Suppose the message 111010 is to be transmitted (beginning with the leftmost bit) using CRC with generator polynomial X 4 +X+1.
Determine the entire message that would be transmitted. Show all your work. Also show how receiver will check and confirm that there is no transmission error.

Explanation:

Given that:

generator polynomial = x⁴ + x + 1

This form can be expressed in binary form as:

= 1             0           0           1            1

=10011

The given message = 111010

Now, we can see that the length of the polynomial generator = 5

The next thing to do is to add (5-1) = 4 0's to the given message end

i.e. message = 1110100000

The next process is to divide the message with the generator to deduce the remainder which ill be added to the message for us to create a transmitted message

∴

10011║1110100000║111

          10011              

           1 1 100

          1 0011              

              1 1 1 1 0

             1 001 1                  

                1 1 0 1 0

                1 0 01 1              

                  1 0 010

                 1 0 01 1              

                     0 010     → Remainder

∴ Transmitted message = 1 1 1 0 1 0 0 0 0 0

                                                           0 0 1 0

To check and confirm that the message transmitted is correct at the receiver's end, we have to divide the transmitted message with the generator.

So, if the end result of the remainder is 0, then the message is correct, otherwise incorrect.

10011║1110100000║1111

          10011              

           1 1 100

          1 0011              

              1 1 1 1 0

             1 001 1                  

                1 1 0 1 0

                1 0 01 1              

                  1 0 010

                 1 0 01 1              

                     00000     → Remainder

Hence, the received message is correct.