Hamming Code Calculator

Encode, decode, and perform error detection & correction using Hamming codes

Hamming Code Operations

Hamming Code Size

Choose code size: total bits - data bits

Operation to Perform

Select the Hamming code operation

Data Bits to Encode (4 bits)

Expected length: 4 bits. Spaces will be ignored.

Current length: 0 bits

Results

Message length must be 4 bits for 7-4 Hamming code

Code Configuration: 7-4 Hamming code with 3 parity bits

Error Correction: Can detect and correct single-bit errors

Efficiency: 4/7 = 57.1% data transmission rate

Code Analysis

Hamming Distance: 3 (can detect up to 2-bit errors, correct 1-bit errors)

Redundancy: 3 parity bits for 4 data bits

Error Detection: Single and double-bit errors

Error Correction: Single-bit errors only

Example: 7-4 Hamming Code

Encoding Process

Data bits: 1011

Positions: _ _ 1 _ 0 1 1

Parity calculations:

P1 (pos 1): 1⊕1⊕1 = 1

P2 (pos 2): 1⊕0⊕1 = 0

P3 (pos 4): 0⊕1⊕1 = 0

Result: 1011001

Error Detection

Received: 1111001 (error in pos 3)

Syndrome: 011 (binary) = 3

Error position: 3

Corrected: 1011001

Decoding

Hamming code: 1011001

Extract data bits: positions 3,5,6,7

Decoded data: 1011

Code Properties

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Linear Code

Based on linear algebra principles

Single Error Correction

Corrects any single-bit error

Double Error Detection

Detects up to two-bit errors

Hamming Distance

Minimum distance of 3 between codewords

Hamming Code Tips

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Parity bits are placed at powers of 2 positions

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Each parity bit covers specific data positions

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Syndrome vector indicates error position

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Widely used in computer memory and storage

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Efficient for single-bit error correction

Understanding Hamming Codes

What are Hamming Codes?

Hamming codes are a family of linear error-correcting codes that can detect and correct single-bit errors in data transmission. Developed by Richard Hamming in 1950, they provide an elegant solution to ensure data integrity in digital communications.

Why are They Important?

•Protect against data corruption in memory and storage
•Enable reliable communication over noisy channels
•Provide optimal ratio of data to redundancy bits
•Form the foundation for modern error correction

How Do They Work?

Parity bits at positions: 2⁰, 2¹, 2², 2³, ...

Data bits at remaining positions

Step 1: Place parity bits at power-of-2 positions

Step 2: Calculate parity for each group of bits

Step 3: Use syndrome to locate errors

Step 4: Flip erroneous bit to correct

Key Insight: The syndrome vector directly gives the position of any single-bit error, making correction straightforward.

Common Hamming Code Sizes

Code Size	Total Bits	Data Bits	Parity Bits	Efficiency
7-4	7	4	3	57.1%
15-11	15	11	4	73.3%
31-26	31	26	5	83.9%
63-57	63	57	6	90.5%

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