$
FL-VAE | $2.29$ | $12.05$
FL-GAN | 1.25 | 11.33
London | VAE | $1.70$ | $14.03$
GAN | $3.29$ | $12.86$
FL-VAE | $1.69$ | $13.95$
FL-GAN | 1.25 | 12.50

##  VI Conclusion

NTNs are anticipated to play a crucial role in future wireless networks due to
their cost efficiency and wide coverage area. In this paper, we present a
comprehensive study that employs a generative framework based on NNs to model
wireless channels in a distributed environment. In order to have a common
model for different cities, we train distributed generative models and combine
them into a unified and adaptable model. Specifically, we propose a channel
model for air-to-ground communication of UAVs in mmWave frequency bands. Our
distributed training method does not require any special knowledge or
technical expertise, as it learns directly from massive raw channel data to
develop a generic channel model. The use of generative NNs, especially GANs
and VAEs, is a suitable method for statistical channel modeling in complex
scenarios. Although both models are capable of capturing data dependencies,
our results show that the proposed FL-GAN approach outperforms the FL-VAE and
centralized baseline methods in terms of learning the accuracy of path loss
parameters. We validate our results with various statistical parameters, and
the resulting model shows effective learning and interesting non-obvious
predictions.

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