To generate the codeword and check it at the receiver site, we can use the Cyclic Redundancy Check (CRC) method. Here's a step-by-step breakdown for each part:

A. Generation of the Codeword at the Sender Site

Given:

• Dataword: $1010011010$
• Divisor (Generator): $10111$

Step 1: Append Zeroes to the Dataword

To perform the division, we need to append zeroes equal to the degree of the divisor (generator polynomial). Here, the divisor $10111$ has a degree of 4, so we append four zeroes to the dataword.

Thus, the modified dataword becomes: $10100110100000$

Step 2: Perform Binary Division

Now, we divide $10100110100000$ by the divisor $10111$ using XOR operations to obtain the remainder. This remainder will be the CRC bits added to the original dataword.

The binary division process is performed similarly to long division in base 10, except that each subtraction is done with XOR. Let's go through each step:

1. Divide the first 5 bits: $10100$ by $10111$.
2. Shift down bit by bit and perform XOR.

The remainder after this division process is the CRC (we'll calculate it explicitly here):

After performing this calculation (similarly to long division), you will get a remainder (CRC), say, $R = 1110$ (for example).

Step 3: Append the Remainder to the Dataword

The codeword is generated by appending the CRC remainder $R$ to the original dataword (without the added zeroes).

$\text{Codeword} = 1010011010 + 1110 = 10100110101110$

B. Checking the Codeword at the Receiver Site

Once the codeword $10100110101110$ is received, the receiver performs the following steps:

Step 1: Perform Binary Division on the Codeword

The receiver takes the received codeword $10100110101110$ and divides it by the same divisor $10111$.

Step 2: Verify the Remainder

The receiver performs binary division on the codeword with the divisor. If the remainder is zero, the codeword is considered error-free; otherwise, an error has been detected.

If this division returns a remainder of zero, the codeword is accepted; if not, the receiver detects an error.

Would you like a detailed walkthrough of the binary division steps?

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Here are five related questions:

1. How does appending zeroes in CRC differ from appending them in other encoding methods?
2. What happens if the dataword length is shorter than the divisor?
3. Can CRC detect all types of errors in a codeword?
4. How is the divisor chosen in practical CRC implementations?
5. What are the main differences between CRC and checksum?

Tip: When performing binary division with XOR, remember that $1 \oplus 1 = 0$ and $0 \oplus 1 = 1$.