Error Detection Capability from Minimum Hamming Distance

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Nov 4, 2026

In coding theory, a communication code’s robustness is determined by its minimum distance $d_{min}$ . The Hamming distance between two codewords measures how many symbol changes are needed to transform one codeword into the other. This directly governs error detection and error correction.

A standard result is:

A code with minimum distance $d_{min}$ can detect up to $d_{min}-1$ errors.
It can correct up to $\left\lfloor \frac{d_{min}-1}{2}\right\rfloor$ errors.

Therefore, if $d_{min}=5$ , the maximum number of errors that can be detected is $5-1=4$ .

We match this to the options:

(i) 2
(ii) 3
(iii) 5
(iv) 4 ✅

type="tip" title="Detection rule to memorize" content="For error detection, use: maximum detectable errors $= d_{min}-1$ (because any error pattern with fewer than $d_{min}$ flips cannot turn one valid codeword into another)."

type="warning" title="Common confusion" content="Do not use the correction bound for detection. Correction is up to $\left\lfloor (d_{min}-1)/2 \right\rfloor$ , which would give $leftarrow$ different (smaller) numbers for $d_{min}=5$ ."

Compute Maximum Detectable Errors

1
Step 1
For a code with minimum Hamming distance $d_{min}$ , the maximum number of detectable errors is $d_{min}-1$ .
2
Step 2
Compute $5-1=4$ .
3
Step 3
The correct choice is (iv) 4.

Key concepts (with quick links)

minimum distance $d_{min}$ sets detection/correction limits.
detected error pattern must be distinguishable from all valid codewords.
pairwise codeword separation is the reason $d_{min}-1$ is the detection ceiling.

Why detection is $d_{min}-1$

From distance to detection limit

Minimum distance

Step 1

Identify $d_{min}$ : smallest Hamming distance between distinct codewords."

Detection capacity

Step 2

Detect up to $d_{min}-1$ errors."

Apply to the problem

Step 3

$d_{min}=5 \Rightarrow 4$ detectable errors."

Knowledge Check

Question 1 of 3

Q1Single choice

A code has minimum Hamming distance $d_{min}=5$ . What is the maximum number of errors it can detect?

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