CCS

Linear Block Codes

Linear block codes are the most important and widely used class of block codes. A
block code is linear if any linear combination of two codewords is a codeword. In the
binary case, this means that the sum of any two codewords is a codeword. In linear
block codes the codewords form a k-dimensional subspace of an n-dimensional space.
Linear block codes are described in terms of a generator matrix G, which is a k x n
binary matrix such that each codeword c can be written in the form

where u is the binary data sequence of length k (the encoder input). Obviously, the
all-0 sequence of length n is always a codeword of an ( n, k) linear block code.

Matlab Coding

 

 

% MATLAB script for Illustrative Problem 10.8.
% Generate U, denoting all information sequences.
k=4;
for i=1:2^k
    for j=k:-1:1
        if rem(i-1,2^(-j+k+1))>=2^(-j+k)
            u(i,j)=1;
        else
            u(i,j)=0;
        end
        echo off ;
    end
end
echo on ;
% Define G, the generator matrix.
g=[1 0 0 1 1 1 0 1 1 1;
    1 1 1 0 0 0 1 1 1 0;
    0 1 1 0 1 1 0 1 0 1;
    1 1 0 1 1 1 1 0 0 1];
% Generate codewords.
c=rem(u*g,2);
% Find the minimum distance.
w_min=min(sum((c(2:2^k,:))'));

>> w_min

w_min =

2

>> c

c =

0 0 0 0 0 0 0 0 0 0
1 1 0 1 1 1 1 0 0 1
0 1 1 0 1 1 0 1 0 1
1 0 1 1 0 0 1 1 0 0
1 1 1 0 0 0 1 1 1 0
0 0 1 1 1 1 0 1 1 1
1 0 0 0 1 1 1 0 1 1
0 1 0 1 0 0 0 0 1 0
1 0 0 1 1 1 0 1 1 1
0 1 0 0 0 0 1 1 1 0
1 1 1 1 0 0 0 0 1 0
0 0 1 0 1 1 1 0 1 1
0 1 1 1 1 1 1 0 0 1
1 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 1 1 0 0
1 1 0 0 1 1 0 1 0 1

>> u

u =

0 0 0 0
0 0 0 1
0 0 1 0
0 0 1 1
0 1 0 0
0 1 0 1
0 1 1 0
0 1 1 1
1 0 0 0
1 0 0 1
1 0 1 0
1 0 1 1
1 1 0 0
1 1 0 1
1 1 1 0
1 1 1 1

>> g

g =

1 0 0 1 1 1 0 1 1 1
1 1 1 0 0 0 1 1 1 0
0 1 1 0 1 1 0 1 0 1
1 1 0 1 1 1 1 0 0 1

In a systematic code, the first k binary symbols in a codeword are the information bits, and the
remaining n - k binary symbols are the parity-check symbols.

Reference,

　　1. <<Contemporary Communication System using MATLAB>> - John G. Proakis