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

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Let us return to the Reed & Solomon's Original View and see if there is another decoding method. We begin in the same way Reed and Solomon did by selecting a set of M indices, S, and nding P (Z) for which degree of P is at most M , 1 and P (x i )=r i for i 2 S The tool is Lagrange interpolation. We ask what is the eect of errors at the selected indices. 12
LAGRANGE INTERPOLATION For each i 2 S we need a polynomial, i (Z), of degree M , 1 which has the value 1 at x i and the value 0 at x j for j 2 S; j 6= i With these, the solution for P is P (Z) = X i2S r i i (Z) If r i includes an error e i then e i i (Z) is added to the correct P (Z). 13
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: Later Berlekamp found a faster algo
Page 15 and 16: Subtracting the interpolated values
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

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

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