Error control coding for wireless communication technologies Background
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
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 0 0 1 1 . .
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 0 0 1 1 . . op
„Big” Field and „Small” Field ops on coefficents according to mod p Algebra over „Big” Field is reduced to the algebra over the „Small” Field !
Algebra over GF(4) Irreducible polynom Field representation Elements of GF(4) Binary representation 0 (00) 1 (01) 2 (10) 3 (11) Polynomial representation
Addition over GF(4) Elements of GF(4) Binary representation 0 (00) 1 (01) 2 (10) 3 (11) + 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 Polynomial representation E. g. :
Multiplication over GF(4) Elements of GF(4) Binary representation 0 (00) 1 (01) 2 (10) 3 (11) * 0 1 2 3 0 0 0 1 2 3 2 0 2 3 1 3 0 3 1 2 Polynomial representation E. g. :
The primitive element of GF(4) and the power table Elements of GF(4) Binary representation 0 (00) 1 (01) 2 (10) 3 (11) Polynomial representation E. g. : Power of the primitive elment 0 0 1 1 2 3
Representation of GF(8) Elements of GF(8) Binary representation 0 (000) 1 (001) 2 (010) 3 (011) 4 (100) 5 (101) 6 (110) 7 (111) Polynomial representation
The power table Elements of GF(8) Polynomial form 0 0 1 1 2 y 3 y+1 4 5 6 7 E. g. Prim. el.
Multiplication by using the power table Elements of GF(4) Polynomial form 0 0 1 1 2 y 3 y+1 4 5 6 7 E. g. Prim. el.
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 y+1 In the next tick of the clock signal From the power table we know that this is
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 1. Given a generator polynom of cyclic RS code and a message vector, generate the correponding codeword by polynom multiplication ! 2. Carry out a multiplication over G(8) by using shift register.
Thank you for your attention !
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