Introduction
The aim of this work is to show how central simple algebras, and their arith-
metics, can be a useful tool in modelling, encoding, decoding and detecting
interesting proprieties of wireless communication.
The basic concept is that the generic signal received through a wireless chan-
nel can be expressed, in a matricial form, as
Y =HX +V
where Y is the received signal, X is the sent signal, V is the noise and H is
the fading matrix.
The main problem, in wireless communication, is given by the presence of
the fading matrix, that is intrinsic in the nature of this kind of trasmissions.
So it is necessary to send X in a particular form. For example we want that
multiplication H X is not zero, assuming X;H6= 0, or that the determi-
nants of two distinct matrices are di erent; these, and other, problems nd
a natural resolution in theory of algebras.
In fact it is possible to embed central simple algebras, in the matrices al-
gebras and nd algebras that do not have zero-divisors and with non-zero
determinant, the division algebras, using the results of this theory.
Another problem arises from the fact that the received signals are points in
the space of matrices, so we want some discrete structures that plays the
same role of the nite-dimensional vector-spaces, over nite elds, in the
standard error-correcting codes theory. To solve this problem we introduce
orders and various concepts of algebraic number theory.
We present also some methods for decoding; we do not concentrate on this
1
Introduction Michele Di Nocera
part because they are principally brute force algorithms that, in the practice,
are supported with probabilistic algorithm.
All the theoretical results that we present will be used principally for the
planning of the problem, and to simplify some calculations in the decoding
that is treated in a \ingegneristic" way that is not very satisfactory from the
theoretical point of view.
As B. A. Sethuraman says: \There is as yet no deep independent \mathe-
matics of space-time codes": the driving force behind the subject consists of
fundamental engineering problems that need to be solved before MIMO wire-
less communication reaches its full practical potential, particularly for three
or more antennas. This author therefore believes that, as things stand now,
isolated mathematical investigations of space-time codes that are not grounded
in concrete engineering questions would very likely lead to sterile results. At
least for now, mathematicians can best contribute to the subject by working
in collaboration with engineers who are motivated by fundamental engineer-
ing questions.[...] There is clearly a lot of work for mathematicians to do:
particularly in decoding systems with large numbers of receive and transmit
antennas, but also in other areas of MIMO communication.".[8]
Let us see more in detail what we are going to show.
This work is divided in three main parts:
Chapter 1 Theoretical part
Chapter 2
Chapter 3
Chapter 4
9
>
=
>
;
Applicative part
Chapter 5 Computational part
In the rst part we give the principal results about the theory of central
simple algebras theory and rings.
We shall see how to embed central simple algebras in matrices algebras over
a suitable eld. Then we shall introduce the important concepts of degree,
index and exponent of a given algebra and we shall see how they work with
the tensor product.
2
Michele Di Nocera Introduction
In the part concerning the rings we shall focus our attention on the costruc-
tions of lattices and orders and on the factorization of primes in integer
extensions.
In the applicative part we shall see how to model wireless communication,
using matrices, and how the theory of algebras and rings can help us to de-
tect interesting proprieties.
In chapter 2 we shall give the de nition of algebra-based code and we intro-
duce the concepts of minimum determinant and of fully-diverse code, and we
do some explicit calculations over some speci c division algebras in order to
see how we can build up a code.
In chapter 3 we shall use all the theoretical part to obtain some important
results about algebra-based codes.
We shall see how to prevent that the determinant of a given algebra of ma-
trices decrease arbitrarily using the orders built up from the rings of integers.
We shall show how to recognize, in a simple computational way, a division
algebra using the factorization of the prime ideals over the rings of integers.
Furthermore we shall study some conditions to have information lossless
code, i.e. codes that maximize the capacity, and we introduce perfect codes,
i.e. codes that are fully-diverse, information lossless, with non-zero determi-
nant and built over a cyclic algebra.
In chapter 4 we shall show how to de ne a distance in the space of matrices
and how to use the previous results for coding and decoding.
We shall present three kind of decoding:
Maximum Likehood
Zero Forcing
Sphere Decoding
In the last chapter we present a list of routines, written in Mathematica,
developed during this work; although it is a minor part in this thesis, this
chapter can be useful in the implementation of the results presented in all
the work.
3
Chapter 1
Preliminaries
In all the work we follow principally [2].
In this section we shall give the principal de nitions and results that we use
in the others section.
1.1 Some results on Algebras
For the rst part of this section we shall refer principally to [1].
We use R to intend a ring with unity.
Let us begin with the de nitions of R-module and R-algebra.
De nition 1.1.1. M is an R-module iff M is an abelian group togheter
with the scalar multiplication for the elements of R:
R M ! M
(r;m) 7! rm
that satis es the following rules:
(r +s)m =rm +sm for all r;s2R and m2M
r(m
1
+m
2
) =rm
1
+rm
2
for all r2R and m
1
;m
2
2M
(rs)m =r(sm) for all r;s2R and m2M
1
R
m =m for all m2M
4
Michele Di Nocera 1.1 Some results on Algebras
It is immediate from the de nition that:
everyK-vector space, withK eld, is a K-module
every group is aZ-module
the additive group of M
n
(R), with R ring, is an R-module
De nition 1.1.2. M
0
is a sub-R-module ofM iff M
0
is a subset ofM and
is an R-module with the structure given by M.
