Question

I am looking into implementing encoding and decoding for devices operating in noisy channel with burst errors. I have found that the most common techniques to overcome burst errors are the following:

1. Reed-Solomon (RS) codes
2. Fire codes
3. Interleaving
4. Concatenation
5. Cascading

I presume in embedded systems, the most efficient way to implement polynomial based techniques to store relevant polynomials in lookup tables. However, the encoding and decoding involves fair amount of computation and just wondering how long these might take.

Any help would be appreciated.

==========

P.S.: Detection and correction both are necessary.

Answer 1

I don’t think there is any substitute for RS to correct block errors. Convolutional codes are quite fast (~15 amortized instructions per decoded bit for viterbi), but are only suitable for sporadic bit changes. Syndrome for BCH (binary) code (31,5) can be calculated even faster, but it has only capability of recovering 2 bits.

This page discusses the complexity of full software implementations of long RS decoding:

 RS(255,251) - 12 Mbps
 RS(255,239) - 2.7 Mbps
 RS(255,223) - 1.1 Mbps

Mapping 2-error syndrome to error locations achieved 12Mbps speed on 166MHz Pentium (using probably a look up table), which could be said to be 5-10% as efficient as todays typical embedded processors. The complexity of larger polynomials is about O(M*N), where M = block length, N=code length for the syndrome calculation and the Berlekamp-Massey algorithm. The Chien search seems to be often done brute force looping over all possible (255) roots and substituting them one by one to the error location polynomial — that has also a complexity of about O(M*N), as M=block size = 255 is about the same as the number of codewords (256).