Admin: Created page with "In coding 1}}, so the expression simplifies to: : $e_j = - \frac{\Omega(X_j^{-1})}{\Lambda'(X_j^{-1})}$ ==Formal derivative== Λ'(''x'') is the [[formal de..."

2019-05-23T15:03:02Z

Created page with "In coding 1}}, so the expression simplifies to: :<math>e_j = - \frac{\Omega(X_j^{-1})}{\Lambda'(X_j^{-1})}</math> ==Formal derivative== Λ'(''x'') is the [[formal de..."

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In [[coding 1}}, so the expression simplifies to:
:<math>e_j = - \frac{\Omega(X_j^{-1})}{\Lambda'(X_j^{-1})}</math>

==Formal derivative==

Λ'(''x'') is the [[formal derivative]] of the error locator polynomial Λ(''x''):
:<math>\Lambda'(x) = \sum_{i=1}^{\nu} i \, \cdot \, \lambda_i \, x^{i-1}</math>


In the above expression, note that ''i'' is an integer, and λ<sub>''i''</sub> would be an element of the finite field. The operator · represents ordinary multiplication (repeated addition in the finite field) and not the finite field's multiplication operator.



==Derivation==
Lagrange interpolation gives a derivation of the Forney algorithm.

==Erasures==
Define the erasure locator polynomial
:<math>\Gamma(x) = \prod (1- x \, \alpha^{j_i})</math>
Where the erasure locations are given by ''j<sub>i</sub>''. Apply the procedure described above, substituting Γ for Λ.

If both errors and erasures are present, use the error-and-erasure locator polynomial
:<math>\Psi(x) = \Lambda(x) \, \Gamma(x)</math>

==See also==
*[[BCH code]]
*[[Reed–Solomon error correction]]

==Source==

[http://wikipedia.org/ http://wikipedia.org/]

[[Category:Error-detecting codes]]
[[Category:Error-correcting codes]]

Forney algorithm - Revision history

Admin: Created page with "In coding 1}}, so the expression simplifies to: :e_j = - \frac{\Omega(X_j^{-1})}{\Lambda'(X_j^{-1})} ==Formal derivative== Λ'(''x'') is the [[formal de..."

Admin: Created page with "In coding 1}}, so the expression simplifies to: : $e_j = - \frac{\Omega(X_j^{-1})}{\Lambda'(X_j^{-1})}$ ==Formal derivative== Λ'(''x'') is the [[formal de..."