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The following code is written for execution via MatLab, and is now intended to be supplemental to the descriptions of Hamming codes already found in this node. It is generally a proof/simulation of the Hamming (7,4) code. The (7,4) incarnation of hamming error correction/detection codes is composed of four data bits and three error check bits. This code calculates BER (bit error rate), and compares to BSC (Binary symmetric channel) efficiency. BSC is simply a single channel capacity and it's corresponding error probability and efficiency. If the probability of error on a channel is 50%, the channel has a capacity of 0 (useless). This is due to the fact that 0 and 1 are equally likely in binary number systems. Capacity is full at error probability of 0% or 100% (inverted).

So the following code is meant as a proof and simulation of Hamming (7,4) code. Its usefulness comes from the resulting two graphs. There is a standard plot of Hamming vs. BSC, and then a log*log plot that shows a better representation of the data comparison (decibels). To run this code, save it into a file, name it "name".m, and move it to the home directoy of any version of MatLab. Then simply type "name" in the MatLab console, and it should all make sense. You can adjust coding variables and table size directly in the program as it is adequately commented. I hope this helps anyone who may be taking classes in digital communications. I think Hamming (7,4) is pretty popular in digital communications courses.

Sources: Ziemer and Peterson: Introduction to Digital Communication, Second Edition. Prentice Hall, 2001.

```% Jordan Rice
% ELEC 476 (Random Signals and Noise)
% Hamming (7,4) Code Simulation

%Defining Variables for Hamming (7,4) simulation

k = 4;		                % Number of message bits per block
n = 7;		                % Number of codeword bits per block
p_vector = 0.1:0.01:1;          % Vector of p values; probability of bit error for BSC
N = length(p_vector);           % Length of p_vector
RUNS = 5000;                    % Number of runs

% Codeword Table

xtable =    [0 0 0 0 0 0 0; ...
1 1 0 1 0 0 0; ...
0 1 1 0 1 0 0; ...
1 0 1 1 1 0 0; ...
1 1 1 0 0 1 0; ...
0 0 1 1 0 1 0; ...
1 0 0 0 1 1 0; ...
0 1 0 1 1 1 0; ...
1 0 1 0 0 0 1; ...
0 1 1 1 0 0 1; ...
1 1 0 0 1 0 1; ...
0 0 0 1 1 0 1; ...
0 1 0 0 0 1 1; ...
1 0 0 1 0 1 1; ...
0 0 1 0 1 1 1; ...
1 1 1 1 1 1 1;];

for (p_i=1:N)

error = 0;                    % Count the number of errors
p=p_vector(p_i);

for (r=1:RUNS)           %Generate a block of 4 message bits

z = unifrnd(0, 1, 1, 4);    %Random 4 bit string of 0's and 1's
w = round(z);               %Round the value of z

%Locate the row index, binary to dec. conversion:

m = w(1) + w(2)*2 + w(3)*4 + w(4)*8;
x = xtable(m + 1, :);
z = unifrnd(0, 1, 1, 7);    % Random 7 bit string of 0's and 1's
zi = find(z <= p);          % Error locations

% Error Vector

e = zeros(1,7);             % Creates a 7 bit string of 0's
e(zi) = ones(size(zi));     % Creates a string of 1's the size of z
y = xor(x,e);               % Exclusive OR your x value and e value

% Approximate the code word

for(q=1:16)
dH(q) = sum(xor(y, xtable(q,:)));   % Compare codeword to received vector

if(dH(q)<=1)
wh = xtable(q, 4:7); % wh matches with distance of 1 or less
end
end

% Count the number of Errors
dHw = sum(xor(w,wh));
error = error + dHw;
end
BER(p_i) = error/(RUNS*4);    % Calculate the Bit Error Rate
P(p_i)=p;                   % Store the value of p
end

Ps=logspace(-4,0,200);
Pb_high = 1 - ((1-Ps).^7 + 7.*(1-Ps).^6.*Ps);
Pb_low = (1 - ((1-Ps).^7 + 7.*(1-Ps).^6.*Ps))/k;

figure(1)
plot(P, BER, 'bx', Ps, Pb_high, 'k-', Ps, Pb_low, 'k-')
legend('Simulated','Analytical')
xlabel('Probability of Error for BSC (p)')
ylabel('BER')
title('Figure 1 - Bit Error Rate for Hamming code over BSC')

figure(2)
loglog(P, BER, 'bx', Ps, Pb_high, 'k-', Ps, Pb_low, 'k-')
legend('Simulated','Analytical')
xlabel('Probability of Error for  BSC (p)')
ylabel('BER')
title('Figure 2 - Bit Error Rate for Hamming code over BSC')

```