Chapter 6 - Binary BCH Codes


Generator Polynomial

Parity Check Matrix

Alternative Syndrome Calculation

Decoding BCH Codes

  1. Compute syndrome S = (S1,S2,S3,...S2t) from received r
  2. Determine Error Polynomial σ(X) from S
  3. Find error location numbers β1,β2,...βv, by finding the roots of σ(X)

Error location polynomial σ(X)

Berlekamp Iterative Algorithm

  1. Find the minimum degree polynomial σμ=1(X) satisfying Newton's μth identity (identities below) for the first iteration, μ = 1
  2. Test if σμ=1(X) also satisfies the second identity S2 + σ1S1 + 2σ2 = 0
  3. Repeat until we have σμ=2t(X); this is our error location polynomial σ(X)
  4. Keep track using a table, which 2 previous starting entries:

BCH Decoding Example

Simplified Berlekamp Iterative Algorithm

Example of simplified Berlekamp Iterative Alg. for binary BCH

Improved error location calculation (instead of 'finding roots of σ(X)')