# A construction of Exceptional Weyl Group $W(F_{4})$ and $W(E_{8})$ using
Quaternion, and the lattice in 16-dimensional Euclidean space

Misaki Ohta,University of the Ryukyus Department of physics, Email:
<EMAIL_ADDRESS>or<EMAIL_ADDRESS>
###### Abstract

It is mentioned that there is a subalgebra isomorphic to the alternating group
$2\cdot A_{4}$ as a subalgebra of the Quaternion over integers and half-
integers called Hurwitz quaternionic integers $\mathscr{H}$ in the book by
J.H.Conway and Neil J. A. Sloane. In this paper, I have followed this book and
extended Quaternion over integers and half-integers to have duality, and
proved that a subalgebra in it isomorphic to Exceptional Weyl group
$W(F_{4})$. I have also found a method of constructing the $16$-dimensional
lattice $\Lambda_{16}$ which seems to be isomorphic to the lattice called the
Barnes-Wall lattice $\Lambda_{\text{Barnes-Wall }}$, which is currently
considered to be very dense (although this remains to be discussed) using the
Dual Quaternion. Lastly, I briefly mention how to construct an exceptional
Weyl group $W(E_{8})$ using an Octonion and Dual Quaternion.

##  1 Introduction

In the history of physics, beautiful symmetries have been found arising from
the algebraic structures they satisfy, such as the Pauli matrices of Quantum
Mechanics and the Gamma matrices of Dirac in relativistic quantum mechanics.
And there is always some inevitability behind the appearance of symmetries. It
is known that in order to understand the structure of symmetry, it is
necessary to consider the structure of the symmetry (i.e., the group) itself
and the object on which it acts in depath, and such studies have actually been
conducted. One of the symmetries that will play a leading role in this paper
is called Quaternions, which is useful in representing the rotation of
coordinates, discovered by Hamilton on the $16$th of October $1843$. It is
frequently used not only in a physical context but also in a mathematical
context and has many applications. The fact that it is widely applied means
that it plays a very profound core role in this world. On the other hand, in
terms of the objects on which the group acts, there is a called Root System,
which will be the main subject of this paper, closely related to Lie algebras
and Lie groups, and essential from the point of view of describing the most
advanced theoretical physics of our time. Root System is simply a set of
vectors generated by reflection, but it is generally known that it can be
defined even in high order coordinates, and over a long period of history, the
classification of the types of reflection was completed. Among the classified
systems, those that deviate from the general classification are called
exceptional Root Systems, such as $G_{2},F_{4}$ and $E_{8}$. In particular,
the $E_{8}$ root system in $8$th dimension is considered to be the largest
among the root systems of exceptional type, and this also appears as a
necessity in the world of theoretical physics, the $E_{8}$ lattice forms an
8-dimensional dense packing as well. In addition, The number of vectors that
are the shortest of a lattice, i.e., the number of the densest packing, tells
us how reasonable and natural the lattice is, because the greatest number of
shortest vectors means that they are most evenly distributed on the sphere of
the space. And for vectors that form a dense packing, it is always possible to
discuss orthogonal transformations that transfer between those vectors, i.e.,
it is a symmetry discussion for that lattice. The orthogonal groups acting on
Root System are called Weyl groups. For example, the Weyl groups of
$G_{2},F_{4}$ and $E_{8}$ are written as $W(G_{2}),W(F_{4})$ and $W(E_{8})$,
respectively. In this paper, I will first give a duality to Quaternion, and
then describe the existence of a subalgebra in the algebraic system
corresponding to the reflection of exceptional type $W(F_{4})$ in
4-dimensional spaces described above. Next, I will briefly study a lattice,
which can be constructed by this subalgebra, with $4320$ shortest vectors
similar (or isomorphic) to the Barnes-Wall lattice, which is said to be the
close-packed of $16$-dimensional Euclidean space through the homomorphism to
the code(vector space) over $\mathbb{F}_{2}$ which is invariant by the action
of the group $2^{4}\cdot A_{8}$, which is the maximal subgroup of Mathieu
group $M_{24}$ called ’Miracle Octad Group’. Lastly, I briefly mention how to
construct an exceptional Weyl group $W(E_{8})$ using an Octonions and Dual
Quaternion.

###  1.1 Quaternion and Hurwitz quaternionic integers

Let $\mathbb{C}$ donate the field of complex numbers
$\\{z=x+iy|x,y\in\mathbb{R}\\}$, where $i^{2}=-1$, and let $\mathbb{H}$ donate
the filed of Quaternions
$\\{z=x_{0}+x_{1}i+x_{2}j+x_{3}k|x_{i}\in\mathbb{R}\\}$ where

$\begin{array}[]{l}i^{2}=j^{2}=k^{2}=ijk=-1\\\ ij=-ji=k,\quad jk=-kj=i,\quad
ki=-ik=j\end{array}$

The conjugate $\bar{z}$ of $z$ is defined to be

$\begin{array}[]{ll}\bar{z}=x-iy&\in\mathbb{C}\\\
\bar{z}=x_{0}-ix_{1}-jx_{2}-kx_{3}&\in\mathbb{H}\end{array}$

and then the norm $N:z\in\mathbb{C},\mathbb{H}\mapsto z\bar{z}\in\mathbb{R}$
of $z$ is

$\begin{array}[]{l}x^{2}+y^{2}\\\
x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\end{array}$

respectively.There are the Gaussian and Eisenstein integers

$\begin{array}[]{l}\mathscr{G}=\\{a+ib:a,b\in\mathbb{Z}\\}\subset\mathbb{C},\\\
\mathscr{E}=\\{a+\omega b:a,b\in\mathbb{Z}\\}\subset\mathbb{C},\end{array}$

Figure 1: Gaussian and Eisenstein integers

where $\omega=(-1+i\sqrt{3})/2$, and the Hurwitz Quaternionic integers

$\mathscr{H}=\\{a+ib+jc+kd:a,b,c,d\in\mathbb{Z}\text{ or
}a,b,c,d\in\mathbb{Z}+1/2\\}\subset\mathbb{H}.$

According to ”Sphere Packings, Lattices and Groups” by John Conway, Neil J. A.
Sloane[1,P53], the set $\mathcal{S}\subset\mathscr{H}\subset\mathbb{H}$
generated by the multiplication of the following generators forms a group

$i,j,k,\frac{1}{2}(-1+i+j+k)\in\mathbb{H}$

and there is isomorphism

$\mathcal{S}\cong 2\cdot A_{4}$

where $A_{4}$ is the alternating group of degree $4$[See Appendix 5.4]. The
fields $\mathscr{G},\mathscr{E}$ and $\mathscr{H}$ can also be interpreted as
the group rings over $\mathbb{Z}$ or $\mathbb{Z}+1/2$ of the following groups
(may divided by the center)

$\begin{array}[]{l}G_{1}=<i>\cong\mathbb{Z}/4\mathbb{Z}\\\
G_{2}=<\omega>\cong\mathbb{Z}/3\mathbb{Z}\\\ G_{3}=\langle i,j\rangle\cong
Q_{8}=\left\langle a,b\mid
a^{4}=e,a^{2}=b^{2},ba=a^{-1}b\right\rangle\end{array}$

In general, an element $x$ of group ring $R[G]$ of $G$ can be written as

$x=\sum_{g_{i}\in G}g_{i}x_{i}\quad(x_{i}\in\mathbb{R})$

if I follow the definition of conjugation above, then the conjugation
$\bar{x}$ of $x$ should be

$\bar{x}=\sum_{g_{i}\in G}g^{-1}_{i}x_{i}\quad\left(x_{i}\in\mathbb{R}\right)$

and if the group is an orthogonal group($G\subset O(n)$), then it is simply
shown that $x$ is Hermitian $\bar{x}={}^{t}x(x=x^{\dagger})$. Because

$\displaystyle\bar{x}$ $\displaystyle=\sum_{g_{i}\in
G}x_{i}g_{i}^{-1}=\sum_{g_{i}\in G}x_{i}^{t}g_{i}$
$\displaystyle={}^{t}\left(\sum_{g_{i}\in G}x_{i}g_{i}\right)={}^{t}x$

and when $N(x)=x\bar{x}=1$, in other words ${}^{t}x=x^{-1}$, it means

$x\in O(n)\quad(x\in R[G]\quad G\subset O(n))$

And on the contrary, if $G$ is an orthogonal group and $x\in R[G]$ is also an
enelent of an orthogonal group, then the norm defined here is determined to be
$1$.

###  1.2 The representation of quaternions

Just as the algebraic structure of Pauli matrices in quantum mechanics

$\displaystyle\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}=1$
$\displaystyle\sigma_{1}\sigma_{2}=-\sigma_{2}\sigma_{1}=i\sigma_{3}$
$\displaystyle\sigma_{2}\sigma_{3}=-\sigma_{3}\sigma_{2}=i\sigma_{1}$
$\displaystyle\sigma_{3}\sigma_{1}=-\sigma_{1}\sigma_{3}=i\sigma_{2}$

allows the matrix to be determined

$\sigma=\left(\begin{array}[]{ll}1&\\\
&1\end{array}\right),\quad\sigma_{1}=\left(\begin{array}[]{ll}&1\\\
1&\end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{cc}&-i\\\
i&\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cc}1&\\\
&-1\end{array}\right)$

or the algebraic structure of Dirac spinor in QFT

$\displaystyle\left(\gamma_{0}\right)^{2}$
$\displaystyle=1,\left(\gamma_{j}\right)^{2}=-1\quad\left(j=1,2,3\right)$
$\displaystyle\left\\{\gamma_{\mu},\gamma_{\nu}\right\\}$
$\displaystyle=\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=0\quad(\mu\neq\nu)$

allows the matrix to be determined

$\begin{array}[]{l}\gamma_{0}=\left(\begin{array}[]{ll}&E\\\
E&\end{array}\right),\quad\gamma_{j}=\left(\begin{array}[]{cc}&\tau_{j}\\\
-\tau_{j}&\end{array}\right)\\\ \text{where
}\quad\tau_{1}=\left(\begin{array}[]{ll}&1\\\
1&\end{array}\right),\quad\tau_{2}=\left(\begin{array}[]{cc}&-i\\\
i&\end{array}\right),\quad\tau_{3}=\left(\begin{array}[]{cc}1&\\\
&-1\end{array}\right)\end{array}$

from the stracture of Quaternions

$\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1$ $\displaystyle ij=-ji=k,\quad
jk=-kj=i,\quad ki=-ik=j$

it allows the matrix to be determined using isomorphism $\rho:<i,j>\rightarrow
O(4,\mathbb{R})$

$\begin{array}[]{l}(i)\rho=\left(\begin{array}[]{cccc}&1&&\\\ -1&&&\\\ &&&1\\\
&&-1&\end{array}\right)\\\ \\\ (j)\rho=\left(\begin{array}[]{cccc}&&1&\\\
&&&-1\\\ -1&&&\\\ &1&&\end{array}\right)\\\ \\\
(k)\rho=\left(\begin{array}[]{cccc}&&&-1\\\ &&-1&\\\ &1&&\\\
1&&&\end{array}\right)\end{array}$

Using this representation, the group
$\mathcal{S}\subset\mathscr{H}\subset\mathbb{H}$ described above yields

$\mathcal{S}\cong\left\langle\left(\begin{array}[]{rrrr}&1&&\\\ -1&&&\\\
&&&1\\\ &&-1&\end{array}\right),\left(\begin{array}[]{rrrr}&&1&\\\ &&&-1\\\
-1&&&\\\
&1&&\end{array}\right)\frac{1}{2}\left(\begin{array}[]{rrrr}-1&1&1&-1\\\
-1&-1&-1&-1\\\ -1&1&-1&1\\\ 1&1&-1&-1\end{array}\right)\right\rangle\cong
2\cdot A_{4}$

###  1.3 Root System

Figure 2: Dynkin diagrams of exceptional root system

####  1.3.1 Root system $F_{4}$

In this paper, I use the concept of root system here.

Let
$\left\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\right\\}$
be the standard basis of $\mathbb{R}^{4}$. I set the notation for $F_{4}$ as
generated by the basis.
$\varepsilon_{2}-\varepsilon_{3},\varepsilon_{3}-\varepsilon_{4},\varepsilon_{4},\left(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4}\right)/2$.
The generator matrix $M(F_{4})$ for $F_{4}$ is

$M({F_{4}})=\left(\begin{array}[]{l}\alpha_{1}\\\ \alpha_{2}\\\ \alpha_{3}\\\
\alpha_{4}\end{array}\right)=\left(\begin{array}[]{cccc}&1&-1&\\\ &&1&-1\\\
&&&1\\\ \frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\end{array}\right)$

and its Gram matrix is

$G=M(F_{4})\cdot{}^{t}M(F_{4})=\left(\begin{array}[]{rrrr}2&-1&&\\\
-1&2&-1&\\\ &-1&1&-\frac{1}{2}\\\ &&-\frac{1}{2}&1\end{array}\right)$

Also its theta function is (See Appendix 5.6 for a description of what a
lattice theta function is.)

$\displaystyle\Theta_{{F_{4}}}(q):=\sum_{x\in\Lambda_{F_{4}}}q^{(x,x)}=1+24q+24q^{2}+96q^{3}+24q^{4}+144q^{5}+\ldots$

however $(x,x)$ to be Euclidean norm and $\Lambda_{F_{4}}$ to be the lattice
generated by the basis of $F_{4}$.

