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h1. Reed-Solomon and Galois Fields

Reed-Solomon codes are non-binary cyclic codes with code symbols from a Galois field. Based on its approach of coding group of bits, this code becomes powerful at dealing with bursts of errors in digital data. So, it is capable of detecting incorrect values then recovers the correct message. 

We distinguish several forms of RS codes. The most important one seems to be the codes symbols from GF(2m). 

Thus, one important feature of RS codes is the fact that the minimum distance of an (n, k) code is n-k+1. 

The codes in this case are called “maximum-distance-separable-codes”.

*RS Codes with Symbols from GF( 2m )*

α is a primitive element in GF( 2m ).

For any positive integer t ≤ 2^m − 1, there exists a t-symbol-error-correcting RS code with symbols from GF( 2m ) and the following parameters:

n = 2^m -1

n-k=2t

k= 2^m -1-2t

dmin= 2t+1 = n-k+1

The generator polynomial is g(x) = (x+α)+ (x+α2)+ (x+α3)+..+ (x+α2t)

Where gi belongs GF(2m ), and α ,α^2 , ,α^2t  are roots for g(x).

Let’s take an example:

For m = 8 and t = 16;

n= 2^m -1=255 and then k = 223

dmin= n-k+1=33,

So we obtain a (255, 223) RS code. This code is NASA standard code for satellite and space communications.