Question

Design (7,3) linear block code with parity check matrix given as

H =
0 1 11 0 0 1
1 0 10 1 0 1
1 1 00 0 1 1

a. Find all the corresponding codewords of the code.

b. What is the error the error-correcting and error-detection capabilities of the code?

c. Find the syndrome for the received vector R = [1101011].

d. Assuming the receiver Maximum likelihood algorithm construct syndrome table for the correctable error patterns

Answer 1

Here in (n,k)=(7,3) .....7 is the Code length and Data length is 3.

data bits is d=n-k ==>7-3==>4

so (d,k)=(4,3)

Given H=0 1 11 0 0 1
1 0 10 1 0 1
1 1 00 0 1 1

Code words of the code are:

d1 d2 d3 d4 p1 p2 p3 ,dmin ...here as parity conditions are not given it is taken as even parity.

c1=0000 0 0 0 =>0

c2=0001 0 1 1 =>3

c3=0010 1 1 1=>4

c4=0011 1 0 0=>3

c5=0100 1 0 1=>3

c6=0101 1 1 0=>4

c7=0110 0 1 0=>3

c8=0111 0 0 1=>4

c9=1000 1 1 0=>3

c10=1001 1 0 1=>4

c11=1010 0 0 1=>3

c12=1011 0 1 0=>4

c13=1100 0 1 1=>4

c14=1101 0 0 0=>3

c15=1110 1 0 0=>4

c16=1111 1 1 1=>7

here the dmin=3

b) The error correcting capacibity is dmin-1=>2

the error correction capability is (dmin-1)/2=>1

c)Received vector is [1101011].

syndrom=R*H^T

[1101011]. 0 1 1
1 0 1
1 1 0

1 0 0

0 1 0

1 1 1

BY SOLVING we Get s=[1 0 0]

There is an error at 4th digit in the received signal.

by adding the error to the codeword we can get the correct patterns (like using 1101100)