####  1.3.2 Root system $E_{8}$

Let
$\left\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4},\varepsilon_{5},\varepsilon_{6},\varepsilon_{7},\varepsilon_{8}\right\\}$
be the standard basis of $\mathbb{R}^{8}$. I set the notation for $E_{8}$ as
generated by the basis $\alpha_{i}:=\varepsilon_{i-1}-\varepsilon_{i-2}(1\leq
i\leq
6),\alpha_{7}:=\varepsilon_{1}+\varepsilon_{2},\alpha_{8}:=\frac{1}{2}\left(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4}-\varepsilon_{5}-\varepsilon_{6}-\varepsilon_{7}+\varepsilon_{8}\right)$
The generator matrix $M(E_{8})$ for $E_{8}$ is

$M({E_{8}})=\left(\begin{array}[]{l}\alpha_{1}\\\ \alpha_{2}\\\ \alpha_{3}\\\
\alpha_{4}\\\ \alpha_{5}\\\ \alpha_{6}\\\ \alpha_{7}\\\
\alpha_{8}\end{array}\right)=\left(\begin{array}[]{cccccccc}1&-1&&&&&&\\\
&1&-1&&&&&\\\ &&1&-1&&&&\\\ &&&1&-1&&&\\\ &&&&1&-1&&\\\ &&&&&1&-1&\\\
&&&&&1&1&\\\ 1/2&-1/2&-1/2&-1/2&-1/2&-1/2&-1/2&-1/2\end{array}\right)$

and its Gram matrix is

$G(E_{8})=M(E_{8})\cdot{}^{t}M(E_{8})=\left(\begin{array}[]{rrrrrrrr}2&-1&&&&&&\\\
-1&2&-1&&&&&\\\ &-1&2&-1&&&&\\\ &&-1&2&-1&&&\\\ &&&-1&2&-1&-1&\\\
&&&&-1&2&&\\\ &&&&-1&&2&-1\\\ &&&&&&-1&2\end{array}\right)$

Also its theta function is

$\displaystyle\Theta_{E_{8}}(z)$
$\displaystyle=\sum_{x\in\Lambda_{E_{8}}}q^{(x,x)}$
$\displaystyle=\frac{1}{2}\left(\theta_{2}(q)^{8}+\theta_{3}(q)^{8}+\theta_{4}(q)^{8}\right)$
$\displaystyle=\theta_{2}(q^{2})^{8}+14\theta_{2}(q^{2})^{4}\theta_{3}(q^{2})^{4}+\theta_{3}(q^{2})^{8}$
$\displaystyle=1+240q^{2}+2160q^{4}+6720q^{6}+17520q^{8}+\cdots$

The theta function of $E_{8}$ is well known as Eisenstein series
$E_{4}(z)=1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n}$ in the theory of modular
form.[1, P122] However,

$\displaystyle\sigma_{k-1}(n)$ $\displaystyle=\sum_{d\mid n}d^{k-1}$
$\displaystyle\theta_{2}(q)$
$\displaystyle=\sum_{n=-\infty}^{\infty}q^{(m+1/2)^{2}}$
$\displaystyle\theta_{3}(q)$
$\displaystyle=\sum_{n=-\infty}^{\infty}q^{m^{2}}$
$\displaystyle\theta_{4}(q)$
$\displaystyle=\sum_{n=-\infty}^{\infty}(-q)^{m^{2}}$

Weyl group $W(F_{4})$ and $W(E_{8})$ are generated by reflections through the
hyperplanes orthogonal to the generators(roots) of $F_{4}$ and $E_{8}$
respectively.In addition the lattices $F_{4},E_{8}$ can be constructed using
Quaternions.[See Appendix 5.4 and 5.5]

##  2 Dual Quaternion and Weyl Group $W\left(F_{4}\right)$

For Hurwitz quaternionic integers $\mathscr{H}$ described above, I studied its
direct product $\mathscr{H}^{2}$, in other words the group ring
$\mathbb{Z}^{\prime}[\mathbb{Q}^{2}_{8}]$ over
$\mathbb{Z}^{\prime}=\mathbb{Z}\oplus(\mathbb{Z}+1/2)\oplus(\mathbb{Z}+1/4)$
of

$\mathbb{Q}^{2}_{8}:=<i,j,p,q|ip=pi,iq=qi,jp=pj>$

where $i,j$ and $p,q$ satisfy

$\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1$ $\displaystyle ij=-ji=k,\quad
jk=-kj=i,\quad ki=-ik=j$ $\displaystyle p^{2}=q^{2}=r^{2}=pqr=-1$
$\displaystyle pq=-qp=r,\quad qr=-rq=p,\quad rp=-pr=q$

$<a,b>$ denotes here the group generated by the multiplication of $a$ and $b$.

There is the permutation representation of $\mathbb{Q}^{2}_{8}$ generated by
following.(See Appendix 5.1.)

$\displaystyle(1,3,15,13)(2,11,16,5)(4,7,14,9)(6,10,12,8)$
$\displaystyle(1,12,15,6)(2,9,16,7)(3,10,13,8)(4,11,14,5)$
$\displaystyle(1,8,15,10)(2,5,16,11)(3,6,13,12)(4,7,14,9)$
$\displaystyle(1,13,15,3),(2,9,16,7),(4,5,14,11),(6,10,12,8)$

or

$\displaystyle(1,2,7,8)(3,4,5,6)$ $\displaystyle(1,5,7,3)(2,4,8,6)$
$\displaystyle(1,6,7,4)(2,3,8,5)$ $\displaystyle(1,8,7,2)(3,4,5,6)$

Figure 3: The permutations isomorphic to the generators of
$\mathbb{Q}^{2}_{8}$

For all elements $x$ in $\mathbb{Z}^{\prime}[\mathbb{Q}^{2}_{8}]$ can be
written as

$x=\left(1,p,q,r\right)\left(\begin{array}[]{cccc}a_{00}&a_{01}&a_{02}&a_{03}\\\
a_{10}&a_{11}&a_{12}&a_{13}\\\ a_{20}&a_{21}&a_{22}&a_{23}\\\
a_{30}&a_{31}&a_{32}&a_{33}\end{array}\right)\left(\begin{array}[]{l}1\\\ i\\\
j\\\ k\end{array}\right)\in\mathscr{H}^{2}$

where

$a_{ij}\in\mathbb{Z}^{\prime}=\mathbb{Z}\oplus(\mathbb{Z}+1/2)\oplus(\mathbb{Z}+1/4)$

and (donate $\mathbb{Q}=<i,j>,\mathbb{Q}^{\prime}=<p,q>$, i.e.
$\mathbb{Q}^{2}_{8}\cong\mathbb{Q}\times\mathbb{Q}^{\prime}$)

$x\cdot x^{\prime}=x^{\prime}\cdot
x\quad({}^{\forall}x\in\mathbb{Q},{}^{\forall}x^{\prime}\in\mathbb{Q}^{\prime})$

The conjugation of $x\in\mathbb{Z}^{\prime}\left[\mathbb{Q}^{2}_{8}\right]$
follows the general one for group rings described above

$\bar{x}=\left(1,p,q,r\right)\left(\begin{array}[]{cccc}a_{00}&-a_{01}&-a_{02}&-a_{03}\\\
-a_{10}&a_{11}&a_{12}&a_{13}\\\ -a_{20}&a_{21}&a_{22}&a_{23}\\\
-a_{30}&a_{31}&a_{32}&a_{33}\end{array}\right)\left(\begin{array}[]{l}1\\\
i\\\ j\\\ k\end{array}\right)\in\mathscr{H}^{2}$

Unlike the Quaternion, the norm $N(x)=x\bar{x}$ now dealing with is not always
real number, but only in some special cases. One of them is Conway and Slone’s
$\mathcal{S}\cong 2\cdot A_{4}$, but it also turns out that there is another
finite group order $1152=2^{7}\cdot 3^{2}$. Lemma.1
$\rho:\mathbb{Q}\rightarrow SO(4,\mathbb{R})$ is isomorphism.

$\rho:i\quad\longmapsto\left(\begin{array}[]{cccc}&1&&\\\ -1&&&\\\ &&&1\\\
&&-1&\end{array}\right)$
$\rho:j\quad\longmapsto\left(\begin{array}[]{cccc}&&1&\\\ &&&-1\\\ -1&&&\\\
&1&&\end{array}\right)$
$\rho:k\quad\longmapsto\left(\begin{array}[]{cccc}&&&-1\\\ &&-1&\\\ &1&&\\\
1&&&\end{array}\right)$

(Proof) Since $\mathbb{Q}\cong\mathrm{Q}_{8}=\left\langle a,b\mid
a^{4}=e,a^{2}=b^{2},ba=a^{-1}b\right\rangle$, To show that they are
isomorphic, it is enough to show that the generators $i,j$ satisfy
$i^{4}=E,\quad i^{2}=j^{\prime 2},ji=i^{-1}j=-ij$. Calculating

$(i)\rho(j)\rho=\left(\begin{array}[]{llll}&1&&\\\ -1&&&\\\ &&&1\\\
&&-1&\end{array}\right)\left(\begin{array}[]{cccc}&&1&\\\ &&&-1\\\ -1&&&\\\
&1&&\end{array}\right)=\left(\begin{array}[]{cccc}&&&-1\\\ &&-1&\\\ &1&&\\\
1&&&\end{array}\right)=(k)\rho$

$((i)\rho)^{2}=((j)\rho)^{2}=((k)\rho)^{2}=\left(\begin{array}[]{rrrr}-1&&&\\\
&-1&&\\\ &&-1&\\\ &&&-1\end{array}\right)$

$(j)\rho(i)\rho=\left(\begin{array}[]{rrrr}&&&1\\\ &&1&\\\ &-1&&\\\
-1&&&\end{array}\right)=-(i)\rho(j)\rho=-(k)\rho$
$((i)\rho)^{-1}(j)\rho=\left(\begin{array}[]{rrrr}&-1&&\\\ 1&&&\\\ &&&-1\\\
&&1&\end{array}\right)\left(\begin{array}[]{rrrr}&&1&\\\ &&&-1\\\ -1&&&\\\
&1&&\end{array}\right)=\left(\begin{array}[]{rrrr}&&&1\\\ &&1&\\\ &-1&&\\\
-1&&&\end{array}\right)$

This result shows that the above is true for the generators.(Q.E.D) Theorem.1
The multiplicative group $G$ order $1152=2^{7}\cdot 3^{2}$ generated by

$\displaystyle e_{1}=\frac{1}{2}(1+iq+jr-kp)$ $\displaystyle
e_{2}=\frac{1}{2}(1-ip-jq-kr)$ $\displaystyle e_{3}=\frac{1}{2}(1-ip-jr+kq)$
$\displaystyle e_{4}=\frac{1}{2}(1-iq-jp+kr)$

is isomorphic to the exceptional Weyl Group$\ W\left(F_{4}\right)$.

$G\cong W\left(F_{4}\right)$

(Proof)
According to ”ATLAS”, the group$W(F_{4})$ is defined by

$W\left(F_{4}\right)=<a,b,c,d\mid
a^{2}=b^{2}=c^{2}=d^{2}=(ab)^{3}=[a,c]=[a,d]=(bc)^{4}=[b,d]=(cd)^{3}=1>$ (1)

to show that the group $G$ is isomorphic to this group, I only need to check
that the generating system of $G$ satisfies the same properties. Using a
linear representation for simplicity.
$\rho:i,j,k\in\mathbb{Q}\mapsto\boldsymbol{i},\boldsymbol{j},\boldsymbol{k}\in
SO(4,\mathbb{R})$ can be written as

$\displaystyle\boldsymbol{i}:=(i)\rho=\left(\begin{array}[]{cccc}&1&&\\\
-1&&&\\\ &&&1\\\ &&-1&\end{array}\right)$
$\displaystyle\boldsymbol{j}:=(j)\rho=\left(\begin{array}[]{cccc}&&1&\\\
&&&-1\\\ -1&&&\\\ &1&&\end{array}\right)$
$\displaystyle\boldsymbol{k}:=(k)\rho=\left(\begin{array}[]{cccc}&&&-1\\\
&&-1&\\\ &1&&\\\ 1&&&\end{array}\right)$

Now determine $p,q$, and $r$ to be commutative with these. Let $(p)\rho$ as

$\boldsymbol{p}=(p)\rho=\left(\begin{array}[]{llll}a_{11}&a_{12}&a_{13}&a_{14}\\\
a_{21}&a_{22}&a_{23}&a_{24}\\\ a_{31}&a_{32}&a_{33}&a_{34}\\\
a_{41}&a_{42}&a_{43}&a_{44}\end{array}\right)$

Since
${}^{\forall}x\in\mathbb{Q},{}^{\forall}x^{\prime}\in\mathbb{Q}^{\prime}$must
be $x\cdot x^{\prime}=x^{\prime}\cdot x$, $\boldsymbol{p},\boldsymbol{i}$ and
$\boldsymbol{j}$must satisfy
$\boldsymbol{p}\cdot\boldsymbol{i}=\boldsymbol{i}\cdot\boldsymbol{p}$ and
$\boldsymbol{p}\cdot\boldsymbol{j}=\boldsymbol{i}\cdot\boldsymbol{p}$. Thus

$\begin{array}[]{c}\boldsymbol{i}^{-1}\cdot\boldsymbol{p}\cdot\boldsymbol{i}=\left(\begin{array}[]{cccc}a_{22}&-a_{21}&a_{24}&-a_{23}\\\
-a_{12}&a_{11}&-a_{14}&a_{13}\\\ a_{42}&-a_{41}&a_{44}&-a_{43}\\\
-a_{32}&a_{31}&-a_{34}&a_{33}\end{array}\right)=\boldsymbol{p}\\\ \\\
\boldsymbol{j}^{-1}\cdot\boldsymbol{p}\cdot\boldsymbol{j}=\left(\begin{array}[]{cccc}a_{33}&-a_{34}&-a_{31}&a_{32}\\\
-a_{43}&a_{44}&a_{41}&-a_{42}\\\ -a_{13}&a_{14}&a_{11}&-a_{12}\\\
a_{23}&-a_{24}&-a_{21}&a_{22}\end{array}\right)=\boldsymbol{p}\end{array}$

Using appropriate $x,y,z,w$, $\boldsymbol{p}$ can be rewiriten as

$\boldsymbol{p}=\left(\begin{array}[]{cccc}x&y&z&-w\\\ -y&x&w&z\\\
-z&-w&x&-y\\\ w&-z&y&x\end{array}\right)$

Also for $\boldsymbol{p}^{2}=-E$.

$\begin{array}[]{l}\left(\begin{array}[]{cccc}x&y&-z&w\\\ -y&x&w&-z\\\
z&-w&x&-y\\\ w&z&y&x\end{array}\right)\left(\begin{array}[]{cccc}x&y&-z&w\\\
-y&x&w&-z\\\ z&-w&x&-y\\\ w&z&y&x\end{array}\right)=\\\
\left(\begin{array}[]{cccc}x^{2}+w^{2}-\left(y^{2}+z^{2}\right),&2(xy+zw),&2(-xz+yw),&2(xw-
yz)\\\ 2(-xy+wz)&x^{2}-y^{2}-z^{2}-w^{2},&2(xw)&-2(yw+xz)\\\
2xz&-2xw&x^{2}-y^{2}-z^{2}-w^{2}&2(wz-xy)\\\
2xw&2(xz)&2xy.&x^{2}-y^{2}-z^{2}+w^{2}\end{array}\right)=-E\end{array}$

where $E$ is the unit matrix of $4\times 4$.To satisfy the above,
$x=w=0$.Rewriting

$\boldsymbol{p}^{2}=\left(\begin{array}[]{cccc}-\left(y^{2}+z^{2}\right)&&&2(-yz)\\\
&-\left(y^{2}+z^{2}\right)&&\\\ &&-\left(y^{2}+z^{2}\right)&\\\
&&&-\left(y^{2}+z^{2}\right)\end{array}\right)$

This shows that either $y,z$ is one of $0,\pm 1$ respectively.From this we
obtain $\boldsymbol{p},\boldsymbol{q},\boldsymbol{r}$ as follows.

