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

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The fact that each pair, (Q k (Z);P k (Z)) and (W k (Z);V k (Z)) satisfy the rst k equations can be expressed as 0 B @ Q k(x i ) P k (x i ) W k (x i ) V k (x i ) 1 0 C A B @ i ,1 1 C A = 0 B @ 0 0 1 C A for i k 28
WHAT ABOUT MINIMALITY This is the dicult part of the theorem and I will just outline the proof with a few comments. I will pattern this proof after that in Tze Hwa Liu's dissertation ( 1984) 29
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 and 14: LAGRANGE INTERPOLATION For each i 2
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: IF d k =0 0 B @ Q k(Z) P k (Z) W k
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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