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

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 .

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

„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

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

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

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

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

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)

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

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.

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

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

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

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

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

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.

Thank you for your attention !