$\begin{array}[]{l}\boldsymbol{p}=(p)\rho=\left(\begin{array}[]{cccc}&1&&\\\
-1&&&\\\ &&&-1\\\ &&1&\end{array}\right)\\\ \\\
\boldsymbol{q}=(q)\rho=\left(\begin{array}[]{cccc}&&1&\\\ &&&1\\\ -1&&&\\\
&-1&&\end{array}\right)\\\ \\\
\boldsymbol{r}=(r)\rho=\left(\begin{array}[]{cccc}&&&1\\\ &&-1&\\\ &1&&\\\
-1&&&\end{array}\right)\end{array}$ (2)

Now we can write the generators as a matrix by $\rho$,

$\begin{array}[]{l}\boldsymbol{e}_{1}=(e_{1})\rho=\frac{1}{2}\left(\begin{array}[]{cccc}1&1&1&1\\\
1&1&-1&-1\\\ 1&-1&1&-1\\\ 1&-1&-1&1\end{array}\right)\\\ \\\
\boldsymbol{e}_{2}=(e_{2})\rho=\left(\begin{array}[]{llll}1&&&\\\ &1&&\\\
&&1&\\\ &&&-1\end{array}\right)\\\ \\\
\boldsymbol{e}_{3}=(e_{3})\rho=\left(\begin{array}[]{llll}1&&&\\\ &1&&\\\
&&&1\\\ &&1&\end{array}\right)\\\ \\\
\boldsymbol{e}_{4}=(e_{4})\rho=\left(\begin{array}[]{llll}1&&&\\\ &&1&\\\
&1&&\\\ &&&1\end{array}\right)\end{array}$ (3)

It can be shown by a simple calculation that these
$\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3},\boldsymbol{e}_{4}$
satisfy (1).Actually

$\boldsymbol{e}_{1}^{2}=\boldsymbol{e}_{2}^{2}=\boldsymbol{e}_{3}^{2}=\boldsymbol{e}_{4}^{2}=\left(\boldsymbol{e}_{1}\boldsymbol{e}_{2}\right)^{2}=\left[\boldsymbol{e}_{1},\boldsymbol{e}_{3}\right]=\left[\boldsymbol{e}_{2},\boldsymbol{e}_{4}\right]=\left[\boldsymbol{e}_{1},\boldsymbol{e}_{4}\right]=\left(\boldsymbol{e}_{2}\boldsymbol{e}_{3}\right)^{4}=\left(\boldsymbol{e}_{3}\boldsymbol{e}_{4}\right)^{3}=1$
(4)

However, the ecommutator is defined to be
$\left[\boldsymbol{e}_{i},\boldsymbol{e}_{j}\right]=\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}\cdot\boldsymbol{e}_{i}^{-1}\cdot\boldsymbol{e}_{j}^{-1}$.
(Q.E.D.)

##  3 The $16$ dimensional lattice $\Lambda_{\mathbb{Q}^{2}_{8}}$ and Barns-
Wall lattice

According to J.H.Conway and Neil J. A. Sloane[1], the set of norm $1$ of
quaternions generated by the multiplication with follwings forms the group
isomorphic to $2\cdot A_{4}$ order $24$.

$<i,j,k,\frac{1+i+j+k}{2}>$

And by the inverse mapping $\phi$ which is the bijection

$\begin{array}[]{llllll}\phi:&(\mathbb{Z}\oplus\mathbb{Z}+1/2)^{4}&\rightarrow&\left\langle
i,j,k,\frac{1+i+j+k}{2}\right\rangle&\\\
&(c_{0},c_{1},c_{2},c_{3})&\mapsto&c_{0}+c_{1}i+c_{2}j+c_{3}k&\end{array}$

we obtain $F_{4}$ lattice[See Appendix $5.4$] whose theta series is

$\Theta_{F_{4}}(q)=2E_{2}(q^{2})-E_{2}(q)=1+24q+24q^{2}+96q^{3}+24q^{4}+144q^{5}+\ldots$

where $E_{2}$ is Eisenstein series

$E_{2}(q)=1-24\sum_{n}\sigma_{1}(n)q^{n}$

The group $G\cong W(F_{4})$ is subring of the group ring
$\mathbb{Z}^{\prime}\left[\mathbb{Q}^{2}_{8}\right]$ of norm 1. So let donate
$\varphi$ the bijection
$\varphi:(\mathbb{Z}\oplus\mathbb{Z}+1/2\oplus\mathbb{Z}+1/4)^{16}{\longrightarrow}\mathbb{Z}^{\prime}\left[\mathbb{Q}^{2}_{8}\right]$
and donate $N$ norm $N:x\rightarrow x\bar{x}$, then $G$ can be also regarded
as the subset of $\text{Ker}(N)$.

In terms of the composition of the following maps,

$\begin{array}[]{llllllll}&(\mathbb{Z}\oplus\mathbb{Z}+1/2\oplus\mathbb{Z}+1/4)^{16}&\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}&\quad\quad\quad\mathbb{Z}^{\prime}\left[\mathbb{Q}^{2}_{8}\right]&\stackrel{{\scriptstyle
N}}{{\rightarrow}}&\quad\quad\mathbb{Z}^{\prime}\left[\mathbb{Q}^{2}_{8}\right]&\\\
&\quad\quad\quad\rotatebox{90.0}{$\in$}&&\quad\quad\quad\quad\quad\rotatebox{90.0}{$\in$}&&\quad\quad\quad\quad\rotatebox{90.0}{$\in$}&\\\
&\left(x_{1},x_{2},\ldots
x_{16}\right)&\longmapsto&a_{00}+a_{01}i+\cdots+a_{33}kr&\mapsto&a_{00}^{\prime}+a_{01}^{\prime}i\cdots
a_{33}^{\prime}kr&\end{array}$

Proposition.1 The freemodule generated by the following elements of $G$ is the
lattice of rank $16$

$\displaystyle\frac{1}{2}(1+i+j+k)$ $\displaystyle\frac{1}{2}(1+i+j-k)$
$\displaystyle\frac{1}{2}(1+i)(1+q)$ $\displaystyle\frac{1}{2}(1+i)(1+r)$
$\displaystyle\frac{1}{2}(1+j)(1+q)$ $\displaystyle\frac{1}{2}(1+j)(1+r)$
$\displaystyle\frac{1}{2}(1+iq+jr-pk)$ $\displaystyle\frac{1}{2}(1-ir+jq-kp)$
$\displaystyle\frac{1}{2}(1-p+q-r)$ $\displaystyle\frac{1}{2}(1-p-q+r)$
$\displaystyle\frac{1}{2}(1-k)(1+q)$ $\displaystyle\frac{1}{2}(1-k)(1+r)$
$\displaystyle\frac{1}{2}(1+ir-jq-kp)$ $\displaystyle\frac{1}{2}(1+j)(1-r)$
$\displaystyle\frac{1}{2}(1+j)(1-p)$
$\displaystyle\frac{1}{4}(1+i+j+k+p+ip+\cdots+kr)$

(Proof)
The above elements of the $G$ can be written by bijection
$\varphi^{-1}:G\rightarrow(\mathbb{Z}\oplus\mathbb{Z}+1/2\oplus\mathbb{Z}+1/4)^{16}$
as

$\varphi^{-1}:(1,p,q,r)\left(\begin{array}[]{llll}a_{00}&a_{01}&a_{02}&a_{03}\\\
a_{10}&a_{11}&a_{12}&a_{13}\\\ a_{20}&a_{21}&a_{22}&a_{23}\\\
a_{30}&a_{31}&a_{32}&a_{33}\end{array}\right)\left(\begin{array}[]{l}1\\\ i\\\
j\\\
k\end{array}\right)\mapsto(a_{00},a_{01},a_{02},a_{03},a_{10},a_{11},a_{12},a_{13},a_{20},...,a_{33})$

using the map, all the generator of the module are shown as

$\left(\begin{aligned} &\frac{1}{2}(1+i+j+k)\\\ &\frac{1}{2}(1+i+j-k)\\\
&\frac{1}{2}(1+i)(1+q)\\\ &\frac{1}{2}(1+i)(1+r)\\\ &\frac{1}{2}(1+j)(1+q)\\\
&\frac{1}{2}(1+j)(1+r)\\\ &\frac{1}{2}(1+iq+jr-pk)\\\ &\frac{1}{2}(1-ir+jq-
kp)\\\ &\frac{1}{2}(1-p+q-r)\\\ &\frac{1}{2}(1-p-q+r)\\\
&\frac{1}{2}(1-k)(1+q)\\\ &\frac{1}{2}(1-k)(1+r)\\\ &\frac{1}{2}(1+ir-jq-
kp)\\\ &\frac{1}{2}(1+j)(1-r)\\\ &\frac{1}{2}(1+j)(1-p)\\\
&\frac{1}{4}(1+i+\cdots+kr)\end{aligned}\right)\mapsto
N=\frac{1}{4}\left(\begin{array}[]{rrrrrrrrrrrrrrrr}2&2&2&2&0&0&0&0&0&0&0&0&0&0&0&0\\\
2&2&2&-2&0&0&0&0&0&0&0&0&0&0&0&0\\\ 2&2&0&0&0&0&0&0&2&2&0&0&0&0&0&0\\\
2&2&0&0&0&0&0&0&0&0&0&0&2&2&0&0\\\ 2&0&2&0&0&0&0&0&2&0&2&0&0&0&0&0\\\
2&0&2&0&0&0&0&0&0&0&0&0&2&0&2&0\\\ 2&0&0&0&0&0&0&-2&0&2&0&0&0&0&2&0\\\
2&0&0&0&0&0&0&-2&0&0&2&0&0&-2&0&0\\\ 2&0&0&0&-2&0&0&0&2&0&0&0&-2&0&0&0\\\
2&0&0&0&-2&0&0&0&-2&0&0&0&2&0&0&0\\\ 2&0&0&-2&0&0&0&0&2&0&0&-2&0&0&0&0\\\
2&0&0&-2&0&0&0&0&0&0&0&0&2&0&0&-2\\\ 2&0&0&0&0&0&0&-2&0&0&-2&0&0&2&0&0\\\
2&0&2&0&0&0&0&0&0&0&0&0&-2&0&-2&0\\\ 2&0&2&0&-2&0&-2&0&0&0&0&0&0&0&0&0\\\
1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\end{array}\right)$

to show the rank of the freemodule, I can confirm it by using line elementary
row transformation of $N$. Actually $N$ can be transformed as

$M=\frac{1}{4}\left(\begin{array}[]{llllllllllllllll}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\
&2&&&&&&2&&&&2&&2&&\\\ &&2&&&&&2&&&&2&&&2&\\\ &&&2&&&&2&&&&2&&&&2\\\
&&&&2&&&2&&&&&&2&2&\\\ &&&&&2&&2&&&&&&2&&2\\\ &&&&&&2&2&&&&&&&2&2\\\
&&&&&&&4&&&&&&&&\\\ &&&&&&&&2&&&2&&2&2&\\\ &&&&&&&&&2&&2&&2&&2\\\
&&&&&&&&&&2&2&&&2&2\\\ &&&&&&&&&&&4&&&&\\\ &&&&&&&&&&&&2&2&2&2\\\
&&&&&&&&&&&&&4&&\\\ &&&&&&&&&&&&&&4&\\\ &&&&&&&&&&&&&&&4\end{array}\right)$

Therefore the rank of the freemodule is $16$.(Q.E.D.)

Proposition.2 Let $\Lambda_{\mathbb{Q}^{2}_{8}}$ be the lattice generated by
the above generator.Thus

$\Lambda_{\mathbb{Q}^{2}_{8}}=\text{Span}_{\mathbb{Z}}\frac{1}{4}\left(\begin{array}[]{llllllllllllllll}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\
&2&&&&&&2&&&&2&&2&&\\\ &&2&&&&&2&&&&2&&&2&\\\ &&&2&&&&2&&&&2&&&&2\\\
&&&&2&&&2&&&&&&2&2&\\\ &&&&&2&&2&&&&&&2&&2\\\ &&&&&&2&2&&&&&&&2&2\\\
&&&&&&&4&&&&&&&&\\\ &&&&&&&&2&&&2&&2&2&\\\ &&&&&&&&&2&&2&&2&&2\\\
&&&&&&&&&&2&2&&&2&2\\\ &&&&&&&&&&&4&&&&\\\ &&&&&&&&&&&&2&2&2&2\\\
&&&&&&&&&&&&&4&&\\\ &&&&&&&&&&&&&&4&\\\ &&&&&&&&&&&&&&&4\end{array}\right)$

then it satisfies

$\Lambda_{16}\subset\Lambda_{\text{Barns-Wall}}$

where $\Lambda_{\text{Barns-Wall}}$ generated below is called the Barnes-Wall
lattice.[1.P131]

$\Lambda_{\text{Barns-
Wall}}=\text{Span}_{\mathbb{Z}}\frac{1}{4}\left(\begin{array}[]{llllllllllllllll}1&&&&&1&&1&&&1&1&&1&1&1\\\
&1&&&&1&1&1&1&&1&&1&1&&\\\ &&1&&&&1&1&1&1&&1&&1&1&\\\
&&&1&&1&&&1&1&&1&1&1&&1\\\ &&&&1&&1&&&1&1&&1&1&1&1\\\ &&&&&2&&&&&&&&&&2\\\
&&&&&&2&&&&&&&&&2\\\ &&&&&&&2&&&&&&&&2\\\ &&&&&&&&2&&&&&&&2\\\
&&&&&&&&&2&&&&&&2\\\ &&&&&&&&&&2&&&&&2\\\ &&&&&&&&&&&2&&&&2\\\
&&&&&&&&&&&&2&&&2\\\ &&&&&&&&&&&&&2&&2\\\ &&&&&&&&&&&&&&2&2\\\
&&&&&&&&&&&&&&&4\end{array}\right)$

(Proof)
For all elements $x\in\Lambda_{\mathbb{Q}^{2}_{8}}$ can be written by the
following linear combination.

