Block codes - Codes and Turbo Codes - page 165

Degree of the polynomial Primitive polynomial

α 2 + α +1

2

α 3 + α +1

3

α 4 + α +1

4

α 5 + α 2 +1

5

α 6 + α +1 6

α 7 + α 3 +1

7

α 8 + α 4 + α 3 + α 2 +1 8

α 9 + α 4 +1 9

α 10 + α 3 +1 10

Table 4.9 - Examples of primitive polynomials

Galois field F 16 being made up of 16 elements, the binary representation of
an element of this field is done with the help of 4 binary symbols belonging to
F 2 . These 4 symbols are equal to the values taken by coecients a , b , c and d
respectively.

Elements
of the
field

Polynomial
representa-
tion

Binary representation

0 0 0000

1 1 0001

α α 0010

α 2 α 2 0100

α 3 α 3

1000

α 4 α +1 0011

α 5 α 2 + α 0110

α 6 α 3 + α 2 1100

α 7 α 3 + α +1 1011

α 8 α 2 +1

0101

α 9 α 3 + α 1010

α 10 α 2 + α +1

0111

α 11 α 3 + α 2 + α

1110

α 12 α 3 + α 2 + α +1

1111

α 13 α 3 + α 2 +1

1101

α 14 α 3 +1

1001

Table 4.10 - Different representations of the elements of Galois field F 16