Forney algorithm

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 derivative of the error locator polynomial Λ(x):

$\Lambda '(x)=\sum _{i=1}^{\nu }i\,\cdot \,\lambda _{i}\,x^{i-1}$

In the above expression, note that i is an integer, and λ_i 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

$\Gamma (x)=\prod (1-x\,\alpha ^{j_{i}})$

Where the erasure locations are given by j_i. Apply the procedure described above, substituting Γ for Λ.

If both errors and erasures are present, use the error-and-erasure locator polynomial

$\Psi (x)=\Lambda (x)\,\Gamma (x)$

See also

• BCH code
• Reed–Solomon error correction

Source

http://wikipedia.org/