1.2.12 Worked Examples: Error Correction  Summary and Q&A
TL;DR
Understanding how encoding can help detect and correct transmission errors using Hamming distance.
Questions & Answers
Q: How does changing a data bit affect the encoding and parity bits?
Changing a data bit results in the data bit itself being flipped, followed by flipping the corresponding row parity bit and column parity bit to maintain odd parity. The parity of the entire message also flips, requiring the bottom right parity bit to be flipped as well.
Q: What is the minimum Hamming distance for this encoding?
The minimum Hamming distance for this encoding is 4, meaning that when a data bit is flipped, a total of 4 entries need to change to maintain the encoding.
Q: Can this encoding detect and correct 1bit errors?
Yes, since the hamming distance is 4, this encoding can both detect and correct 1bit errors by checking the parity of each row, column, and the full message.
Q: Can this encoding correct 2bit errors?
No, this encoding can only detect 2bit errors but cannot correct them. The hamming distance of 4 falls short of the requirement for correcting 2bit errors, which would require a hamming distance of at least 5.
Summary & Key Takeaways

This content explains the encoding of messages with 9 data bits and 7 parity bits, exploring how changing one data bit affects the encoding and parity bits.

The minimum Hamming distance for this encoding is determined to be 4, indicating the number of entries that need to change to maintain the parity of the entire message when flipping a data bit.

It is established that this encoding can detect and correct 1bit errors, but cannot correct 2bit errors due to the hamming distance of 4.