$x=(c_{1},c_{2},...,c_{16})\cdot M$

where $(c_{1},c_{2},...,c_{16})\in\mathbb{Z}^{16}$ and $M$ is the generator
matrix of $\Lambda_{\mathbb{Q}^{2}_{8}}$. Let $M^{\prime}$ be the generator
matrix of $\Lambda_{\text{Barns-Wall}}$ and if
$\Lambda_{16}\subset\Lambda_{\text{Barns-Wall}}$ then for all
$c\in\mathbb{Z}^{16}$, there exists $(a_{1},a_{2},...,a_{16})$ such that

$(c_{1},c_{2},...,c_{16})M=(a_{1},a_{2},...,a_{16})M^{\prime}$

in other words, the following must always be met if
$\Lambda_{\mathbb{Q}^{2}_{8}}\subset\Lambda_{\text{Barns-Wall}}$

$(c_{1},c_{2},...,c_{16})\cdot M\cdot(M^{\prime})^{-1}\in\mathbb{Z}^{16}$ (5)

By actually performing the calculation of $M\cdot(M^{\prime})^{-1}$

$M\cdot(M^{\prime})^{-1}=\left(\begin{array}[]{rrrrrrrrrrrrrrrr}1&1&1&1&1&-1&-1&-1&-1&-1&-1&-1&-1&-2&-1&5\\\
&2&&&&-1&-1&&-1&&-1&1&-1&&&2\\\ &&2&&&&-1&&-1&-1&&&&-1&&2\\\
&&&2&&-1&&1&-1&-1&&&-1&-1&&2\\\ &&&&2&&-1&1&&-1&-1&&-1&&&1\\\
&&&&&1&&1&&&&&&1&&-1\\\ &&&&&&1&1&&&&&&&1&-1\\\ &&&&&&&2&&&&&&&&-1\\\
&&&&&&&&1&&&1&&1&1&-2\\\ &&&&&&&&&1&&1&&1&&-1\\\ &&&&&&&&&&1&1&&&1&-1\\\
&&&&&&&&&&&2&&&&-1\\\ &&&&&&&&&&&&1&1&1&-1\\\ &&&&&&&&&&&&&2&&-1\\\
&&&&&&&&&&&&&&2&-1\\\ &&&&&&&&&&&&&&&1\end{array}\right)$

Obviously, equation (5) is satisfied since
$(c_{1},c_{2},...,c_{16})\in\mathbb{Z}^{16}$.Therefore
$\Lambda_{\mathbb{Q}^{2}_{8}}\subset\Lambda_{\text{Barns-Wall}}$
(Q.E.D)

By the above, the following is shown at the same time.

$\Lambda_{\text{Barns-Wall}}\not\subset\Lambda_{\mathbb{Q}^{2}_{8}}$

According to Conway and Slone,As the theta series shows the lattice
$\Lambda_{\text{Barns-Wall}}$ is known to be a 16-dimensional dense
packing(however this is considered controversial.) and its kissing number is
4320.

$\displaystyle\Theta_{\Lambda_{\text{Barnes-Wall}}}(z)$
$\displaystyle=1/2\left\\{\theta_{2}(q^{2})^{16}+\theta_{3}(q^{2})^{16}+\theta_{4}(q^{2})^{16}+30\theta_{2}(q^{2})^{8}\theta_{3}(q^{2})^{8}\right\\}$
$\displaystyle=1+4320q^{4}+61440q^{6}+\cdots.$

It is also mentioned that $\Lambda_{\text{Barns-Wall}}$ may constructed from
the Leech lattice $\Lambda_{24}$ which is invariant under the action of the
Conwy group.

The Gram matrix of $M$ which is the generator matrix of
$\Lambda_{\mathbb{Q}^{2}_{8}}$ is following by the way

$G:=M\cdot{}^{t}M=\frac{1}{4}\left(\begin{array}[]{llllllllllllllll}4&2&2&2&2&2&2&1&2&2&2&1&2&1&1&1\\\
2&4&2&2&2&2&1&2&2&2&1&2&1&2&&\\\ 2&2&4&2&2&1&2&2&2&1&2&2&1&&2&\\\
2&2&2&4&1&2&2&2&1&2&2&2&1&&&2\\\ 2&2&2&1&4&2&2&2&2&1&1&&2&2&2&\\\
2&2&1&2&2&4&2&2&1&2&1&&2&2&&2\\\ 2&1&2&2&2&2&4&2&1&1&2&&2&&2&2\\\
1&2&2&2&2&2&2&4&&&&&&&&\\\ 2&2&2&1&2&1&1&&4&2&2&2&2&2&2&\\\
2&2&1&2&1&2&1&&2&4&2&2&2&2&&2\\\ 2&1&2&2&1&1&2&&2&2&4&2&2&&2&2\\\
1&2&2&2&&&&&2&2&2&4&&&&\\\ 2&1&1&1&2&2&2&&2&2&2&&4&2&2&2\\\
1&2&&&2&2&&&2&2&&&2&4&&\\\ 1&&2&&2&&2&&2&&2&&2&&4&\\\
1&&&2&&2&2&&&2&2&&2&&&4\end{array}\right)$ (6)

For the density of the Barns-wall lattice and the lattice
$\Lambda_{\mathbb{Q}^{2}_{8}}$ is as follows.

$\frac{\det(M)}{\det(M^{\prime})}=256=(\sqrt{2})^{16}$ (7)

Proposition.3

For any element of
$\Lambda^{\prime}_{\mathbb{Q}^{2}_{8}}=\\{x\in\Lambda_{\mathbb{Q}^{2}_{8}}|4\cdot
x\\}$, the components are either all even or all odd. (Proof)
The generator matrix $M_{4}$ of $\Lambda^{\prime}_{\mathbb{Q}^{2}_{8}}$ is
defined to be

$M_{4}=\left(\begin{array}[]{c}e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\\\ \vdots\\\
e_{16}\end{array}\right)=\left(\begin{array}[]{llllllllllllllll}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\
&2&&&&&&2&&&&2&&2&&\\\ &&2&&&&&2&&&&2&&&2&\\\ &&&2&&&&2&&&&2&&&&2\\\
&&&&2&&&2&&&&&&2&2&\\\ &&&&&2&&2&&&&&&2&&2\\\ &&&&&&2&2&&&&&&&2&2\\\
&&&&&&&4&&&&&&&&\\\ &&&&&&&&2&&&2&&2&2&\\\ &&&&&&&&&2&&2&&2&&2\\\
&&&&&&&&&&2&2&&&2&2\\\ &&&&&&&&&&&4&&&&\\\ &&&&&&&&&&&&2&2&2&2\\\
&&&&&&&&&&&&&4&&\\\ &&&&&&&&&&&&&&4&\\\ &&&&&&&&&&&&&&&4\end{array}\right)$

We can find that the generators of the lattice $e_{1}\in(2\mathbb{Z}+1)^{16}$
and others $e_{i}\in(2\mathbb{Z})^{16}\ (i=1,2,...,16)$.Therefore, all of the
elements generated by their addition are also even or odd.
(Q.E.D)

From now, $\Lambda^{\prime}_{\mathbb{Q}^{2}_{8}}$ will be simply written as
$\Lambda$. Proposition.4 The following homomorphism exist.

$f:\Lambda\rightarrow 2\Lambda$

where

$2\Lambda:=\left\\{x\in\Lambda\mid\sum_{i=1}^{16}c_{i}e_{i},\quad c_{1}\in
2\mathbb{Z},\quad c_{i}\in\mathbb{Z}\quad(i=2,3,\ldots,16)\right\\}$

(Proof)
All the elements $x\in\Lambda$ can be written as the linear combination of the
basis $\sum_{i=1}^{16}c_{i}x_{i}\quad(c_{i}\in\mathbb{Z})$, and whether the
$x$ is even or odd is determined by the evenness of $c_{1}$. By defining the
homomorphism as follows so the map to be $f:\Lambda\rightarrow 2\Lambda$.

$\begin{array}[]{lllll}f:&\mathbb{Z}&\rightarrow&2\mathbb{Z}&\\\
&c_{1}&\mapsto&c^{\prime}_{1}&\end{array}$

($\mathbb{Z}/\text{Ker}f=\mathbb{Z}/(2\mathbb{Z})\cong\mathbb{F}_{2}$.) In
other words

$\begin{array}[]{lllll}f:&\mathbb{Z}^{16}&\rightarrow&(2\mathbb{Z})\oplus\mathbb{Z}^{15}&\\\
&(c_{1},c_{2},...,c_{16})&\mapsto&(2c_{1},c_{2},...,c_{16})&\end{array}$

it is easily to show the map is homomrphic. For any $x,y\in\Lambda$ can be
written as

$\displaystyle x$ $\displaystyle=c_{1}e_{1}+\sum_{i=2}^{16}c_{i}e_{i}$
$\displaystyle y$ $\displaystyle=a_{1}e_{1}+\sum_{i=2}^{16}a_{i}e_{i}$

where $c,a\in\mathbb{Z}$. And from the definition

$\displaystyle f(x)$ $\displaystyle=2c_{1}e_{1}+\sum_{i=2}^{16}c_{i}e_{i}$
$\displaystyle f(y)$ $\displaystyle=2a_{1}e_{1}+\sum_{i=2}^{16}a_{i}e_{i}$

on the other hand

$f(x+y)=2(c_{1}+a_{1})e_{1}+\sum_{i=2}^{16}(c_{i}+a_{1})e_{i}\\\ $

Thus

$f(x)+f(y)=f(x+y)$

(Q.E.D) Proposition.5 For $x\in\Lambda$, the map $g$ follwing is the
isomorphism.

$g:x\mapsto x/2$

(Proof)
It is obvious that $g$ is the bijection. For any $x,y\in\Lambda$,

$g(x)+g(y)=\frac{x}{2}+\frac{y}{2}$

on the other and

$g(x+y)=\frac{x+y}{2}$

Thus

$g(x)+g(y)=g(x+y)$

hence $g$ is the homomorphism. Therefore $g$ is the isomorphism.
(Q.E.D)

Also for $x\in\Lambda$, donate $h$ to be the homomorphism

$h:x\mapsto x\quad(\text{mod}\ 2)$

then

Proposition.6 The composition $\pi=h\circ g\circ f$ is the homomorphism to the
code (vector space of degree $16$ and dimension $11$ over Finite Field
$\mathbb{F}_{2}$) $\mathcal{C}_{16}$ generated below from $\Lambda$.

$\mathcal{C}_{16}=\text{Span}_{\mathbb{F}_{2}}\left(\begin{array}[]{llllllllllllllll}1&0&0&0&0&0&0&1&0&0&0&1&0&1&1&1\\\
0&1&0&0&0&0&0&1&0&0&0&1&0&1&0&0\\\ 0&0&1&0&0&0&0&1&0&0&0&1&0&0&1&0\\\
0&0&0&1&0&0&0&1&0&0&0&1&0&0&0&1\\\ 0&0&0&0&1&0&0&1&0&0&0&0&0&1&1&0\\\
0&0&0&0&0&1&0&1&0&0&0&0&0&1&0&1\\\ 0&0&0&0&0&0&1&1&0&0&0&0&0&0&1&1\\\
0&0&0&0&0&0&0&0&1&0&0&1&0&1&1&0\\\ 0&0&0&0&0&0&0&0&0&1&0&1&0&1&0&1\\\
0&0&0&0&0&0&0&0&0&0&1&1&0&0&1&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1\end{array}\right)$

(Proof)
From the Proposition.3, the lattice $\Lambda$ can be decomposed directly
$2\Lambda\oplus(2\Lambda+1)$, where

$\displaystyle 2\Lambda$
$\displaystyle:=\left\\{x\in\Lambda\mid\sum_{i=1}^{16}c_{i}e_{i},\quad
c_{1}\in 2\mathbb{Z},\quad c_{i}\in\mathbb{Z}\quad(i=2,3,\ldots,16)\right\\}$
$\displaystyle 2\Lambda+1$
$\displaystyle:=\left\\{x\in\Lambda\mid\sum_{i=1}^{16}c_{i}e_{i},\quad
c_{1}\in(2\mathbb{Z}+1),\quad
c_{i}\in\mathbb{Z}\quad(i=2,3,\ldots,16)\right\\}$

and as I mentioned in Proposition.4 and 5 there are the homomorphism
$f:\Lambda\rightarrow 2\Lambda$, and the isomorphism $g$, the homomorphism as
well. Donate the images produced by these mappings to be as follows.

$\begin{array}[]{lllllllll}&\mathbb{Z}^{16}&\stackrel{{\scriptstyle
f}}{{\longrightarrow}}&(2\mathbb{Z})^{16}&\stackrel{{\scriptstyle
g}}{{\longrightarrow}}&\mathbb{Z}^{16}&\stackrel{{\scriptstyle
h}}{{\longrightarrow}}&(\mathbb{F}_{2})^{16}&\\\ &\cup&&\cup&&\cup&&\cup&\\\
&\Lambda&\stackrel{{\scriptstyle\text{Hom}}}{{\longrightarrow}}&2\Lambda&\stackrel{{\scriptstyle\text{Iso}}}{{\longrightarrow}}&\Lambda^{\prime}&\stackrel{{\scriptstyle\text{Hom}}}{{\longrightarrow}}&\Lambda(\mathbb{F}_{2})&\\\
\end{array}$

where $\mathbb{F}_{2}$ is the finite field.(The operation rules for this field
is $1+1=0,1+0=1,0+0=0$)

For example the element $e_{13}$ will be the following by composition of the
maps $\pi$.

$(0,0,0,...,0,2,2,2,2)\mapsto(0,0,0,...,0,2,2,2,2)\mapsto(0,0,0,...,0,1,1,1,1)\mapsto(0,0,0,...,0,1,1,1,1)$

Also $e_{16}$ and $e_{1}$ becomes

$\displaystyle(0,0,0,...,0,0,0,0,4)\mapsto(0,0,0,...,0,0,0,0,4)\mapsto(0,0,0,...,0,0,0,0,2)\mapsto(0,0,0,...,0,0,0,0,0)$
$\displaystyle(1,1,1,...,1,1,1,1,1)\mapsto(2,2,2,...,2,2,2,2,2)\mapsto(1,1,1,...,1,1,1,1,1)\mapsto(1,1,1,...,1,1,1,1,1)$

So the generators become as follows by the homomorphism $\pi$.

$\pi:\left(\begin{array}[]{c}e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\\\ e_{5}\\\
\vdots\\\
e_{16}\end{array}\right)\mapsto\left(\begin{array}[]{llllllllllllllll}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\
0&1&&&&&&1&&&&1&&1&&\\\ \vdots&&1&&&&&1&&&&1&&&1&\\\ &&&1&&&&1&&&&1&&&&1\\\
&&&&1&&&1&&&&&&1&1&\\\ &&&&&1&&1&&&&&&1&&1\\\ \vdots&&&&&&1&1&&&&&&&1&1\\\
0&\cdots&&&&&&&&&&&&&\cdots&0\\\ \vdots&&&&&&&&1&&&1&&1&1&\\\
&&&&&&&&&1&&1&&1&&1\\\ \vdots&&&&&&&&&&1&1&&&1&1\\\
0&\cdots&&&&&&&&&&&&&\cdots&0\\\ &&&&&&&&&&&&1&1&1&1\\\ &&&&&&&&&&&&&&&0\\\
\vdots&&&&&&&&&&&&&&&\vdots\\\
0&\cdots&&&&&&&&&&&&&\cdots&0\end{array}\right)$

from elementary row transformation, I found
$\mathcal{C}_{16}:=\Lambda(\mathbb{F}_{2})$ is the vector space of degree $16$
and dimension $11$ over Finite Field of size 2 with the following generators.
And the number of the elements is $2^{11}=2048$ as the vector space is 11
dimensional over $\mathbb{F}_{2}$

$\mathcal{C}_{16}=\text{Span}_{\mathbb{F}_{2}}\left(\begin{array}[]{llllllllllllllll}1&&&&&&&1&&&&1&&1&1&1\\\
0&1&&&&&&1&&&&1&&1&&\\\ \vdots&&1&&&&&1&&&&1&&&1&\\\ &&&1&&&&1&&&&1&&&&1\\\
&&&&1&&&1&&&&&&1&1&\\\ &&&&&1&&1&&&&&&1&&1\\\ &&&&&&1&1&&&&&&&1&1\\\
&&&&&&&&1&&&1&&1&1&\\\ &&&&&&&&&1&&1&&1&&1\\\ \vdots&&&&&&&&&&1&1&&&1&1\\\
0&\cdots&&&&&&&&&\cdots&0&1&1&1&1\end{array}\right)$

(Q.E.D)

Since all the number of the components, which is 1, are even of the generator
of $\mathcal{C}_{16}$, the number of the components of the elements of it must
be also even. And also by calculating all the combination of
$\mathcal{C}_{16}$, I found the follwing result. Fact.1 The elements of the
linear space $\mathcal{C}_{16}$ has at least $4$ $1$’s as components ($140$
vectors). And the list of the distribution is following.[See Appendix 5.3 for
details.]

