Reed Solomon Code Doug Young Suh 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  n : codeword size in digits (data + parity) t-digit correcting RS (n,k) Encoder 2t = n-k Input data of k digits m(X) Output codeword of n digits U(X),

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

Reed-Solomon Decoding 1.syndrome computation 2.error location 3.error value 4.error correction

Syndrome computation Note that

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

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

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

Error values

Error correction (1)

Error correction(2)

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

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

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