---
title: "Checksum error detection"
subject: "Data Communication"
chapter: 8
key: "semester_5/data_communication:checksum-12-15-15-10-5-2"
source: https://www.bctengineer.com/q/semester_5/data_communication/checksum-12-15-15-10-5-2
site: BCT Engineering
---

# Checksum error detection

Data Communication · Chapter 8: Information Theory and Coding

- Asked in: 2078 Bhadra, 8 marks
- Video: [Checksum](https://www.youtube.com/watch?v=AtVWnyDDaDI) — Neso Academy
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/checksum-12-15-15-10-5-2
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-8

## Question

Demonstrate how checksum is used to detect errors while sending a data word of 12, 15, 15, 10, 5, 2.

## Short answer

In 4-bit one’s complement checksum, sum of data = 59 (111011). Wrap carry (11) into low 4 bits: 1011+11=1110. Complement is 0001, so checksum = 1. Receiver adds all words (including checksum), wraps, and complements; 0 means no error.

## Answer

The **checksum** method adds the data words using **one's complement arithmetic**; the complement of the sum is sent as an extra word. The receiver adds all words including the checksum and complements; a result of 0 means no error.

Data words are 12, 15, 15, 10, 5, 2, so the largest is 15 and a 4-bit word size is the natural choice (sums wrap around in 4 bits).

#### Sender
Binary values: 12 = 1100, 15 = 1111, 15 = 1111, 10 = 1010, 5 = 0101, 2 = 0010.

Sum = 12 + 15 + 15 + 10 + 5 + 2 = 59 = 111011 (6 bits).
- Wrap around: the carry bits (11 = 3) are added to the low 4 bits (1011 = 11): 11 + 3 = 14 = 1110.
- Checksum = complement of 1110 = **0001 = 1**.

Transmitted: 12, 15, 15, 10, 5, 2, **1**.

#### Receiver (no error)
Sum = 59 + 1 = 60 = 111100. Wrap: 1100 + 11 = 12 + 3 = 15 = 1111. Complement = **0000**, so the data is accepted.

#### Receiver (with error)
If 10 becomes 11 in transit: sum = 61 = 111101 → 1101 + 11 = 13 + 3 = 16 = 10000 → wrap again: 0000 + 1 = 0001. Complement = 1110 ≠ 0, so the error is detected and the data is rejected.

**Answer: checksum = 1 (binary 0001); receiver result = 0 means accepted.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What arithmetic does the checksum use?
   A. Two's complement without wrap-around
   B. One's complement addition with carry wrapped around
   C. Modulo-2 polynomial division
   D. Subtraction of words
   Answer: B. Words are added in one's complement, carries are wrapped back into the low bits, and the sum is complemented. (Key point: Checksum arithmetic)

2. The sum of 12, 15, 15, 10, 5, 2 is 59 = 111011. After wrapping in 4 bits, what is the result?
   A. 1011
   B. 1111
   C. 1110 (14)
   D. 0001
   Answer: C. Low 4 bits 1011 (11) plus carry 11 (3) gives 14 = 1110. (Key point: Sender calculation)

3. What checksum is sent?
   A. 14
   B. 1 (complement of 1110 = 0001)
   C. 15
   D. 0
   Answer: B. The checksum is the complement of the wrapped sum 1110, which is 0001 = 1. (Key point: Checksum value)

4. What result at the receiver means the data is accepted?
   A. The complement of the wrapped sum is 0000
   B. The sum equals 1
   C. The sum equals 59
   D. Any non-zero result
   Answer: A. Sum with checksum wraps to 1111, whose complement is 0000, meaning no error detected. (Key point: Receiver decision)

## Related in Data Communication

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