Hamming (7,4) Code with Even Parity for Error Detection and Correction

The Hamming (7,4) code is a forward error correction code that encodes 4 data bits into 7 bits by adding 3 parity bits. It can detect and correct single-bit errors. Here's how the process works step-by-step:

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Step 1: Understand the Hamming Code Structure

The Hamming (7,4) code uses 7 bits structured as:

1. Bits 1, 2, 4: Parity bits $P_1, P_2, P_4$
2. Bits 3, 5, 6, 7: Data bits $D_1, D_2, D_3, D_4$

The encoded message $M = [P_1, P_2, D_1, P_4, D_2, D_3, D_4]$.

Step 2: Transmitted Codeword

The given transmitted codeword is:

$\text{Transmitted Codeword: } 1011001$

However, an error occurred during transmission, and the 5th bit was flipped. We'll find and correct this error.

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Step 3: Error Detection with Syndrome

To detect errors, compute the parity for specific bit groups using syndrome bits:

1. Syndrome 1 ($S_1$) checks positions $1, 3, 5, 7$
2. Syndrome 2 ($S_2$) checks positions $2, 3, 6, 7$
3. Syndrome 4 ($S_4$) checks positions $4, 5, 6, 7$

The syndromes are calculated as:

$S = [S_4, S_2, S_1]$

where each syndrome is the parity of the corresponding group.

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Step 4: Calculate the Syndrome Values

Split the Codeword into Parity Groups

• $P_1 = 1, P_2 = 0, P_4 = 0$
• $D_1 = 1, D_2 = 0, D_3 = 0, D_4 = 1$

Compute the Parities

Using even parity:

1. $S_1 = (P_1 \oplus D_1 \oplus D_2 \oplus D_4) = (1 \oplus 1 \oplus 0 \oplus 1) = 1$
2. $S_2 = (P_2 \oplus D_1 \oplus D_3 \oplus D_4) = (0 \oplus 1 \oplus 0 \oplus 1) = 0$
3. $S_4 = (P_4 \oplus D_2 \oplus D_3 \oplus D_4) = (0 \oplus 0 \oplus 0 \oplus 1) = 1$

[ S = [S_4, S_2, S_1] = [1, 0, 1] = 5 ]

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Step 5: Identify and Correct the Error

The syndrome value $S = 5$ indicates the 5th bit is incorrect. Flipping the 5th bit of the codeword:

$\text{Received Codeword: } 1011001 \quad \longrightarrow \quad \text{Corrected Codeword: } 1010001$

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Step 6: Retrieve the Original 4-bit Message

The corrected codeword $1010001$ maps to:

[ [D_1, D_2, D_3, D_4] = [1, 0, 0, 1] ]

Thus, the original 4-bit message is:

$\boxed{1001}$

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Summary

1. Error detection is performed using parity checks to compute the syndrome.
2. The syndrome $S = 5$ points to the 5th bit.
3. Correcting the 5th bit restores the codeword.
4. The original 4-bit message is $1001$.

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Would you like further explanation on any of these steps or examples? Here are related questions to explore:

1. What are the roles of parity bits in the Hamming (7,4) code?
2. How does the Hamming code handle multiple-bit errors?
3. Can the Hamming code be extended for larger data packets?
4. How is the syndrome calculated for an 8-bit message using the Hamming code?
5. What is the difference between even and odd parity in error correction?

Tip: Understanding syndrome calculation is key to decoding and correcting errors in Hamming codes efficiently!