2. (10 points) In an engine management system, the sixteen binary messages 0000

Question

2. (10 points) In an engine management system, the sixteen binary messages 0000 through 1111 are to be transmitted over a data link. Each message is to be protected by a CRC generated using the polynomial: x3 + x2+ 1

a) Derive the check bits for the following messages: 0000, 0001, 0010.

b) What is the (minimum) Hamming distance for the code?

c) Let the transmitted message be 1111, show how a single and double bit error are decoded.

d) Give an example of an invalid codeword that will not be detected.

Explanation / Answer

2

a)

Given polynomial x3+x2+1

It can be written as 1101

i)

For the message 0000 append 0000 to the message because the length of the polynomial is 4.

Now divide message with CRC polynomial

In the above message 0000 are the check bits are 0000.

ii)

For the message 0001 append 0000 to the message because the length of the polynomial is 4.

Now divide message 0001 with CRC polynomial

In the above message 0001 are the check bits are 111.

iii)

For the message 0010 append 0000 to the message because the length of the polynomial is 4.

Now divide message 0010 with CRC polynomial

In the above message 0010 are the check bits are 011.

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