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Linear Binary Codes

You can not use the definition Hermitian inner product to solve this problem.

You need to use the definition of weight.

We need to generalize the equation.

1-Define the product, x*y , of two binary vectors of the same length to be the vector whose ith component is the product of the ith components of x and y.
Show that wt (x +y ) =wt (x ) +wt (y )- 2wt (x*y ).

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7. The parity check matrix of a binary [n, k, d] = [15, 11, 3]
Hamming code whose columns are in numeric order is

a) Find the corresponding generator matrix.
Use the method of permuting and unpermuting the columns for the [7, 4, 3]
binary Hamming code.
b) Encode the information vector 10001000001. Explain your steps.
c) What codeword was most likely sent if the received vector is 100010101000000? Explain your steps.
d) What information vector was most likely sent if the received vector is as given in part c? Explain your steps
Note: Assume that the check bits are in columns 1, 2, 4 and 8.

Solution 62
a)
n=15
k=11
d=3
Therefore m=n-k=15-11=4

Generator G= [ Ik (kxk) P (kxm)]

Parity check Matrix H= [ PT Im]

To bring H in the standard form by permuting the rows and columns perform the following operations strictly in the order I have mentioned below.
C15 ----> C15 - C14
C14-----> C14 - C12
C13-----> C13 - C9
C12-----> C12 - C4
After this we get H in the standard form. ie. H= [ PT ...

Solution Summary

The definition of Hermitian inner products to solve this problem is determined.

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