Reed-Solomon Encoder: Basics and MATLAB Implementation

This section describes the basics and provides MATLAB source code for a Reed-Solomon (RS) encoder. Reed-Solomon codes are primarily used for providing FEC (Forward Error Correction) to received data blocks that may contain errors. The RS encoder is used at the transmitting end to add redundancy bytes to the input data before transmission. Conversely, the RS decoder is employed at the receiver to correct errors introduced during transmission by leveraging this redundant information.

RS codes are a type of systematic linear block code. They are considered “block codes” because the encoding process involves splitting the original message into fixed-length blocks. An RS encoder is typically designated as (N, K, T), where:

• N is the total number of bytes after encoding.
• K is the number of data bytes before encoding.
• T is the number of data bytes that the RS encoder and RS decoder combination can correct. T is equal to (N-K)/2.

Most common error correction codes operate on finite fields, also known as Galois Fields. These fields are constructed using a primitive polynomial p(x). This document describes an RS encoder based on the following polynomials:

• Code generator polynomial: g(x) = (x + λ⁰)(x + λ¹)(x + λ²)…(x + λ^2T-1), where λ = 02 in hexadecimal.
• Field Generator polynomial: p(x) = x⁸ + x⁴ + x³ + x² +1

Reed Solomon Encoder MATLAB Code

input_data= [113    44    42   179    77   236   149   177   105   250     0    81    64 62 148   223   123    98    95    90   193   216    17     0];
n=32;
k=24;
t=4;

g_f=0;
g_f(2)=1;
for i=1:7
    g_f(2+i)=bitshift(g_f(2+i-1),1);
end
x=2+i;
x1=2;x2=4;x3=5;x4=6;
for i=1:247
    g_f(x+i)=bitxor(bitxor(g_f(x1),g_f(x2)),bitxor(g_f(x3),g_f(x4)));
    x1=x1+1;x2=x2+1;x3=x3+1;x4=x4+1;
end

%------------- code generator polynomial ---------------%
r=[1 59 13 104 189 68 209 30 8 163 65 41 229 98 50 36 59];

for j=2:17
    for i=1:256
        if(r(j)==g_f(i))
            indx(j-1)=i-2;
            break
        end
    end
end

lfsr_op(1:16)=0;

for dc=1:k
    lfsr_input=bitxor(lfsr_op(1),input_data(dc));

    %get index of data with respect to Galois Field
    for i=1:256
        if(lfsr_input==g_f(i))
            lfsr_input=i-2;
            break
        end
    end

    temp1=0;
    reg_in=2+mod((indx+lfsr_input),255);

    for j=16:-1:1
        temp=lfsr_op(j);
        if(lfsr_input==-1)
            lfsr_op(j)=bitxor(0,temp1);
        else
            lfsr_op(j)=bitxor(g_f(reg_in(j)),temp1);
        end
        temp1=temp;
    end
end

encoded_data_output=[lfsr_op(1:(n-k)) input_data]

RS Encoder Input and Output