$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr 0&4&6&8&10&12&16\\\ \hline\cr
1&140&448&870&448&140&1\\\ \hline\cr\end{array}$

Since $\pi$ is the homomorphism $\pi:\Lambda\rightarrow\mathcal{C}_{16}$. it
can be also said Proposition.7 For the type of the vectors
$(2^{n},-2^{m},0^{l})$ in $\Lambda$,

$n+m\geqslant 4$

where $\left(a^{n},b^{m}\right)$ means a vector of type containing $n$ a’s and
$m$ b’s. (Proof)
If there exists the type $(2^{n},-2^{m},0^{l})$ of the vectors such that
$0<n+m<4$ in $\Lambda$, then $\mathcal{C}_{16}$ must have the vectors type
$(1^{k},0)$ such that $k<4$, as $\pi$ is the homomorphism. But actually
not.Therefore, no such type of vector exists in $\Lambda$.(Q.E.D)

From this proposition, $\Lambda_{\mathbb{Q}^{2}_{8}}$ does not have the
vectors, whose norm are $1/2$, such that

$\frac{1}{4}(2,2,0,0,...,0)$

By counting up the vectors that are the shortest of
$\Lambda_{\mathbb{Q}^{2}_{8}}$, I find the following facts. Fact.2 The lattice
$\Lambda_{\mathbb{Q}^{2}_{8}}$ also has $4320$ shortest vectors. The details
of the above are as follows. Fact.3 The vectors whose norm is 1 in
$\Lambda_{\mathbb{Q}^{2}_{8}}$ are classified into the following types.

$\begin{array}[]{|l|l|}\hline\cr\text{Types}&\text{Number of the vectors.}\\\
\hline\cr\frac{1}{4}(-1^{a},1^{b})&2048\quad(a+b=16)\\\
\frac{1}{4}\left(2^{4},0^{12}\right)&140\\\
\frac{1}{4}\left(-2^{4},0^{12}\right)&140\\\
\frac{1}{4}\left(4^{1},0^{15}\right)&16\\\
\frac{1}{4}\left(-4^{1},0^{15}\right)&16\\\
\frac{1}{4}\left(-2^{2},2^{2},0^{12}\right)&840\\\
\frac{1}{4}\left(-2^{1}\cdot 2^{3},0^{12}\right)&560\\\
\frac{1}{4}\left(-2^{3},2^{1},0^{12}\right)&560\\\ \hline\cr\end{array}$

where $(a^{n},b^{m})$ means a vector of type containing $n$ a’s and $m$ b’s.

Also vectors at the second distance are classified into the following types.

$\begin{array}[]{|l|l|}\hline\cr\frac{1}{4}\left(2^{6},0^{10}\right)&448\\\
\frac{1}{4}\left(-2^{6},0^{10}\right)&448\\\
\frac{1}{4}\left(-2^{1},2^{5},0^{10}\right)&2688\\\
\frac{1}{4}\left(-2^{2},2^{4},0^{10}\right)&6720\\\
\frac{1}{4}\left(-2^{3},2^{3},0^{10}\right)&8960\\\
\frac{1}{4}\left(-2^{4},2^{2},0^{10}\right)&6720\\\
\frac{1}{4}\left(-2^{5},2^{1},0^{10}\right)&2688\\\
\frac{1}{4}\left(-3^{1},1^{a},-1^{b}\right)&16384\quad(a+b=15)\\\
\frac{1}{4}\left(3^{1},1^{a},-1^{b}\right)&16384\quad(a+b=15)\\\
\hline\cr\end{array}$

On the Other hand, the shortest vectors in $\Lambda_{\text{Barns-Wall}}$ are
classified into follwoing types.(Total $4320$ vectors)

$\begin{array}[]{|l|l|}\hline\cr\text{Types}&\text{Number of the vectors}\\\
\hline\cr\frac{1}{4}\left(\pm 1^{8},0^{8}\right)&30\\\
\frac{1}{4}\left(-1^{2},1^{6},0^{8}\right)&30\\\
\frac{1}{4}\left(-1^{4},1^{4},0^{8}\right)&30\cdot\left(\begin{array}[]{l}8\\\
2\end{array}\right)=840\\\
\frac{1}{4}\left(-1^{4},1^{4},0^{8}\right)&30\cdot\left(\begin{array}[]{l}8\\\
4\end{array}\right)=2100\\\
\frac{1}{4}\left(-1^{6},1^{4},0^{8}\right)&30\cdot\left(\begin{array}[]{l}8\\\
6\end{array}\right)=840\\\ \frac{1}{4}\left(\pm 2^{2},0^{14}\right)&240\\\
\frac{1}{4}\left(-2^{1},2^{1},0^{14}\right)&2\cdot 120=240\\\
\hline\cr\end{array}$

Some vectors in the lattice $\Lambda_{\mathbb{Q}^{2}_{8}}$ with the elements
in $G\subset\mathbb{Z}^{\prime}\left[\mathbb{Q}^{2}_{8}\right]$ are shown
below.

$\displaystyle\frac{(1+i+j+k)}{2}$ $\displaystyle\mapsto$
$\displaystyle(1/2,1/2,1/2,1/2,0,0,0,0,0,0,0,0,0,0,0,0)$
$\displaystyle\frac{(1+i+j-k)}{2}$ $\displaystyle\mapsto$
$\displaystyle(1/2,1/2,1/2,-1/2,0,0,0,0,0,0,0,0,0,0,0,0)$ $\displaystyle
e_{1}=\frac{(1+iq+jr-kp)}{2}$ $\displaystyle\mapsto$
$\displaystyle(1/2,0,0,0,0,0,0,-1/2,0,1/2,0,0,0,0,1/2,0)$ $\displaystyle
e_{2}=\frac{(1-ip-jq-kr)}{2}$ $\displaystyle\mapsto$
$\displaystyle(1/2,0,0,0,0,-1/2,0,0,0,0,-1/2,0,0,0,0,-1/2)$ $\displaystyle
e_{3}=\frac{(1-ip-jr+kq)}{2}$ $\displaystyle\mapsto$
$\displaystyle(1/2,0,0,0,0,-1/2,0,0,0,0,0,1/2,0,0,-1/2,0)$ $\displaystyle
e_{4}=\frac{(1-iq-jp+kr)}{2}$ $\displaystyle\mapsto$
$\displaystyle(1/2,0,0,1/2,0,0,0,0,-1/2,0,0,-1/2,0,0,0,0)$
$\displaystyle\frac{(1+i+j+k+p+ip+jp+\cdots+kr)}{4}$ $\displaystyle\mapsto$
$\displaystyle\frac{1}{4}(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)$

I empirically predicted the following. Conjecture.1 For the image
$\text{Im}(\varphi^{-1})$ of the inverse mapping
$\varphi^{-1}:G\rightarrow\\{\mathbb{Z}\oplus(\mathbb{Z}+1/2)\oplus(\mathbb{Z}+1/4)\\}^{16}$,

$\text{Im}(\varphi^{-1})\subset\Lambda_{\mathbb{Q}^{2}_{8}}$

##  4 Octanion and Weyl Group$\ W\left(E_{8}\right)$

Octonion
$\mathbb{O}=\left\\{u_{0}\zeta_{0}+u_{1}\zeta_{1}+\cdots+u_{7}\zeta_{7}\mid
u_{i}\in\mathbb{R}\right\\}$is defined as $\zeta_{i}$ satisfying the following
operations.

$\begin{array}[]{|c||c|c|c|c|c|c|c|c|}\hline\cr\times&\zeta_{0}&\zeta_{1}&\zeta_{2}&\zeta_{3}&\zeta_{4}&\zeta_{5}&\zeta_{6}&\zeta_{7}\\\
\hline\cr\hline\cr\zeta_{0}&\zeta_{0}&\zeta_{1}&\zeta_{2}&\zeta_{3}&\zeta_{4}&\zeta_{5}&\zeta_{6}&\zeta_{7}\\\
\hline\cr\zeta_{1}&\zeta_{1}&-\zeta_{0}&\zeta_{3}&-\zeta_{2}&\zeta_{5}&-\zeta_{4}&-\zeta_{7}&\zeta_{6}\\\
\hline\cr\zeta_{2}&\zeta_{2}&-\zeta_{3}&-\zeta_{0}&\zeta_{1}&\zeta_{6}&\zeta_{7}&-\zeta_{4}&-\zeta_{5}\\\
\hline\cr\zeta_{3}&\zeta_{3}&\zeta_{2}&-\zeta_{1}&-\zeta_{0}&\zeta_{7}&-\zeta_{6}&\zeta_{5}&-\zeta_{4}\\\
\hline\cr\zeta_{4}&\zeta_{4}&-\zeta_{5}&-\zeta_{6}&-\zeta_{7}&-\zeta_{0}&\zeta_{1}&\zeta_{2}&\zeta_{3}\\\
\hline\cr\zeta_{5}&\zeta_{5}&\zeta_{4}&-\zeta_{7}&\zeta_{6}&-\zeta_{1}&-\zeta_{0}&-\zeta_{3}&\zeta_{2}\\\
\hline\cr\zeta_{6}&\zeta_{6}&\zeta_{7}&\zeta_{4}&-\zeta_{5}&-\zeta_{2}&\zeta_{3}&-\zeta_{0}&-\zeta_{1}\\\
\hline\cr\zeta_{7}&\zeta_{7}&-\zeta_{6}&\zeta_{5}&\zeta_{4}&-\zeta_{3}&-\zeta_{2}&\zeta_{1}&-\zeta_{0}\\\
\hline\cr\end{array}$

The above rules of operation follow the arrows in the figure below.

Figure 4: Figure representing the multiplication rules of Octantion.

For example, an element$u\in\mathbb{O}$ can be written by the following linear
combination.

$u=u_{0}\zeta_{0}+u_{1}\zeta_{1}+u_{2}\zeta_{2}+u_{3}\zeta_{3}+u_{4}\zeta_{4}+u_{5}\zeta_{5}+u_{6}\zeta_{6}+u_{7}\zeta_{7}\\\
$

For the sake of simplicity, I will simply write the following.

$:=(u_{0},u_{1},u_{2},u_{3},u_{4},u_{5},u_{6},u_{7})$

then multiplying by $\zeta_{1}$ from the left will be

$\displaystyle\zeta_{1}u$
$\displaystyle=-u_{1}\zeta+u_{0}\zeta_{1}-u_{3}\zeta_{2}+u_{2}\zeta_{3}-u_{5}\zeta_{4}+u_{4}\zeta_{5}+u_{7}\zeta_{6}-u_{6}\zeta_{7}$
$\displaystyle=(-u_{1},u_{0},-u_{3},u_{2},-u_{5},u_{4},u_{7},-u_{6})$
$\displaystyle=\left(\begin{array}[]{cccc|cccc}&-1&&&&&&\\\ 1&&&&&&&\\\
&&&-1&&&&\\\ &&1&&&&&\\\ \hline\cr&&&&&-1&&\\\ &&&&1&&&\\\ &&&&&&&1\\\
&&&&&&-1&\end{array}\right)\left(\begin{array}[]{l}u_{0}\\\ u_{1}\\\ u_{2}\\\
u_{3}\\\ u_{4}\\\ u_{5}\\\ u_{6}\\\ u_{7}\end{array}\right)$

Thus the above operation is the $8$-dimensional orthogonal transformation
$\zeta_{1}:\mathbb{R}^{8}\rightarrow\mathbb{R}^{8}$ for
$u=(u_{0},u_{1},...,u_{7})$. From now on, I will write this $8\times 8$ matrix
simply as $\boldsymbol{\zeta}_{1}$.This can be written using the matrices
described in the previous section

$\boldsymbol{\zeta}_{1}=\left(\begin{array}[]{c|c}-\boldsymbol{i}&\\\
\hline\cr&-\boldsymbol{p}\end{array}\right)$

Others can also be written similarly as follows

$\begin{array}[]{l}\boldsymbol{\zeta}_{2}=\left(\begin{array}[]{c|c}-\boldsymbol{j}&\\\
\hline\cr&-\boldsymbol{q}\end{array}\right)\\\ \\\
\boldsymbol{\zeta}_{3}=\left(\begin{array}[]{c|c}\boldsymbol{k}&\\\
\hline\cr&\boldsymbol{r}\end{array}\right)\\\ \\\
\boldsymbol{\zeta}_{4}=\left(\begin{array}[]{c|c}&\frac{1}{2}(\boldsymbol{i}\boldsymbol{p}+\boldsymbol{j}\boldsymbol{q}-\boldsymbol{k}\boldsymbol{r}+E)\\\
\hline\cr-\frac{1}{2}(\boldsymbol{i}\boldsymbol{p}+\boldsymbol{j}\boldsymbol{q}-\boldsymbol{k}\boldsymbol{r}+E)&\end{array}\right)\\\
\\\
\boldsymbol{\zeta}_{5}=\left(\begin{array}[]{c|c}&\frac{1}{2}(\boldsymbol{i}-\boldsymbol{jr}-\boldsymbol{kq}-\boldsymbol{p})\\\
\hline\cr\frac{1}{2}(\boldsymbol{i}+\boldsymbol{jr}+\boldsymbol{kq}-\boldsymbol{p})&\end{array}\right)\\\
\\\
\boldsymbol{\zeta}_{6}=\left(\begin{array}[]{c|c}&\frac{1}{2}(\boldsymbol{ir}+\boldsymbol{j}+\boldsymbol{kp}-\boldsymbol{q})\\\
\hline\cr\frac{1}{2}(-\boldsymbol{ir}+\boldsymbol{j}-\boldsymbol{kp}-\boldsymbol{q})&\end{array}\right)\\\
\\\
\boldsymbol{\zeta}_{7}=\left(\begin{array}[]{c|c}&\frac{1}{2}(-\boldsymbol{iq}+\boldsymbol{jp}-\boldsymbol{k}-\boldsymbol{q})\\\
\hline\cr\frac{1}{2}(\boldsymbol{iq}-\boldsymbol{jp}-\boldsymbol{k}-\boldsymbol{q})&\end{array}\right)\end{array}$

