CRC Code Calculation for Data Word 110101010

The goal is to find the Cyclic Redundancy Check (CRC) codeword for a given data word using a specific divisor (generator polynomial).

Steps for CRC Calculation:

1. Identify Data and Divisor:
   • Data Word (D): $110101010$
   • Divisor (P): $10101$
2. Append Zeros: Determine the number of zero bits to append. This is equal to the degree of the divisor polynomial, which is one less than its bit length.
   • Divisor length = 5 bits
   • Degree (n) = 5 - 1 = 4
   • Append $n=4$ zeros to the data word: $110101010$0000$ = $1101010100000$
3. Perform Binary Division: Divide the appended data word ($1101010100000$) by the divisor ($10101$) using modulo-2 (XOR) arithmetic.
4. Determine Remainder: The result of the division is a quotient and a remainder. The remainder is the CRC code. For this specific calculation, the remainder obtained is $1011$.
5. Form the CRC Codeword: Append the calculated remainder ($1011$) to the original data word ($110101010$).
   • CRC Codeword = Data Word + Remainder
   • CRC Codeword = $110101010$ + $1011$ = $1101010101011$

This resulting codeword $1101010101011$ matches Option D.