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

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DECODING AS PRESENTED IN THE REED-SOLOMON PAPER Reed and Solomon described the following decoding procedure. Receive (r 1 ;r 2 ;;r N ) Select M indices in all possible ways (Lagrange Interpolation)For each selection, nd P (Z) with deg(P )=M,1 P(x i )=r i at these indices. The coecients of P form a potential message. As all possible selections are made, the message which gives a codeword closest to the received word occurs more often than any other. 8
An alternative is to form a codeword corresponding to each potential message and stop when the result disagrees with the received message in at most (N , M)=2 places. If none is found, failure to decode occurs. These procedures are very inecient. return to this later. We will 9
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: Alternatively, dividing Z N ,M P (Z
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 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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