Now I define

$\boldsymbol{\eta}:=\frac{\boldsymbol{\zeta}_{1}+\boldsymbol{\zeta}_{2}+\boldsymbol{\zeta}_{3}+\boldsymbol{\zeta}_{4}}{2}=\left(\begin{array}[]{c|c}\frac{1}{2}(-\boldsymbol{i}-\boldsymbol{j}+\boldsymbol{k})&\frac{1}{4}(\boldsymbol{ip}+\boldsymbol{jq}-\boldsymbol{kr}+E)\\\
\hline\cr-\frac{1}{4}(\boldsymbol{ip}+\boldsymbol{jq}-\boldsymbol{kr}+E)&\frac{1}{2}(-\boldsymbol{p}-\boldsymbol{q}+\boldsymbol{r})\end{array}\right)$

And

$\begin{array}[]{l}\boldsymbol{\xi}_{1}:=\left(\begin{array}[]{c|c}-\boldsymbol{p}&\\\
\hline\cr&\boldsymbol{p}\end{array}\right)\\\
\boldsymbol{\xi}_{2}:=\left(\begin{array}[]{c|c}-\boldsymbol{q}&\\\
\hline\cr&\boldsymbol{q}\end{array}\right)\\\
\boldsymbol{\xi}_{3}:=\left(\begin{array}[]{c|c}-\boldsymbol{r}&\\\
\hline\cr&\boldsymbol{r}\end{array}\right)\end{array}$

then I observed the following. Observation The group $G^{\prime}$ defined
below is the orthogonal group of order $348364800=2^{13}\cdot 3^{5}\cdot
5^{2}\cdot 7$

$\displaystyle G^{\prime}$
$\displaystyle=\left\langle\boldsymbol{\zeta}_{1},\boldsymbol{\zeta}_{2},\boldsymbol{\zeta}_{3},\boldsymbol{\eta},\boldsymbol{\xi}_{1},\boldsymbol{\xi}_{2}\right\rangle$

and satisfies

$\simeq O_{8}^{+}(2)=W\left(E_{8}\right)$

##  5 Appendex

##### matrixies

$\begin{array}[]{l}{e}_{1}=\frac{1}{2}(1,i,j,k)\left(\begin{array}[]{cccc}1&0&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ 0&-1&0&0\end{array}\right)\left(\begin{array}[]{l}1\\\
p\\\ q\\\
r\end{array}\right)\stackrel{{\scriptstyle\rho}}{{\longmapsto}}\frac{1}{2}\left(\begin{array}[]{cccc}1&1&1&1\\\
1&1&-1&-1\\\ 1&-1&1&-1\\\ 1&-1&-1&1\end{array}\right)\\\
e_{2}=\frac{1}{2}(1,i,j,k)\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\
0&0&-1&0\\\ 0&0&0&-1\end{array}\right)\left(\begin{array}[]{l}1\\\ p\\\ q\\\
r\end{array}\right)\stackrel{{\scriptstyle\rho}}{{\longmapsto}}\left(\begin{array}[]{rrrr}1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&-1\end{array}\right)\\\
e_{3}=\frac{1}{2}(1,i,j,k)\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\
0&0&0&-1\\\ 0&0&1&0\end{array}\right)\left(\begin{array}[]{l}1\\\ p\\\ q\\\
r\end{array}\right)\stackrel{{\scriptstyle\rho}}{{\longmapsto}}\left(\begin{array}[]{llll}1&0&0&0\\\
0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{array}\right)\\\
e_{4}=\frac{1}{2}(1,i,j,k)\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&-1&0\\\
0&-1&0&0\\\ 0&0&0&1\end{array}\right)\left(\begin{array}[]{l}1\\\ p\\\ q\\\
r\end{array}\right)\stackrel{{\scriptstyle\rho}}{{\longmapsto}}\left(\begin{array}[]{llll}1&0&0&0\\\
0&0&1&0\\\ 0&1&0&0\\\ 0&0&0&1\end{array}\right)\end{array}$

$\boldsymbol{\zeta}_{1}=\left(\begin{array}[]{rrrrrrrr}0&-1&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0\\\ 0&0&0&-1&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&-1&0&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&-1&0\end{array}\right)$

$\boldsymbol{\zeta}_{2}=\left(\begin{array}[]{rrrrrrrr}0&0&-1&0&0&0&0&0\\\
0&0&0&1&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 0&-1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&-1&0\\\
0&0&0&0&0&0&0&-1\\\ 0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&1&0&0\end{array}\right)$

$\boldsymbol{\zeta}_{3}=\left(\begin{array}[]{rrrrrrrr}0&0&0&-1&0&0&0&0\\\
0&0&-1&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&-1\\\
0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&-1&0&0\\\ 0&0&0&0&1&0&0&0\end{array}\right)$

$\boldsymbol{\zeta}_{4}=\left(\begin{array}[]{rrrrrrrr}0&0&0&0&-1&0&0&0\\\
0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&1\\\ 1&0&0&0&0&0&0&0\\\
0&-1&0&0&0&0&0&0\\\ 0&0&-1&0&0&0&0&0\\\ 0&0&0&-1&0&0&0&0\end{array}\right)$

$\boldsymbol{\zeta}_{5}=\left(\begin{array}[]{rrrrrrrr}0&0&0&0&0&-1&0&0\\\
0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&-1&0\\\ 0&1&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&-1&0&0&0&0&0\end{array}\right)$

$\boldsymbol{\zeta}_{6}=\left(\begin{array}[]{rrrrrrrr}0&0&0&0&0&0&-1&0\\\
0&0&0&0&0&0&0&-1\\\ 0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&1&0&0&0&0&0\\\
0&0&0&-1&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\end{array}\right)$
$\boldsymbol{\zeta}_{7}=\left(\begin{array}[]{rrrrrrrr}0&0&0&0&0&0&0&-1\\\
0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&-1&0&0\\\ 0&0&0&0&-1&0&0&0\\\ 0&0&0&1&0&0&0&0\\\
0&0&1&0&0&0&0&0\\\ 0&-1&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0\end{array}\right)$

$\boldsymbol{\eta}=\frac{1}{2}\left(\begin{array}[]{rrrrrrrr}0&1&1&1&1&0&0&0\\\
-1&0&1&-1&0&-1&0&0\\\ -1&-1&0&1&0&0&-1&0\\\ -1&1&-1&0&0&0&0&-1\\\
-1&0&0&0&0&1&1&1\\\ 0&1&0&0&-1&0&-1&1\\\ 0&0&1&0&-1&1&0&-1\\\
0&0&0&1&-1&-1&1&0\end{array}\right)$

$\boldsymbol{\xi}_{1}=\left(\begin{array}[]{rrrrrrrr}0&-1&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&-1&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0\\\
0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&0&0&-1\\\ 0&0&0&0&0&0&1&0\end{array}\right)$

$\boldsymbol{\xi}_{2}=\left(\begin{array}[]{rrrrrrrr}0&0&-1&0&0&0&0&0\\\
0&0&0&-1&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&1\\\ 0&0&0&0&-1&0&0&0\\\ 0&0&0&0&0&-1&0&0\end{array}\right)$

$\boldsymbol{\xi}_{3}=\left(\begin{array}[]{rrrrrrrr}0&0&0&-1&0&0&0&0\\\
0&0&1&0&0&0&0&0\\\ 0&-1&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&-1&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&0&0&-1&0&0&0\end{array}\right)$

###  5.1 The permutation representation of $\mathbb{Q}^{2}_{8}$

By the way, we can find that $i,j,k,p,q,r$ as a subgroup of the group
$C\subset Sym(16)$ order $192$ generated by the following permutations of the
rotations in three orthogonal directions.

$\displaystyle C=$
$\displaystyle<(1,2,10,9)(3,11,12,4)(5,6,14,13)(7,15,16,8),$
$\displaystyle(1,4,12,9)(2,3,11,10)(5,8,16,13)(6,7,15,14),$
$\displaystyle(1,5,13,9)(2,6,14,10)(3,7,15,11)(4,8,16,12)>$

Figure 5: The permutation of the generator $g_{1}$ of the group $C$

The figure shows one of the generator which is

$g_{1}=(1,2,10,9)(3,11,12,4)(5,6,14,13)(7,15,16,8)\in C$

and $C$ include subset $C^{\prime}$ order $32$ generated by

$\displaystyle(1,3,15,13)(2,11,16,5)(4,7,14,9)(6,10,12,8)$
$\displaystyle(1,12,15,6)(2,9,16,7)(3,10,13,8)(4,11,14,5)$
$\displaystyle(1,8,15,10)(2,5,16,11)(3,6,13,12)(4,7,14,9)$
$\displaystyle(1,13,15,3),(2,9,16,7),(4,5,14,11),(6,10,12,8)$

and there is isomorphism $\tau:\mathbb{Q}^{2}_{8}\cong<i,j,p,q>\rightarrow
C^{\prime}$ that is

$\displaystyle(i)\tau=(1,3,15,13)(2,11,16,5)(4,7,14,9)(6,10,12,8)$
$\displaystyle(j)\tau=(1,12,15,6)(2,9,16,7)(3,10,13,8)(4,11,14,5)$
$\displaystyle(k)\tau=(1,10,15,8)(2,14,16,4)(3,6,13,12)(5,9,11,7)$
$\displaystyle(p)\tau=(1,8,15,10)(2,5,16,11)(3,6,13,12)(4,7,14,9)$
$\displaystyle(q)\tau=(1,13,15,3),(2,9,16,7),(4,5,14,11),(6,10,12,8)$
$\displaystyle(r)\tau=(1,6,15,12)(2,14,16,4)(3,10,13,8)(5,7,11,9)$
$\displaystyle(-1)\tau=(1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10)$

As $-1$ is commutative with all the elements, actually
$(-1)\tau=(1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10)$ is the center of
the group $C$. Graphically

Figure 6: The center of $C$

###  5.2 A constraction of Quotanion

$\mathcal{I}=(\sqrt{-1})\sigma=\left(\begin{array}[]{cc}&1\\\
-1&\end{array}\right)$

and let $\mathcal{J}$ to be

$\mathcal{I}\mathcal{J}=-\mathcal{J}\mathcal{I}$

then $\mathcal{J}$ is determined as

$\mathcal{J}=\left(\begin{array}[]{ll}1&0\\\ 0&-1\end{array}\right)$

$\mathcal{I}$ and $\mathcal{J}$ satisfy

$\mathcal{I}^{2}=-1,\quad\mathcal{I}^{4}=\mathcal{J}^{2}=1,\quad\mathcal{I}\mathcal{J}=-\mathcal{J}\mathcal{I}$

using above

$\begin{array}[]{l}\boldsymbol{i}=\left(\begin{array}[]{ll}\mathcal{J}&\\\
&\mathcal{I}\end{array}\right)\quad\boldsymbol{k}=\left(\begin{array}[]{ll}&-1\\\
1&\end{array}\right)\\\
\boldsymbol{j}=\left(\begin{array}[]{cc}&\mathcal{I}\\\
-\mathcal{I}&\end{array}\right)\end{array}$

Also the multiplicative group generated by $\mathcal{I},\mathcal{J}$ order $8$
is isomorphic to $W(B_{2})\cong W(C_{2})$

###  5.3 The code $\mathcal{C}_{16}$ in $(\mathbb{F}_{2})^{16}$

The generator matrix of $\mathcal{C}_{16}$ is

$\left(\begin{array}[]{c}u_{1}\\\ u_{2}\\\ u_{3}\\\ \vdots\\\ \\\ \\\ \\\ \\\
\vdots\\\
u_{11}\end{array}\right)=\left(\begin{array}[]{llllllllllllllll}1&0&0&0&0&0&0&1&0&0&0&1&0&1&1&1\\\
0&1&0&0&0&0&0&1&0&0&0&1&0&1&0&0\\\ 0&0&1&0&0&0&0&1&0&0&0&1&0&0&1&0\\\
0&0&0&1&0&0&0&1&0&0&0&1&0&0&0&1\\\ 0&0&0&0&1&0&0&1&0&0&0&0&0&1&1&0\\\
0&0&0&0&0&1&0&1&0&0&0&0&0&1&0&1\\\ 0&0&0&0&0&0&1&1&0&0&0&0&0&0&1&1\\\
0&0&0&0&0&0&0&0&1&0&0&1&0&1&1&0\\\ 0&0&0&0&0&0&0&0&0&1&0&1&0&1&0&1\\\
0&0&0&0&0&0&0&0&0&0&1&1&0&0&1&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1\end{array}\right)$

For all vectors $x\in\mathcal{C}_{16}$ can be written as

$x=a_{1}u_{1}+a_{2}u_{2}+\cdots+a_{11}u_{11}$

where $a_{i}\in\mathbb{F}_{2}$.The operation is then performed as follows.

$\displaystyle u_{10}+u_{11}$
$\displaystyle=(0,0,\cdots,0,1,1,0,0,1,1)+(0,0,\cdots,0,0,0,1,1,1,1)$
$\displaystyle=(0,0,\cdots,0,1,1,1,1,0,0)$

This generates the vector space of degree $16$ and dimension $11$ over Finite
Field of size $2$.

The weight distribution of $\mathcal{C}_{16}$ is as follows.

$\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr 0&4&6&8&10&12&16\\\ \hline\cr
1&140&448&870&448&140&1\\\ \hline\cr\end{array}$

and at least, $\mathcal{C}_{16}$ is invariant by the action of the group
$G_{\mathcal{C}}$ order $322560=2^{10}\cdot 3^{2}\cdot 5\cdot 7$ defined by
the following

$G_{\mathcal{C}}:=\langle(0,1)(2,7,9,11,13,4,14,3,6,8,10,12,5,15),(0,15,8,11,14,1,6,5)(2,3,10,7,12,13,4,9)\rangle$

and I also found $G_{\mathcal{C}}\cong 2^{4}\cdot A_{8}$ which is called
’Miracle Octad Group’ of Mathieu group $M_{24}$, the subgroup of
$G_{\mathcal{C}}$ generated by following is isomorphic to $A_{8}$.

$\displaystyle\tau_{1}=$ $\displaystyle(1,4,2,7)(5,6)(8,13,11,14)(12,15)$
$\displaystyle\tau_{2}=$ $\displaystyle(1,6)(2,14)(3,8)(4,15)(5,9)(10,13)$

For any elements of $\mathcal{C}_{16}$(which has four $1$s) share $2$
coordinates for each of the $32$ generators(which has four $1$s), and $1$
coordinates for each of the $64$ generators(which has four $1$s). Below are
140 vectors in $\mathcal{C}_{16}$ with four 1’s in the coordinates.

