Hamming Distance Calculator – Calculating Hamming Distance Between Binary Strings

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.

Hamming Distance Calculator

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. 1 vs 1 (match)
2. 1 vs 0 (differ)
3. 1 vs 0 (differ)
4. 1 vs 0 (differ)
5. 1 vs 0 (differ)
6. 0 vs 1 (differ)
7. 0 vs 1 (differ)
8. 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. 1 vs 1 (match)
2. 0 vs 1 (differ)
3. 1 vs 1 (match)
4. 0 vs 1 (differ)
5. 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. 1 vs 1 (match)
2. 0 vs 0 (match)
3. 1 vs 0 (differ)
4. 0 vs 1 (differ)
5. 1 vs 0 (differ)
6. 0 vs 0 (match)
7. 1 vs 1 (match)
8. 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

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