Hamming Distance Calculator

The Richard Hamming, hamming distance calculator is a powerful tool used in information theory and coding to measure the divergence between two binary strings.

Consider the following example:

Binary String 1: 1010
Binary String 2: 1100

To calculate the Hamming distance, we compare each bit:

First bit: 1 vs 1 (match)
Second bit: 0 vs 1 (differ)
Third bit: 1 vs 0 (differ)
Fourth bit: 0 vs 0 (match)

The Hamming distance in this case is 2, as there are two positions where the bits differ.

Binary String A	Binary String B	Hamming Distance
1100110	1010101	3
11110000	00001111	8
10101010	01010101	8
11001100	11110000	4
10001	10011	2
00000000	11111111	8
11111111	00000000	8
10111010	10010011	4
00110011	11001100	4
01010101	10101010	8
11100011	11000101	3
01101100	10111010	5
00101010	00100001	2
11010101	11100111	4
10011001	01100110	6

Hamming Distance Calculation Formula

For two binary strings A and B of equal length, the Hamming distance H(A, B) is defined as:

H(A, B) = Σ(A[i] ⊕ B[i])

Where:

Σ represents the sum
⊕ denotes the XOR operation
i ranges from 1 to the length of the strings

This formula counts the number of ‘1’s in the XOR result of the two strings.

Let’s break it down with an example:

A: 1011
B: 1101

Perform XOR: 1011 ⊕ 1101 = 0110

Count ‘1’s in the result: The result has 2 ‘1’s.

The Hamming distance between 1011 and 1101 is 2.

How is Hamming distance calculated?

Calculating Hamming distance involves a bit-by-bit comparison of two binary strings. Here’s a step-by-step process:

Ensure both strings are of equal length.
Initialize a counter to zero.
Compare corresponding bits from left to right.
For each position where bits differ, increment the counter.
The final counter value is the Hamming distance.

Let’s apply this to an example:

String 1: 10110
String 2: 11001

Comparison:

1|0|1|1|0
1|1|0|0|1
X| |X|X|X

Counting these, we get a Hamming distance of 4.

What is the Hamming distance between 11111000 and 10000111?

Let’s analyze this pair of 8-bit strings:

11111000
10000111

Comparing bit by bit:

1 vs 1 (match)
1 vs 0 (differ)
1 vs 0 (differ)
1 vs 0 (differ)
1 vs 0 (differ)
0 vs 1 (differ)
0 vs 1 (differ)
0 vs 1 (differ)

Counting the differences, we find that the Hamming distance is 7.

What is the Hamming distance between 10101 and 11110?

For this 5-bit pair:

10101
11110

Bit-wise comparison:

1 vs 1 (match)
0 vs 1 (differ)
1 vs 1 (match)
0 vs 1 (differ)
1 vs 0 (differ)

The Hamming distance here is 3.

What is the Hamming distance between 10101010 and 10010010?

Examining this 8-bit pair:

10101010
10010010

Comparison:

1 vs 1 (match)
0 vs 0 (match)
1 vs 0 (differ)
0 vs 1 (differ)
1 vs 0 (differ)
0 vs 0 (match)
1 vs 1 (match)
0 vs 0 (match)

The Hamming distance in this case is 3.

Sample Calculations Using the Hamming Distance Calculator

To further illustrate the concept, let’s create a table of sample calculations:

References

Hamming, R. W. (1950). Error detecting and error correcting codes. The Bell System Technical Journal, 29(2), 147-160. https://ieeexplore.ieee.org/document/6772729
MacKay, D. J. (2003). Information theory, inference and learning algorithms. Cambridge University Press. https://www.cambridge.org/core/books/information-theory-inference-and-learning-algorithms/2E26154E6D44BDCC0F6C78CE2F8AEB9F
Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms. MIT Press. https://mitpress.mit.edu/books/introduction-algorithms-third-edition

Related Tools:

Hamming Distance Calculator – Calculating Hamming Distance Between Binary Strings