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Reed Solomon Code

Reed Solomon Code. Doug Young Suh suh@khu.ac.kr Last updated : Aug 1, 2009. Basic number theory GF(2 3 ). Primitive function. GF(2 3 ) in 2 Operations. Addition table multiplication table. Reed-Solomon Encoding(1). k : number of data digits

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Reed Solomon Code

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  1. Reed Solomon Code Doug Young Suh suh@khu.ac.kr Last updated : Aug 1, 2009

  2. Basic number theory GF(23) Primitive function

  3. GF(23) in 2 Operations Addition table multiplication table

  4. Reed-Solomon Encoding(1) • k : number of data digits • n : codeword size in digits (data + parity) Input data of k digits m(X) Output codeword of n digits U(X) t-digit correcting RS (n,k) Encoder 2t = n-k ,

  5. Reed-Solomon Encoding(2) Let Then (The MSB is the most right bit.) Since ,

  6. Reed-Solomon Decoding • syndrome computation • error location • error value • error correction

  7. Syndrome computation Note that

  8. Error location(1) v errors at Let , for simplicity Then, …………………………………………….

  9. Error location(2) Make an error location polynomial. Therefore, Then, it is proved that the coefficients satisfy,

  10. Error location(3) Starting from v=t, find the largest v for which . and

  11. Error values

  12. Error correction (1)

  13. Error correction(2)

  14. Usage In Internet(1) Shortened RS code • n-k =(number of parity packets), • coding rate R=k/n • Numbers of data packets and/or parity packets can be flexibly determined before transmission • According to network condition, it is possible to control number of parity and data packets 255 2t k

  15. Usage In Internet(2) Erasures : RTP Sequence numbers of lost packets N data parity K lossy channel recovery • RTP Based FEC • The more parity packets, the more packets are recovered

  16. Usage In Internet(3) data parity parity Recover the less packets data parity parity Recover the more packets RS(4,2) code vs. RS(8,4) code : Tradeoff between delay and loss

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