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

More documents

Recommendations

Info

MA<strong>THE</strong>MATICALLY: Let G(Z) = N,M Y i=1 (Z , i ) Let (m 1 ;m 2 ;;m M ) be a message where m i 2 GF (2 n ). Dene a polynomial by P (Z) =m 1 +m 2 Z+m 3 Z 2 ++m M Z M,1 and dene C(Z) = P (Z)G(Z) X = N ,1 i=0 c i Z i Then the codeword is (c 0 ;c 1 ;;c N,1 ) 6
Alternatively, dividing Z N ,M P (Z) byG(Z)to get Z N ,M P (Z) =Q(Z)G(Z)+R(Z) Then C(Z) =Z N,M P(Z),R(Z) is divisible by G(Z) and its coecients form a systematic codeword. 7
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: THE CLASSICAL VIEW Reed Solomon Cod
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 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 ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?