1. **Stating the problem:** We need to generate the CRC code for the message $1101010101$ using the given generator polynomial $g(x) = x^4 + x + 1$. 2. **Understanding CRC generation:** CRC (Cyclic Redundancy Check) is generated by dividing the message polynomial (augmented with zeros) by the generator polynomial using modulo-2 division (XOR operations). 3. **Convert the message to polynomial form:** The message $1101010101$ corresponds to the polynomial $$M(x) = x^9 + x^8 + x^6 + x^4 + x^2 + 1$$ 4. **Augment the message:** Append $n$ zeros to the message, where $n$ is the degree of $g(x)$. Here, degree of $g(x)$ is 4, so append 4 zeros: $$1101010101\,0000$$ 5. **Perform modulo-2 division:** Divide the augmented message by $g(x)$ using XOR operations step-by-step to find the remainder $R(x)$. 6. **Division steps (summary):** - Initial dividend: $11010101010000$ - Divisor: $10011$ (binary for $x^4 + x + 1$) - Perform XOR division until the last 4 bits remain. 7. **Result:** The remainder after division is the CRC code. For this message and polynomial, the remainder is $1000$. 8. **Final transmitted message:** Append the CRC code to the original message: $$1101010101\,1000$$ **Answer:** The CRC code for the message $1101010101$ with generator polynomial $x^4 + x + 1$ is **1000**.