# Q2(b) · CRC encoding

Subject: Computer Networks

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/cn/#tee/papers/paper-2023-ai-partial/q2b

Canonical: https://deep-learning-study-room.vercel.app/read/cn/tee/papers/paper-2023-ai-partial--q2b/

Exact historical variant. Current-syllabus questions solved; excluded topics and source limitations stated.

### Q2(b). Given message 1011011010 and generator G(x) = x³ + x + 1, calculate the CRC code and transmission message. [5 marks]

### Answer

### Given

Data = 1011011010; generator bits = 1011; degree r = 3.

### To find

CRC remainder, complete transmitted frame and receiver check.

### Formula and why

Append r zeros, divide using XOR (modulo 2), and replace those zeros with the r-bit remainder. XOR has no borrowing or carrying. A valid received codeword has remainder zero; the receiver does not append new zeros.

### Step 1. Append zeros

1011011010 → 1011011010000

### Step 2. Perform XOR division

| Leading position | Before XOR | Aligned generator | After XOR |
| --- | --- | --- | --- |
| 1 | 1011011010000 | 1011000000000 | 0000011010000 |
| 6 | 0000011010000 | 0000010110000 | 0000001100000 |
| 7 | 0000001100000 | 0000001011000 | 0000000111000 |
| 8 | 0000000111000 | 0000000101100 | 0000000010100 |
| 9 | 0000000010100 | 0000000010110 | 0000000000010 |

### Step 3. Build the frame

Remainder = **010**. Transmit **1011011010010**.

### Step 4. Divide the received frame

| Position | Before | Generator | After |
| --- | --- | --- | --- |
| 1 | 1011011010010 | 1011000000000 | 0000011010010 |
| 6 | 0000011010010 | 0000010110000 | 0000001100010 |
| 7 | 0000001100010 | 0000001011000 | 0000000111010 |
| 8 | 0000000111010 | 0000000101100 | 0000000010110 |
| 9 | 0000000010110 | 0000000010110 | 0000000000000 |

Receiver remainder = 000. No error is detected; this is not a guarantee that every possible corruption is detectable.