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   Reed-Solomon Coder (RSCOD)       

Reed-Solomon Coder (RSCOD)

Limits

Notes

This model is used to perform Reed-Solomon (RS) encoding. RS codes are a non-binary sub­class of the BCH block codes. The code format is RS(n,k) defined on Galois Field (2^m). The error correcting capability of the RS(n,k) code is defined by t = (n-k)/2, which means this code can correct up to t = (n-k)/2 errors.

The RS coder adds a sequence of 2t parity code symbols to each k input data symbols to form a codeword with n = 2^m-1 symbols. However, RS(n,k) can be shorten into RS(n-j, k-j) by simply forcing j leading input data symbols to be zeros, and then deleting these zero symbols from a systematic codeword.

For example, the digital satellite communications Reed-Solomon (RS) coder/decoder RS(204,188), is the shortened length of the RS(255,239) Reed-Solomon code. This shortened code is formed by 188 input data symbols, plus 51 additional zero-padded symbols, before encoding with the RS(255,239). The transmitted codeword doesn’t contain the padding sequence which is rebuilt in RS decoder.

At the output of the RS coder, the data is left unchanged and the parity codes are appended as shown in Fig.1. In this model, the parity codes are transmitted first.
Fig.1: Codeword for systematic Reed-Solomon code

(1) Galois Field Arithmetic

Reed-Solomon codes are based on a specific area of mathematics known as Galois Field, which is set up according to the number of bits per symbol and the number of symbols per block (i.e., codeword). The elements of the Galois Field GF(2^m) are generated from the mth degree irreducible primitive polynomial with the smallest number of terms. The primitive polynomial of degree m can be written in the form:

(1)

Table I. List of primitive polynomials
The elements if Galois Field GF(2^m) can have two representations: power representation and polynomial representations. Let a represent the root of the primitive polynomial P(X). Each power representation aⁱ, where

,
for elements of Galois Field GF(2^m) can be expressed as

(2)
where the binary vector {a_i0, a_1i, ..., a_i,m-1} is the polynomial representation of aⁱ. The power representation is convenient for multiplication and the polynomial representation is convenient for addition.

(2) Generator polynomial

The generator polynomial of Reed-Solomon code is generally defined as
(3)
where t is the correctable error number. For the special case of b₀ =1, the above equation will be simplified into
(4)
The generator polynomial can also be expressed as a 2t order of polynomial
(5)

(3) Encoding

2. In the Reed-Solomon code, all generated codewords are exactly divisible by the generator polynomial. Let

(6)
be the input data to be encoded, where k= n-2t. The parity check codes will be the coefficients of the remainder, b(X) = b₀ +b₁X +...+b_2t-1X^2t-1 resulting from dividing the input data polyno­mial X^2t a(X) by the generator polynomial g(X). These parity codes are then joined to the data symbols to form the transmitted codeword. The RS encoding procedure can be accomplished by using a division circuit as shown in Fig.2

Fig.2 Encoding diagram for systematic Reed-Solomon code

Netlist Form

RSCOD:NAME n1 n2 N=val K=val M =val [B0 =val] [P0=val . . . PM=val] [RIN=val] [ROUT=val]

Netlist Example

RSCOD:1 1 2 N=204 K=188 M =8 B0 =1 {P0,…P8} = {1,0,1,1,1,0,0,0,1}

References

1. James J. Spilker, Digital Communications by satellite, Prentice-Hall, 1977.

2. Shu Lin and D.J.Costello, Error Control Coding: Fundamentals and applications, Prentice-Hall, 1983.

3. Y.Shayan, T.Le-Ngoc and V.Bhargava, “A versatile time-domain Reed-Solomon decoder,” IEEE Journal on Selected Areas in Communications, vol. 8, No.8, pp.1535-1542, Oct. 1990.

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