THE ORIGINAL VIEW OF REED-SOLOMON CODES THE ORIGINAL ...

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SYNDROMES Let T (Z) be the message polynomial and e k be the error at position k Then r k = T (x k )+e k Given an index selection, S, the Lagrange Interpolation polynomial evaluated at x k will be P (x k ) = X r i i (x k ) i2S = X T (x i ) i (x k )+ X e i i (x k ) i2S i2S 14
Subtracting the interpolated values from the received values: r k , P (x k ) = T (x k ) , X +e k , X i2S i2S T (x i ) i (x k ) e i i (x k ) Since T has degree at most M , 1, and we have T (x k )= X i2S T(x i ) i (x k ) k =r k ,P(x k )=e k , X for k 2 S c i2S which is a new family of syndromes. e i i (x k ) Given k for k 2 S c and assuming the number of errors is within the R-S bound, we would like to solve for the e j 's. 15
Page 1 and 2: L.R.Welch THE ORIGINAL VIEW OF REED
Page 3 and 4: MATHEMATICALLY: Let (x 1 ;x 2 ;;x N
Page 5 and 6: THE CLASSICAL VIEW Reed Solomon Cod
Page 7 and 8: Alternatively, dividing Z N ,M P (Z
Page 9 and 10: An alternative is to form a codewor
Page 11 and 12: Later Berlekamp found a faster algo
Page 13: LAGRANGE INTERPOLATION For each i 2
Page 17 and 18: The syndromes can now be expressed
Page 19 and 20: For k 2 S c , either e k =0orQ C (x
Page 21 and 22: We introduce an auxillary pair of p
Page 23 and 24: When we have generated Q k ;P k ;W
Page 25 and 26: IF d k 6=0 We form another quantity
Page 27 and 28: IF d k =0 0 B @ Q k(Z) P k (Z) W k
Page 29 and 30: WHAT ABOUT MINIMALITY This is the d
Page 31 and 32: THEOREM: The algorithm described ab
Page 33 and 34: Lemmas 5,6 L(Q k (Z);P k (Z)) + L(W

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