---
title: "Short note: Error detection and correction capability"
subject: "Communication Systems"
chapter: 8
key: "semester_6/communication_systems:short-note-error-detect-correct-capability"
source: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-error-detect-correct-capability
site: BCT Engineering
---

# Short note: Error detection and correction capability

Communication Systems · Chapter 8: Error Detection and Correction Coding

- Asked in: 2081 Bhadra (CS II), 4 marks
- Video: [Error Detection & Error Correction Capability of Linear Block Code in Digital Communication](https://www.youtube.com/watch?v=xkJZY-vrIco) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/short-note-error-detect-correct-capability
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-8

## Question

Write a short note on error detection and correction capability of a codeword.

## Short answer

Error detection capability is up to d_min-1 errors; error correction capability is up to floor((d_min-1)/2) errors. d_min is the minimum Hamming distance of the code.

## Answer

The **error detection and correction capability** of a block code depends on its **minimum Hamming distance** $d_{min}$, the smallest number of bit positions in which any two codewords differ. For a linear code, $d_{min}$ equals the smallest weight of a non-zero codeword.

#### Detection

If fewer than $d_{min}$ bits are changed, a codeword cannot turn into another valid codeword, so the error is noticed:

$$
s\le d_{min}-1
$$

#### Correction

The decoder chooses the nearest codeword. This is always right if the received word is closer to the sent codeword than to any other, which needs

$$
t\le\left\lfloor\frac{d_{min}-1}{2}\right\rfloor\quad(d_{min}\ge 2t+1)
$$

To correct $t$ and also detect $s>t$ errors: $d_{min}\ge t+s+1$.

```text
  c1 o----x----x----o c2     d_min = 3
       radius t = 1 spheres do not overlap
```

| Code | $d_{min}$ | Detects | Corrects |
|---|---|---|---|
| Single parity | 2 | 1 | 0 |
| (7,4) Hamming | 3 | 2 | 1 |
| Repetition (3,1) | 3 | 2 | 1 |
| Extended (8,4) Hamming | 4 | 3 | 1 |
| (5,1) repetition | 5 | 4 | 2 |

Larger $d_{min}$ needs more redundancy (lower code rate $k/n$).

## Check yourself

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

1. A code with minimum distance dmin can detect up to how many errors?
   A. dmin
   B. dmin - 1
   C. dmin / 2
   D. dmin + 1
   Answer: B. Fewer than dmin changed bits cannot turn a codeword into another codeword, so s <= dmin - 1. (Key point: Detection capability)

2. What condition on dmin lets a code correct t errors?
   A. dmin >= t + 1
   B. dmin >= 2t + 1
   C. dmin >= t
   D. dmin >= 3t
   Answer: B. Radius-t spheres around codewords must not overlap, so t <= floor((dmin-1)/2). (Key point: Correction capability)

3. To correct t errors and also detect s > t errors, which condition is needed?
   A. dmin >= s + t + 1
   B. dmin >= 2s + 1
   C. dmin >= s - t
   D. dmin >= 2t
   Answer: A. The combined requirement is dmin >= t + s + 1. (Key point: Correct and detect together)

4. An extended (8,4) Hamming code has dmin = 4. What does it detect and correct?
   A. Detects 4, corrects 2
   B. Detects 3, corrects 1
   C. Detects 2, corrects 1
   D. Detects 3, corrects 2
   Answer: B. Detects dmin-1 = 3 and corrects floor(3/2) = 1. (Key point: Table values)

## Related in Communication Systems

- [Error detection vs correction; rate ½ convolutional encoder](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aerror-detect-correct-conv-half-rate) — Communication Systems, chapter 8
- [Phase error in synchronous DSB-SC detection; synchronized receiver](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-error-dsbsc-costas) — Communication Systems, chapter 2
- [Phase error in coherent DSB-SC detection; AM demodulation by PLL](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-error-dsbsc-am-pll) — Communication Systems, chapter 2
- [Threshold effect in non-coherent FM detection and its correction](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Athreshold-noncoherent-fm-correction) — Communication Systems, chapter 9
- [Costas loop detector and phase error correction](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acostas-loop-phase-error) — Communication Systems, chapter 2
- [Synchronous detection of DSB-SC; phase and frequency error](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asynchronous-dsbsc-phase-freq-error) — Communication Systems, chapter 2

## Related in other subjects

- [Dataword vs codeword; error coding techniques](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Adataword-codeword-error-coding) — Data Communication, chapter 8
- [Error correction vs detection, bit stuffing](https://www.bctengineer.com/question.md?key=semester_7%2Fcomputer_networks_and_security%3Aerror-correct-detect-bitstuff) — Computer Networks and Security, chapter 3
- [Forward error correction using Hamming code](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afec-hamming) — Data Communication, chapter 8
- [Capability of a code with dmin = 4](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ahamming-distance-dmin4) — Data Communication, chapter 8