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)