1. Error control coding for wireless communication technologies
Background material for Reed- Solomon and cyclic codes
EU-USA Atlantis Programme
FIT & Budapest University of Technology and Economics

2. p is prime number and given an irreducible polynom p(y) of degree m
Algebra over GF(pm)
p is prime number and given an irreducible polynom p(y) of degree m
Field representation
Elements
p-ary representations
Polynom 1 .

3. p is prime number and given an irreducible polynom p(y) of degree m
Algebra over GF(pm)
p is prime number and given an irreducible polynom p(y) of degree m
Field representation
Elements
p-ary representations
Polynom 1 . op

4. „Big” Field and „Small” Field
Algebra over „Big” Field is reduced to the algebra over the „Small” Field !
ops on coefficents according to mod p

5. Algebra over GF(4) Field representation Irreducible polynom
Elements of GF(4)
Binary representation
Polynomial representation
(00) 1
(01) 2
(10) 3
(11)

6. Addition over GF(4) E.g.: Elements of GF(4) Binary representation
Polynomial representation
(00) 1
(01) 2
(10) 3
(11)
E.g.: + 1 2 3

7. Multiplication over GF(4)
Elements of GF(4)
Binary representation
Polynomial representation
(00) 1
(01) 2
(10) 3
(11)
E.g.: * 1 2 3

8. The primitive element of GF(4) and the power table
Elements of GF(4)
Binary representation
Polynomial representation
(00) 1
(01) 2
(10) 3
(11)
E.g.:
Power of the primitive elment 1 2 3

9. Representation of GF(8)
Elements of GF(8)
Binary representation
Polynomial representation
(000) 1
(001) 2
(010) 3
(011) 4
(100) 5
(101) 6
(110) 7
(111)

10. The power table Elements of GF(8) Polynomial form Prim.el. 1 2 y 3 y+11 2 y 3
y+1 4 5 6 7
E.g.

11. Multiplication by using the power table
Elements of GF(4)
Polynomial form
Prim.el. 1 2 y 3
y+1 4 5 6 7
E.g.

12. Multiplication by Shift Registers over GF(8)
E.g. multiply two with a general element
From the power table we know that this is y+1
In the next tick of the clock signal

13. Example: multiplying 2 with 6 over GF(8)
In the next tick of the clock signal
Indeed: 2*6=7 over GF(8)

14. Multiplication by Shift Registers over GF(8)
E.g. multiply four with a general element
From the power table we know that this is
From the power table we know that this is y+1
In the next tick of the clock signal

15. Multiplication of 4 with 6 over GF(8)
In the next tick of the clock signal
Indeed 4*6=5 over GF(8)

16. Suggested readings
D. Costello: Error control codes, Wiley, 2005, Chapter 2

17. Expected Quiz questions
Given a generator polynom of cyclic RS code and a message vector, generate the correponding codeword by polynom multiplication !
Carry out a multiplication over G(8) by using shift register.

18. Thank you for your attention !