rs 解码的一点资料

.2 Decoder architecture

A general architecture for decoding Reed-Solomon codes is shown in the following diagram.

Key

r(x) Received codeword Si Syndromes L(x) Error locator polynomial Xi Error locations Yi Error magnitudes c(x) Recovered code word v Number of errors

The received codeword r(x) is the original (transmitted) codeword c(x) plus errors:

r(x) = c(x) + e(x)

A Reed-Solomon decoder attempts to identify the position and magnitude of up to t errors (or 2t erasures) and to correct the errors or erasures.

Syndrome Calculation

This is a similar calculation to parity calculation. A Reed-Solomon codeword has 2t syndromes that depend only on errors (not on the transmitted code word). The syndromes can be calculated by substituting the 2t roots of the generator polynomial g(x) into r(x).

Finding the Symbol Error Locations

This involves solving simultaneous equations with t unknowns. Several fast algorithms are available to do this. These algorithms take advantage of the special matrix structure of Reed-Solomon codes and greatly reduce the computational effort required. In general two steps are involved:

Find an error locator polynomial

This can be done using the Berlekamp-Massey algorithm or Euclid’s algorithm. Euclid’s algorithm tends to be more widely used in practice because it is easier to implement: however, the Berlekamp-Massey algorithm tends to lead to more efficient hardware and software implementations.

Find the roots of this polynomial

This is done using the Chien search algorithm.

Finding the Symbol Error Values

Again, this involves solving simultaneous equations with t unknowns. A widely-used fast algorithm is the Forney algorithm.