Extended Golay¶

After several attempts to improve this code, Golay Extended code is reached. In this version, we can detect 4-bit errors thanks to adding a parity check bit to the Golay code which allows all combinations of 4 bit errors to be detected but not corrected, and then all odd numbers of bit errors can be detected in codeword. We noted it as Golay (24, 12) and it is used for deep space communication.
Therefore, a trade-off should be taken into account between correct and detect error. Since the maximum number of error detected is 6 in any 24-codeword, 3 bits maximum can be corrected.

So, in any application, depending on what we favorite between error detection or error correction, we get different result;

Error detection case (per 24-bit) in extended Golay

• 100% of one- to six-bit errors detected, any pattern
• 100% of odd bit-errors detected, any pattern
• 99.988% of other errors detected

Using the error correction facilities of the code, these are the data reliability rates:

• 100% of one- to three-bit errors corrected, any pattern
• 100% of four-bit errors detected, any pattern
• 100% of odd numbers of bit errors detected, any pattern
• 0.24% of other errors corrected (1/4096)

We still have researches on this kind of algorithm especially on designing topic.
We will try to implement a simple decoding algorithm using Golay (24, 12, 8)
We distinguish 2 types of this code

• Ternary
• Binary

While ternary has codewords equal 11, Binary has codewords equal 23.
We do mostly the ternary code, but this project will focus more on the Binary version of Golay.

References¶

Article Using The Golay Error Detection And Correction Code
http://aqdi.com/articles/using-the-golay-error-detection-and-correction-code-3/

Article IEEE Simplified Algorithm and Hardware Implementation for the (24, 12, 8) Extended Golay Soft
http://www.ccis2k.org/iajit/PDF/vol.11,no.2/4478.pdf