De nition 1.1.3. ’ is an R-module homomorphism iff ’ : M
1
! M
2
is
a group homomorphism between two R-modules with the additive rule
’(rm) =r’(m) with r2R;m2M
1
De nition 1.1.4. LetM be anR-module, andS a subset ofM. S generates
M iff there exist r
1
;:::;r
n
2R, s
1
;:::;s
n
2S such that every m2M can
be written as m =
P
n
i=1
r
i
s
i
De nition 1.1.5. Let X be a set. M is a free R-module with basis X iff
M =f’ :X!R :’(x) = 0 for almost all x2Xg
with the scalar multiplication given by (r’)(x) =r(’(x)).
Furthermore we can de ne the rank of M as rk(M) =card(X).
De nition 1.1.6. Let M;N;P be R-modules, and ’ :M N!P a func-
tion. We say that ’ is R-bilinear iff
’(m
1
+m
2
;n) =’(m
1
;n) +’(m
2
;n) for m
1
;m
2
2M;n2N
’(m;n
1
+n
2
) =’(m;n
1
) +’(m;n
2
) for m2M;n
1
;n
2
2N
’(rm;n) =’(m;rn) =r’(m;n) for m2M;n2N;r2R
De nition 1.1.7. A is an R-algebra iff A is an R-module with an appli-
cation
: A A ! A
(a;b) 7! a b
such that
5
1.1 Some results on Algebras Michele Di Nocera
(a b) c =a (b c) for a;b;c2A
a (b +c) = (a b) + (a c) for a;b;c2A
(a +b) c = (a c) + (b c) for a;b;c2A
r(a b) = (ra) b =a (rb) for r2R;a;b2A
91
A
2A such that a 1
A
= 1
A
a =a for a2A
In other words an R-algebra is an R-module that is also a ring.
It is clear from the de nition that:
every ring R is an R-algebra
every ring is aZ-algebra
M
n
(R) is an R-algebra
Remark 1.1.8. We shall talk about an ideal of anR-algebraA meaning that
is an ideal ofA as ring. It is clear that every ideal ofA is a sub-R-module
ofA as R-module.
De nition 1.1.9. A
0
is a sub-R-algebra ofA iff A
0
is a subset ofA and
is an R-algebra with the structure given byA.
De nition 1.1.10. ’ is an homomorphism of R-algebras iff ’ is an ho-
momorphism of R-modules and of rings.
We can also de ne:
’ is an isomorphism iff ’ is an homomorphism and is bijective
’ is an endomorphism iff ’ is an homomorphism and ’ :A!A
withA R-algebra
’ is an automorphism iff ’ is an endomorphism and is bijective
Remark 1.1.11. Given anR-moduleM, we can considerEnd
R
(M), the set
of the R-linear endomorphisms of M.
It is easy to verify that it is anR-module, inheriting the scalar multiplication
6
Michele Di Nocera 1.1 Some results on Algebras
thanks to theR-linearity, and it is anR-algebra considering the composition.
If M is a nitely generated R-module of rank n, we can build up an isomor-
phism of R-algebras between End
R
(M) and M
n
(R).
Furthermore, ifA is an R-algebra and a2A, we can de ne the left multi-
plication for a by
m
a
: A ! A
b 7! ab
that is an endomorphism R-linear ofA since m
ab
=m
a
m
b
.
So we can de ne the map
m : A ! End
R
(A)
a 7! m
a
that is an injection.
De nition 1.1.12. Given an R-algebraA we de ne
Z(A) =fa2A :ab =ba 8b2Ag and we call it center ofA.
Given an R-algebraA and a sub-R-algebraB we de ne
Z
B
(A) =fb2B :ab =ba 8a2Ag and we call it centralizer ofA inB.
We shall consider principally algebras over elds and we underline that
using the notationK;L for elds.
Remark 1.1.13. Given aK-algebraA, it is immediate thatK Z (A).
De nition 1.1.14. GivenA aK-algebra:
A is central iff K =Z(A)
A is simple iff A does not have proper bilaterals ideals
A is a division algebra iff every non-zero element ofA is invertible
A is split iff A
=M
n
(K) for some n2N
Remark 1.1.15. In the same way we say that an R-module M is simple
iff there are not proper sub-R-modules of M.
7
1.1 Some results on Algebras Michele Di Nocera
Example 1.1.16. We want to show that M
n
(K) is a central simple K-
algebra. Notice that a K-basis is given by the setfE
ij
g
n
i;j=1
, where E
ij
is
the matrix with one in the position (i;j) and zero in the other.
E
ij
=
0
B
B
B
B
B
B
B
B
B
B
B
@
j
0 ::: ::: 0 ::: ::: 0
.
.
.
.
.
.
.
.
.
.
.
. 0
.
.
.
i 0 ::: 0 1 0 ::: 0
.
.
. 0
.
.
.
.
.
.
.
.
.
.
.
.
0 ::: ::: 0 ::: ::: 0
1
C
C
C
C
C
C
C
C
C
C
C
A
The multiplication over this basis is de ned by:
E
ij
E
kt
=
(
0 iff j6=k
E
it
iff j =k
So the generic matrix M2M
n
(K) can be written as
M =
n
X
i;j=1
ij
E
ij
with ij
2K
Let us see the multiplication of the generic matrix with an element of the
basis
M E
kt
=
n
X
i;j=1
ij
E
ij
E
kt
=
n
X
i=1
ik
E
it
E
kt
M =
n
X
i;j=1
ij
E
kt
E
ij
=
n
X
j=1
tj
E
kj
Let us prove the centrality.
We have thatZ(M
n
(K)) K, remembering that we use the natural embed-
ding
K ,! M
n
(K)
7! I
n
8
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