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Error Control Coding in Wireless Technologies: Reed-Solomon and Cyclic Codes Explained

This document serves as background material on error control coding, focusing on Reed-Solomon and cyclic codes within wireless communication technologies. It covers algebra over finite fields (GF(pm)), including representations, irreducible polynomials, and the operations of addition and multiplication in GF(4) and GF(8). Key examples, including multiplication using power tables and shift registers, illustrate practical applications of these codes. The document also includes expected quiz questions for reinforcement of the concepts presented.

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Error Control Coding in Wireless Technologies: Reed-Solomon and Cyclic Codes Explained

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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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