$\displaystyle(0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0)$
$\displaystyle(1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1)$
$\displaystyle(0,0,1,0,0,0,0,1,0,0,0,1,0,0,1,0)$
$\displaystyle(1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,1)$
$\displaystyle(0,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0)$
$\displaystyle(0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1)$
$\displaystyle(1,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0)$
$\displaystyle(0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1)$
$\displaystyle(0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,1)$
$\displaystyle(1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0)$
$\displaystyle(1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1)$
$\displaystyle(0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0)$
$\displaystyle(0,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0)$
$\displaystyle(0,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0)$
$\displaystyle(1,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1)$
$\displaystyle(1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0)$
$\displaystyle(0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1)$
$\displaystyle(1,0,1,0,0,1,0,1,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0)$
$\displaystyle(0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1)$
$\displaystyle(1,1,0,0,1,1,0,0,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1)$
$\displaystyle(1,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0)$
$\displaystyle(1,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,1)$
$\displaystyle(0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0)$
$\displaystyle(0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,1)$
$\displaystyle(1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0)$
$\displaystyle(0,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0)$
$\displaystyle(0,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0)$
$\displaystyle(1,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0)$
$\displaystyle(0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,0)$
$\displaystyle(1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1)$
$\displaystyle(0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0)$
$\displaystyle(0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0)$
$\displaystyle(0,1,1,0,0,0,0,0,1,0,0,1,0,0,0,0)$
$\displaystyle(1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0)$
$\displaystyle(0,0,0,0,1,0,0,1,1,0,0,1,0,0,0,0)$
$\displaystyle(0,1,0,0,1,0,0,0,1,0,0,0,0,1,0,0)$
$\displaystyle(0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0)$
$\displaystyle(0,0,0,1,1,0,0,0,1,0,0,0,0,0,0,1)$
$\displaystyle(1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0)$
$\displaystyle(0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,1)$
$\displaystyle(0,0,0,1,0,1,0,0,1,0,0,0,0,0,1,0)$
$\displaystyle(1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0)$
$\displaystyle(0,1,0,0,0,0,1,0,1,0,0,0,0,0,0,1)$
$\displaystyle(0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0)$
$\displaystyle(0,0,0,0,0,1,1,0,1,0,0,1,0,0,0,0)$
$\displaystyle(0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1)$
$\displaystyle(1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0)$
$\displaystyle(0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1)$
$\displaystyle(1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0)$
$\displaystyle(0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0)$
$\displaystyle(0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0)$
$\displaystyle(1,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0)$
$\displaystyle(0,0,1,0,1,0,0,0,0,1,0,0,0,0,0,1)$
$\displaystyle(0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,0)$
$\displaystyle(0,0,0,0,0,1,0,1,0,1,0,1,0,0,0,0)$
$\displaystyle(0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0)$
$\displaystyle(0,0,1,0,0,1,0,0,0,1,0,0,0,0,1,0)$
$\displaystyle(0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1)$
$\displaystyle(1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1)$
$\displaystyle(0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0)$
$\displaystyle(0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1)$
$\displaystyle(0,0,0,1,0,0,0,1,0,0,1,0,0,0,1,0)$
$\displaystyle(0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0)$
$\displaystyle(1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0)$
$\displaystyle(0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1)$
$\displaystyle(0,0,0,1,1,0,0,0,0,0,1,0,0,1,0,0)$
$\displaystyle(1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1)$
$\displaystyle(0,1,0,0,0,1,0,0,0,0,1,0,0,0,1,0)$
$\displaystyle(0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0)$
$\displaystyle(0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0)$
$\displaystyle(0,0,0,0,0,0,1,1,0,0,1,1,0,0,0,0)$
$\displaystyle(0,1,0,0,0,0,1,0,0,0,1,0,0,1,0,0)$
$\displaystyle(0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0)$
$\displaystyle(0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,1)$
$\displaystyle(0,0,0,0,0,0,0,0,1,0,1,0,0,1,0,1)$
$\displaystyle(1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0)$
$\displaystyle(0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,0)$
$\displaystyle(0,0,0,0,0,1,0,1,1,0,1,0,0,0,0,0)$
$\displaystyle(0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0)$
$\displaystyle(0,0,0,0,0,0,0,0,0,1,1,0,0,1,1,0)$
$\displaystyle(0,1,1,0,0,0,0,0,0,1,1,0,0,0,0,0)$
$\displaystyle(1,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0)$
$\displaystyle(0,0,0,0,1,0,0,1,0,1,1,0,0,0,0,0)$
$\displaystyle(0,0,0,0,0,1,1,0,0,1,1,0,0,0,0,0)$
$\displaystyle(0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0)$
$\displaystyle(0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1)$
$\displaystyle(1,0,0,0,0,0,0,1,0,0,0,1,1,0,0,0)$
$\displaystyle(1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0)$
$\displaystyle(1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0)$
$\displaystyle(0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,1)$
$\displaystyle(1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1)$
$\displaystyle(0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0)$
$\displaystyle(0,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0)$
$\displaystyle(0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1)$
$\displaystyle(0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0)$
$\displaystyle(0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0)$
$\displaystyle(0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,0)$
$\displaystyle(0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0)$
$\displaystyle(0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,0)$
$\displaystyle(0,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0)$
$\displaystyle(0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0)$
$\displaystyle(0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,1)$
$\displaystyle(0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1)$
$\displaystyle(0,0,0,1,0,0,0,1,1,0,0,0,1,0,0,0)$
$\displaystyle(1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0)$
$\displaystyle(0,1,0,0,0,1,0,0,1,0,0,0,1,0,0,0)$
$\displaystyle(0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0)$
$\displaystyle(0,0,0,0,0,0,0,0,0,1,0,1,1,0,1,0)$
$\displaystyle(0,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0)$
$\displaystyle(0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0)$
$\displaystyle(1,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0)$
$\displaystyle(0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0)$
$\displaystyle(0,0,0,0,0,0,0,0,1,1,0,0,1,1,0,0)$
$\displaystyle(0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0)$
$\displaystyle(0,1,0,0,0,0,0,1,0,0,1,0,1,0,0,0)$
$\displaystyle(0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0)$
$\displaystyle(0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0)$
$\displaystyle(1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0)$
$\displaystyle(0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0)$
$\displaystyle(0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,1)$

###  5.4 The group $2\cdot A_{4}$ with Quaternions and its lattice

As mentioned previously, Quaternions $\mathbb{H}$ is defined by

$\mathbb{H}=\left\\{z=x_{0}+x_{1}i+x_{2}j+x_{3}k\mid
x_{i}\in\mathbb{R}\right\\}$

And there is the subalgebra called Hurwitz Quaternionic integers

$\mathscr{H}=\\{a+ib+jc+kd:a,b,c,d\in\mathbb{Z}\text{ or
}a,b,c,d\in\mathbb{Z}+1/2\\}\subset\mathbb{H}$

The group generated below whose norm is $1$ is isomorphic to $2\cdot A_{4}$.

$<i,j,k,\omega>$

where $\omega=\frac{1+i+j+k}{2}$. Using the representation defined in the
previous section, the group can also be written as

$\left\langle\left(\begin{array}[]{rrrr}&1&&\\\ -1&&&\\\ &&&1\\\
&&-1&\end{array}\right),\left(\begin{array}[]{rrrr}&&1&\\\ &&&-1\\\ -1&&&\\\
&1&&\end{array}\right),\left(\begin{array}[]{rrrr}&&&-1\\\ &&-1&\\\ &1&&\\\
1&&&\end{array}\right),\frac{1}{2}\left(\begin{array}[]{rrrr}1&1&1&-1\\\
-1&1&-1&-1\\\ -1&1&1&1\\\ 1&1&-1&1\end{array}\right)\right\rangle$

Now consider the inverse map of the bijection defined below for each elements
of the group.

$\begin{array}[]{llllll}\phi:&(\mathbb{Z}\oplus\mathbb{Z}+1/2)^{4}&\rightarrow&<i,j,k,\omega>&\\\
&(c_{0},c_{1},c_{2},c_{3})&\mapsto&c_{0}+c_{1}i+c_{2}j+c_{3}k\end{array}$

for example

$\begin{array}[]{lllll}&\phi^{-1}:&i&\mapsto&(0,1,0,0)\\\
&\phi^{-1}:&j&\mapsto&(0,0,1,0)\\\ &\phi^{-1}:&k&\mapsto&(0,0,0,1)\\\
&\phi^{-1}:&\omega&\mapsto&1/2(1,1,1,1)\\\ \end{array}$

and the list of the elements is as follows

$\begin{array}[]{lll}\pm i\\\ \pm j\\\ \pm k\\\
\pm\omega&=\pm\frac{1}{2}(1+i+j+k)&=\mp\omega^{4}\\\
\pm\omega^{2}&=\pm\frac{1}{2}(-1+i+j+k)&=\mp\omega^{5}\\\ \pm\omega^{3}&=\pm
1\\\ \pm i\omega&=\pm\frac{1}{2}(-1+i-j+k)&=\pm\omega k\\\ \pm
j\omega&=\pm\frac{1}{2}(-1+i+j-k)&=\pm\omega i\\\ \pm
k\omega&=\pm\frac{1}{2}(-1-i+j+k)&=\pm\omega j\\\ \pm
i\omega^{2}&=\pm\frac{1}{2}(-1-i-j+k)&=\pm\omega^{2}j\\\ \pm
j\omega^{2}&=\pm\frac{1}{2}(-1+i-j-k)&=\pm\omega^{2}k\\\ \pm
k\omega^{2}&=\pm\frac{1}{2}(-1-i+j-k)&=\pm\omega^{2}i\end{array}$

This yields the following list of $24$ vectors.

$\displaystyle\left(\begin{array}[]{llll}1&0&0&0\end{array}\right)$
$\displaystyle\left(\begin{array}[]{llll}-1&0&0&0\end{array}\right)$
$\displaystyle\left(\begin{array}[]{llll}0&0&0&1\end{array}\right)$
$\displaystyle\left(\begin{array}[]{llll}0&0&0&-1\end{array}\right)$
$\displaystyle\left(\begin{array}[]{llll}0&0&1&0\end{array}\right)$
$\displaystyle\left(\begin{array}[]{llll}0&0&-1&0\end{array}\right)$
$\displaystyle\left(\begin{array}[]{llll}0&-1&0&0\end{array}\right)$
$\displaystyle\left(\begin{array}[]{llll}0&1&0&0\end{array}\right)$

$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}-1&-1&-1&-1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}1&1&1&1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}1&-1&1&-1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}-1&1&-1&1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}1&1&-1&-1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}-1&-1&1&1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}-1&1&1&-1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}1&-1&-1&1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}-1&1&1&1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}1&-1&-1&-1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}-1&1&-1&-1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}1&-1&1&1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}-1&-1&-1&1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}1&1&1&-1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}1&1&-1&1\end{array}\right)$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{llll}-1&-1&1&-1\end{array}\right)$

And these vectors form the $F_{4}$ lattice is spanned by the following
generator matrix

$M(F_{4})=\left(\begin{array}[]{rrrr}&1&-1&\\\ &&1&-1\\\ &&&1\\\
\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\end{array}\right)$

can also be spanned by

$\left(\begin{array}[]{cccc}\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\\
&1&&\\\ &&1&\\\ &&&1\end{array}\right)$

###  5.5 The icosian group order $120$ and $E_{8}$ Lattice

I introduced above the subalgebra that is isomorphic to the group $2\cdot
A_{4}$, actually we can also construct the group $2\cdot A_{5}$ with
quaternions (John Conway, Neil J. A. Sloane[1.P207]) defining $\tau$ and
$\sigma$ as

$\sigma=\frac{1-\sqrt{5}}{2}\quad\tau=\frac{1+\sqrt{5}}{2}$

the $\sigma$ and $\tau$ always satisfy

$\begin{array}[]{l}\sigma+\tau=-\sigma\tau=1\\\
\sigma^{2}=\sigma+1\quad\tau^{2}=\tau+1\end{array}$

as they are the answers of the equation $\lambda^{2}-\lambda-1=0$. Let
$x,\omega$ be the following.

$x=\frac{1}{2}(i+j\sigma+k\tau)\quad\omega=\frac{1}{2}(1+i+j+k)$

then there is homomorphism such that

$\begin{array}[]{lll}i&\mapsto&(1,2)(3,4)\\\ j&\mapsto&(1,3)(4,2)\\\
k&\mapsto&(1,4)(2,3)\\\ \omega&\mapsto&(2,3,4)\\\
x&\mapsto&(5,2)(3,4)\end{array}$

whose kernel is $\\{\pm 1\\}$.By simple calculation, you can easily verify the
following.

$x^{4}=\omega^{6}=i^{4}=j^{4}=k^{4}=(x\cdot i\cdot\omega)^{5}=(i\cdot
x\cdot\omega^{2})^{10}=1$

Let $\mathscr{I}$ be the ring formed by icosian, for all the elements
$v\in\mathscr{I}$ can be written as

$v=\alpha+\beta i+\gamma j+\delta k$

where $\alpha,\beta,\gamma,\delta\in\mathbb{Q}[\sqrt{5}]$. Thus it also can be
rewritten as

$v=(a_{1}+a_{2}\sqrt{5})+(a_{3}+a_{4}\sqrt{5})i+(a_{5}+a_{6}\sqrt{5})j+(a_{7}+a_{8}\sqrt{5})k$

where $a_{i}\in\mathbb{Q}$. Now consider the following mapping.

$\begin{array}[]{lllll}F:&\mathscr{I}&\rightarrow&\mathbb{R}^{4}\\\
&v=(a_{1}+a_{2}\sqrt{5})+(a_{3}+a_{4}\sqrt{5})i+\cdots+(a_{7}+a_{8}\sqrt{5})k&\mapsto&F(v)=(a_{1}+a_{2},a_{3}+a_{4},a_{5}+a_{6},a_{7}+a_{8})\end{array}$

In addition, the same mapping is applied to $\sigma v=(1-\sqrt{5})v/2$, then
the lattice formed by the following mapping is isomorphic to $E_{8}$

$\begin{array}[]{lllll}&\mathscr{I}^{2}&\rightarrow&\Lambda_{\mathscr{I}}\subset\mathbb{R}^{8}\\\
&(v,\sigma v)&\mapsto&F(v)\oplus F(\sigma v)\end{array}$

whoes the generator matrix is

$\frac{1}{2}\left(\begin{array}[]{llllllll}1&&&1&&1&&1\\\ 0&1&&1&&&1&1\\\
\vdots&&1&1&&1&1&\\\ &&&2&&&&\\\ &&&&1&1&1&1\\\ &&&&&2&&\\\ \vdots&&&&&&2&\\\
0&\cdots&&&&\cdots&0&2\end{array}\right)$

Here is some example, as for $\omega=\frac{1}{2}(1+i+j+k)$

$F(\omega)=(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$

and
$\sigma\omega=\frac{1}{4}\\{(1-\sqrt{5})+(1-\sqrt{5})i+(1-\sqrt{5})j+(1-\sqrt{5})k\\}$,
hence

$F(\sigma\omega)=(0,0,0,0)$

therefore

$F(\omega)\oplus
F(\sigma\omega)=(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},0,0,0,0)$

For $x$,from the definition

$\displaystyle x$ $\displaystyle=\frac{1}{2}(i+\sigma j+\tau k)$
$\displaystyle=\frac{1}{2}\left\\{i+\left(\frac{1-\sqrt{5}}{2}\right)j+\left(\frac{1+\sqrt{5}}{2}\right)k\right\\}$

Thus

$F(x)=(0,\frac{1}{2},0,\frac{1}{2})$

And

$\sigma
x=\left(\frac{1}{4}-\frac{\sqrt{5}}{4}\right)i+\left(\frac{3}{4}-\frac{\sqrt{5}}{4}\right)j-\frac{k}{2}$

Thus

$F(\sigma x)=(0,0,\frac{1}{2},-\frac{1}{2})$

therefore

$F(x)\oplus F(\sigma
x)=(0,\frac{1}{2},0,\frac{1}{2},0,0,\frac{1}{2},-\frac{1}{2})$

In the above way, we obtain the $120$ vectors whose norm is $1$.

