COMM 604
Channel Coding
Lecture 1
Algebra of Finite Fields
Dr. Engy Aly Maher
Spring 2020
Binary Operation
G is a set of elements “*” A binary operation
on G is a rule that
assigns to each pair of
elements a and b a
uniquely defined element
c
a ,b G
c a* b
if c G G is closed under “*”
Groups
A set G on which a binary operation “*” is defined is called a Group if:
i. The binary operation is associative
ii. G contains an identity element e
(a *e = e *a = a)
iii. For any element a in G, there exists an inverse element a’
in G
(a *a’ = a’ *a = e)
Commutative Group G if for any a and b in G : a *b = b *a
Theorems
The identity element in a group G is unique
Proof If we have two identity elements e and e’ in G, Then,
e’ =e’ * e =e e, e’ are identical
The inverse of any element in a group G is unique
If we have two inverse elements a’ and a’’ for a in
G, Then,
a’ =a’ *e =a’ *(a*a’’) = (a’ *a) *a’’ = e *a’’ = a’’ a’, a’’ are identical
Proof
Examples
The set of all integers is a commutative group under
addition.
Identity element = 0. Inverse of a = -a.
The set of all rational numbers except zero is a commutative
group under multiplication.
Identity element = 1. Inverse of a = a-1 = 1/a.
The above groups contain infinite number of elements.
Example: Modulo-2 Addition
The set G ={0,1} is a group of order 2 under modulo-2 addition
Modulo-2 Addition
0 0 0
0 1 1
1 0 1
1 1 0
i. Modulo-2 addition is
associative
ii. The identity element is 0
iii. The inverse of 0 is 0 in G
The inverse of 1 is 1 in G
Example: Modulo-m Addition
The set G ={0,1,2,…,m-1} is a group of order munder modulo-m addition
– mod-5 addition
Example: Modulo-m Addition
The set G ={0,1,2,…,m-1} is a group of order munder modulo-m addition
Modulo-mAddition
i+j= qm+r,
0≤ r< m-1
+i j =r
i. Modulo-m addition is
associative
ii. The identity element is 0
iii. The inverse of i is m-i in G
Example: Modulo-p Multiplication
G={1,2,…,p-1}, p is a prime number, is a group
of order p under modulo-p multiplication
Modulo-p Multiplication
i.j =qp +r, 0 ≤ r
Subgroup
Define a set G as a group under a binary operation *, A subset H is called a subgroup if
i. H is closed under the binary operation *
ii. For any element a in H, the inverse of a is also in H
Example:
Let G be the set of rational numbers constitute a group under
real addition. Therefore,
The set of integers H is a proper (i.e., H ≠G) subgroup
under real addition
Example
G ={0,1,2,…,15} under modulo-16 addition
H ={0,4,8,12} is a subgroup of G why?
The coset
+3 H ={3,7,11,15}= +7 H
Four Distinct and Disjoint Cosets of H
+0 H ={0,4,8,12}
+1 H ={1,5,9,13}
+2 H ={2,6,10,14}+3 H ={3,7,11,15}
Theorem (Read Only)
Let H be a subgroup of a group G with binary operation *.
No two elements in a Coset of H are identical
* *
Suppose * , * are identical where
Given denotes the inverse of , then
* * * *
* *
(Contradiction)
-
a G
a H a h : h H
a h a h' h h'
a a
a a h a a h'
e h e h'
h h'
1
1 1
Theorem (Read Only)
No two elements in two
different Cosets of a
subgroup H of a group G
are identical
* *
Suppose * * * *
If * = *
* * * *
* *
* * *
* * *
* *
* * Contradiction
a H b H , a , b G
a h a H , b h' b H
a h b h'
a h h b h' h
a b h'', h'' h' h
a H b h'' H
a H b h'' h : h H
a H b h''' : h''' H
a H b H
1 1
1
Properties of Cosets
i. Every element in G appears in one and only one of distinct
Cosets of H
ii. All the distinct Cosets of H are disjoint
iii. The union of all distinct Cosets of H forms the group G
Fields
Let F be a set of elements on which two binary operations
called addition “+” and multiplication “.” are defined. The
set F and the two binary operations represent a field if:
i. F is a commutative group under addition. The identity element with
respect to addition is called the zero element (denoted by 0)
ii. The set of nonzero elements in F is a commutative group under
multiplication. The identity element with respect to multiplication is
called the unit element (denoted the 1 element)
iii. Multiplication is distributive over addition:
a.(b+c) = a.b + a.c, a, b, c in F
Fields
Let F be a set of elements on which two binary operations
called addition “+” and multiplication “.” are defined. The
set F and the two binary operations represent a field if:
Basic Properties of Fields
a.0=0.a=0
If a,b≠0, a.b≠0
a.b=0 and a≠0 imply that b=0
-(a.b)=(-a).b=a.(-b)
If a≠0, a.b=a.c imply that b=c
Binary Field GF(2)
+ 0 1
0 0 1
1 1 0
. 0 1
0 0 0
1 0 1
Modulo-2 Addition Modulo-2 Multiplication
F={0,1} is a Finite field of order 2 under modulo-2 addition and modulo-
2 multiplication
Galois Field of the order 2
GF (7)