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Error control coding for wireless communication technologies

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( p m ) .

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Error control coding for wireless communication technologies

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  1. Errorcontrolcodingforwireless communicationtechnologies Backgroundmaterialfor Reed- Solomon and cycliccodes EU-USA Atlantis Programme FIT & Budapest University of Technology and Economics

  2. Algebra over GF(pm) p is primenumber and given an irreduciblepolynomp(y) of degreem Fieldrepresentation

  3. Algebra over GF(pm) p is primenumber and given an irreduciblepolynomp(y) of degreem Fieldrepresentation op

  4. „Big” Field and „Small” Field Algebra over „Big” Field is reducedtothe algebra over the „Small” Field ! opsoncoeffficentsaccrodingtomodp

  5. Algebra over GF(4) Irreduciblepolynom Fieldrepresentation

  6. Addition over GF(4) E.g.:

  7. Multiplication over GF(4) E.g.:

  8. The primitive element of GF(4) and the power table E.g.:

  9. Representation of GF(8)

  10. The power table E.g.

  11. Multiplication by using the power table 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. Suggestedreadings D. Costello: Errorcontrolcodes, Wiley, 2005, Chapter2

  17. ExpectedQuizquestions Given a generatorpolynom of cyclic RS code and a messagevector, generatethecorrepondingcodewordbypolynommultiplication ! Carry out a multiplication over G(8) byusing shift register.

  18. Thank you for your attention !

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