The information bit sequence \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1\}\) is to be transmitted by encoding with Cyclic Redundancy Check 4 (CRC-4) code, for which the generator polynomial is \(C(x) = x^4 + x + 1\). The encoded sequence of bits is \(\_\_\_\_\).
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- \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 0\}\)
- \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 1\}\)
- \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 1 \, 0\}\)
- \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 0\}\)
The Correct Option is A
Solution and Explanation
The given information bit sequence is \(\{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1\}\). Add 4 zeros to the end of the sequence, resulting in: \[ \{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 1 \, 0 \, 0 \, 0 \, 0\}. \] Step 2: Perform modulo-2 polynomial division.
Divide the appended sequence by the generator polynomial \(C(x) = x^4 + x + 1\) using modulo-2 arithmetic. The resulting remainder is: \[ \{1 \, 1 \, 0 \, 0\}. \] Step 3: Create the encoded sequence.
Add the remainder to the appended sequence, replacing the appended zeros. The encoded sequence becomes: \[ \{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 0\}. \] Final Answer: \[ \boxed{{(1) } \{1 \, 1 \, 1 \, 0 \, 1 \, 0 \, 1 \, 1 \, 1 \, 0 \, 0\}} \]
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