###  5.6 On the Theta Functions of Lattices

As one of the simplest example, consider the lattice generated by

$M=\left(\begin{array}[]{ll}1&0\\\
0&1\end{array}\right)=\left(\begin{array}[]{l}e_{1}\\\
e_{2}\end{array}\right)$

the lattice is simply $\mathbb{Z}^{2}$

Figure 7: The lattice generated by $M$ which is isomorphic to $\mathbb{Z}^{2}$

The lattice spanned by the generator means that for all elements $x$ in the
laatice can be written as the combinations

$x=c_{1}e_{1}+c_{2}e_{2}=(c_{1}\quad c_{2})\cdot M$

where $c_{i}\in\mathbb{Z}$ and let $c=(c_{1}\quad c_{2})$. As the norm is
$N(x)=x\cdot{}^{t}x=c\cdot M\cdot{}^{t}(c\cdot M)$, for all norm of the
elements of the lattice are written as

$N(x)=c\cdot G\cdot{}^{t}c$

where $G$ is gram matrix $G:=M\cdot{}^{t}M$. In the case of the example, the
gram matrix is the identitly matrix. So the norm of all the elements in the
lattice can be written as

$N(x)=c\cdot{}^{t}c=c^{2}_{1}+c^{2}_{2}$

functions with the same number of norms of the lattice as coefficients can be
defined as follows

$\Theta(q)=\sum_{x\in\Lambda}q^{N(x)}$

where $\Lambda$ is the lattice spanned by the generators, and we call the
function is theta function of lattices.In this case, the function will be

$\Theta(q)=\sum_{x\in\Lambda}q^{N(x)}=\sum^{\infty}_{c_{1}=-\infty}\sum^{\infty}_{c_{2}=-\infty}q^{c^{2}_{1}+c^{2}_{2}}$

using Jacobi theta functions

$\displaystyle\theta_{2}(q)$
$\displaystyle=\sum_{n=-\infty}^{\infty}q^{(m+1/2)^{2}}$
$\displaystyle\theta_{3}(q)$
$\displaystyle=\sum_{n=-\infty}^{\infty}q^{m^{2}}$
$\displaystyle\theta_{4}(q)$
$\displaystyle=\sum_{n=-\infty}^{\infty}(-q)^{m^{2}}$

the theta function of the lattice is rewritten as

$\Theta(q)=\theta_{3}(q)^{2}=1+4q+4q^{2}+4q^{4}+8q^{5}+4q^{8}+4q^{9}+8q^{10}+\cdots$

from the definition, the coefficients are equal to the numbers of coordinates
on the same circle in the following figure.

Figure 8: Relation between vectors of the same distance.

## References

  * [1] Sphere Packings, Lattices and Groups [John Conway, Neil J. A. Sloane]
  * [2] ATLAS: Weyl group W(F4), Orthogonal group GO4+(3) (https://brauer.maths.qmul.ac.uk/Atlas/misc/WF4/)
  * [3] Orthogonal group O8+(2) (https://webspace.maths.qmul.ac.uk/j.n.bray/web/Pres/O8p2.html)
  * [4] Introduction to elliptic curves and modular forms [N Koblitz]


# Serverless Architecture for Bulk Email Management

Bazaru Priyatham Sai Chand  Dept. of Computer Science
Mahatma Gandhi Institute of Technology
Hyderabad, Telangana
Email<EMAIL_ADDRESS>
###### Abstract

Sending emails in large quantities can be tedious considering free services do
not cover bulk email and paid services can be costly and are not easy to
customize. Traditional email client used for basic emailing services fail to
be useful in larger volumes of emails to target people or spread information
to consented individuals. This paper proposes a serverless architecture to
tackle such problems by using one such offering from the Amazon Web
Services(AWS) which can be easily replaced by a software architects choice of
service. The constraints help to make an architecture using components that
can fit most of the needs of a serverless backend and extend it to scenarios
such mobile notifications, One Time Password (OTP) systems or other means of
communication to minimize single point of failure and also decrease the
dependency on physical servers for such operations offering a comparable
solution within the cloud. The architecture proposed is tested to find the
time taken to send the emails of various quantities and see how it affects the
cost. The architecture was successful able to send multiple emails in a quick
and single invocation and has demonstrated a higher level of scalability
compared to conventional methods.

##  I Introduction

Email has become one of the most important means of communications.Sending
emails through conventional software like Gmail is limited to a mere thousands
which is not sufficient to fit into the business needs and managing the
insights of sent emails is crucial to companies to further enhance their
approach.

Serverless computing is any computing platform that hides server usage from
developers and runs code on-demand automatically scaled and billed only for
the time the code is running[1].This reduces the time and knowledge required
to deploy pieces of software while offering significant reduction in pricing
compared to hosting the software on dedicated hardware.Various cloud platforms
such as AWS, Microsoft Azure and Google Cloud Platform have adapted to this
way of thinking making it easily accessible.A robust serverless architecture
for sending bulk emails of the magnitude of thousands to millions using
scalable model which is implemented using AWS lambda[2] and Simple Email
Service(SES)[3] for the SMTP protocol. Challenges such as able to trace the
bounce emails as well as the statistics for the emails sent. The prototype is
built using the React framework along with underlying services from AWS. These
tightly integrated services offer the best way to simulate the architecture
and help in sending customized for every recepient based on email templates
along with storing bounce emails.

The effectiveness of the scalability offered by the architecture is analyzed
using X-ray traces to simulate and test the bottlenecks and how to overcome
them is discussed in the paper.With serverless technology, the cloud provider
abstracts away the server management, provisioning servers with fine
granularity, on demand, and with a pay-per-use model[4].

This paper proposes an architecture for sending and management of the emails
through serverless application using the AWS services. The paper outlines the
accuracy and the reliability of the design and future aspects for this
architecture.

##  II Overview of bulk email management system

The bulk email management system is used to send emials of magnitude in
thousands and more and broadcast the message to multiple email addresses with
a single operation. All the required fields such as the senders’ email
addresses, subject, the html body and to addresses are selected and the
sending is initiated with a button. Upon successful sending to the API
endpoint the emails are processed and send through the AWS simple email
services to all the users and the emails that were bounced are returned after
the sending operation.It makes use of a Javascript based framework called
ReactJS to input the fields and send it to the API hosted on the AWS service
where an array of services are used to scale and perform the sending and
retrieving of bounces.

Figure 1: Architecture for bulk email management

##  III Architecture Design

The architecture demonstrated in the Fig 1 with the main functionalities
include sending the emails and tracking the bounces. Initially, all the data
sent to the lambda using the API gateway which handles the request and
response for the whole operation. Upon reaching the lambda, JavaScript Object
Notation (JSON) objects from the string parameters preprocessed, the html part
of the email along with the subject made into SES templates accessed by other
lambdas and the emails batched no more than 50 emails per batch and placed in
a Simple queue Service[5] queue. The ordering of the messages is important
hence, we use a First In First Out queue, which triggers another lambda that
sends emails to all the 50 recipients using a templated email function by
accessing the template that we created earlier. We set a flag to mark the end
of the batches deleting the template by the end of the execution.

The emails use a configuration set that triggers events such as
bounce,complaints or seen to other services for logging and other analysis.
The configuration set sends all the bounce receipts to SNS which triggers a
lambda function that stores the receipts useful information as a document in a
NoSQL database such as dyanamoDB once all the emails sent and all bounce
receipts stored in the database. Another lambda triggers using an API
retrieves all the records and sends the bounced emails as response to API
gateway which forwards it the application shown to the user.

The emails that are bounced are identified by the configuration set and are
sent to the SNS with email address and the type of bounce which invokes a
lambda trigger that stores the Simple Notification
Service(SNS)[6]notifications after filtering the message into the dynamo DB.

All the bounced emails are automatically updated to a email supression list
which saves the names of bounced emails and accepts them but doesn’t send the
emails after it is declared bounce once. Thereby, saving the bounce rate of
the service.

Additionally, All the other statistics are sent to Cloudwatch metrics which
can be used to analysis the statistics of sending emails at different time
frames and track the bounce rate and can be useful in finding out the
effectiveness of the campaign by the number of people who opened the email.

The architecture keeping in mind the constraints on SES service. The
advantages of this is two folds. It helps to work within the limits of the
service. Morever, these constraints can be fine tuned to deal with the limits
of SMTP server if an independent server or other services are used.

##  IV Constraints

The architecture is based a few constraints from the SES service and other
bottlenecks that affect the sending of emails. The SES through which the
emails are initially sent through sandbox to test and later needs to be
upgraded by the AWS authority to permit the user to request increase in the
number of emails to be sent. For the purpose of this architecture 50,000
emails have been requested which was granted by the authority.

  * •

14 emails/second

  * •

50,000 emails per day

  * •

request data 5kb

There are two performance monitoring services available to AWS application
developers: CloudWatch and X-ray.

##  V Analysis

Basic message transfer from Author to Recipients is accomplished by using an
asynchronous store-and-forward communication infrastructure in a sequence of
independent transmissions through some number of mail transfer agents[7]. Two
of the most common ways to analyze the architecture was to consider the time
required to send all the emails and the cost associated with it. Both of the
metrics are analysed in the following section.

###  V-A Performance

There are two performance monitoring services available to AWS application
developers: CloudWatch and X-ray.[8] The application is analyzed to see the
time required to send the emails and the time taken to run the functions
between the request and response to the API endpoint. This can be seen through
the cloudwatch[9] logs and X-ray to find out the issues with latency over
various regions and finding the best regions and how to optimize the
functions.In this case, we used X-Ray as a monitoring and display service that
automatically samples the entry and exit of function instances, called
segments, using unique trace identifiers (trace_id)[8][10].

Here we take the case of sending multiple emails and the time taken is sampled
over 30 data points to represent in Figure 2. The time is calculated by
considering the cloudwatch logs that can be used to determine the time taken
to execute the last function of lambda as show in Figure 1 to send the email
and the time leading to that function from the start of the API gateway
request log that is stored in the logs. This experimented several times to
cancel out the variations of cold start that is a common phenomenon in
serverless functions. All the calculations are done to avoid any overlap
between different invocations and only a single request is sent within the
same region.

Figure 2: Time taken to execute and return the response to the number of
emails that are sent.

The number of emails sent is directly proportional to the time taken. The time
taken for 10,00 emails averaging at around 30 minutes is sufficient and
satisfactory considering that the time taken is to derived from the last email
that is sent in the list of emails.

###  V-B Cost

The cost of the services can also be a motivating factor to implement such an
architecture. In our use case The following architecture can be tested with
the AWS free tier and later expanded to the needs of the users. With $0.01
dollars per 1,000 emails but one needs to be wary about the associated
services like the SES, SNS and the cloudwatch metrics to analyze the emails
add up to cost but since they are calculated based on the GB of data it needs
to pre-process the rates will not exceed the cost of emails.

Here we take the example, provided by the SES platform to estimate the cost
required to implement the architecture.

You use Amazon SES to send about 250,000 emails per month. You receive 1,000
emails per month. You don’t use dedicated IP addresses. Every message you send
and receive is 32KB in size which results in a total of $25.98 per month [11]
which is significantly less than competitors such as SendGrid or MailChimp who
offer their own SMTP server or schedule emails to fit into the constraints of
other providers to carry out email campaigns.

Amazon provides sample pricing calculations,but as your workload varies, so
will the billing.[12] The SQS requests can exceed the free tier if not
monitored carefully and add up to additional costs for the next 1000 requests
or more based on the usage of the system.

##  VI Conclusion

To conclude, the architecture has been effective in sending bulk emails of
various volumes with runtime that can be considered as sufficient or optimal
and cost-effective. The primary goal of the architecture to be able to manage
the entire system without a server being provisioned directly i.e. serverless
is achieved and is promising to further develop and enhance its capabilities.
The statistics displayed can be used for reach through the customers and the
key idea of the architecture is to make use of serverless functions and
services to manage emails with an optional fronted interface which was the
case in this paper. Thus, making it a fast application to deliver emails to
the users’ inboxes in a short period of time. Without provisioning any IP
addresses and acquiring the compute power needed to send the emails beforehand
thereby making it more effective and possesses a higher level of abstraction
that the conventional method.

## References

  * [1] P. Castro, V. Ishakian, V. Muthusamy, and A. Slominski, “The rise of serverless computing,” _Communications of the ACM_ , vol. 62, no. 12, pp. 44–54, 2019.
  * [2] Aws lambda. [Online]. Available: https://aws.amazon.com/lambda/
  * [3] Amazon simple email service. [Online]. Available: https://aws.amazon.com/ses/
  * [4] S. Eismann, J. Scheuner, E. Van Eyk, M. Schwinger, J. Grohmann, N. Herbst, C. L. Abad, and A. Iosup, “Serverless applications: Why, when, and how?” _IEEE Software_ , vol. 38, no. 1, pp. 32–39, 2020.
  * [5] Amazon simple queue service. [Online]. Available: https://aws.amazon.com/sqs/
  * [6] Amazon simple notification service. [Online]. Available: https://aws.amazon.com/sns/
  * [7] D. Crocker, “Internet Mail Architecture,” RFC 5598, Jul. 2009. [Online]. Available: https://rfc-editor.org/rfc/rfc5598.txt
  * [8] W.-T. Lin, C. Krintz, R. Wolski, M. Zhang, X. Cai, T. Li, and W. Xu, “Tracking causal order in aws lambda applications,” in _2018 IEEE International Conference on Cloud Engineering (IC2E)_. IEEE, 2018, pp. 50–60.
  * [9] Amazon cloudwatch. [Online]. Available: https://aws.amazon.com/cloudwatch/
  * [10] Amazon x-ray. [Online]. Available: https://aws.amazon.com/xray/
  * [11] Amazon ses pricing. [Online]. Available: https://aws.amazon.com/ses/pricing/
  * [12] A. Eivy and J. Weinman, “Be wary of the economics of” serverless” cloud computing,” _IEEE Cloud Computing_ , vol. 4, no. 2, pp. 6–12, 2017.