# Q2(b)(ii) · CRC and syndrome

Subject: Computer Networks

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/cn/#tee/papers/paper-2023-jan-ai-reexam/q2bii

Canonical: https://deep-learning-study-room.vercel.app/read/cn/tee/papers/paper-2023-jan-ai-reexam--q2bii/

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

### Q2(b)(ii). Data 1001 is transmitted using G(x)=x³+x+1. (a) Find the transmitted codeword. (b) Explain the role of syndrome bits at the receiver. [5 marks]

### Answer

### Given

Data = 1001; 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

1001 → 1001000

### Step 2. Perform XOR division

| Leading position | Before XOR | Aligned generator | After XOR |
| --- | --- | --- | --- |
| 1 | 1001000 | 1011000 | 0010000 |
| 3 | 0010000 | 0010110 | 0000110 |

### Step 3. Build the frame

Remainder = **110**. Transmit **1001110**.

### Step 4. Divide the received frame

| Position | Before | Generator | After |
| --- | --- | --- | --- |
| 1 | 1001110 | 1011000 | 0010110 |
| 3 | 0010110 | 0010110 | 0000000 |

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