Using a Golay(24,12) decoder for Golay(23,12)

Yesterday I explained an algebraic decoding algorithm for Golay(24,12) and commented that it was not easy to adapt it to decode Golay(23,12). Today I’ve thought of a simple way to use any Golay(24,12) decoder to decode Golay(23,12).

Recall that a systematic Golay(23,12) code is obtained from a systematic Golay(24,12) by omitting the last component of each codeword (i.e., the codeword \((c_1,\ldots,c_{24})\) from the Golay(24,12) code gives the codeword \((c_1,\ldots,c_{23})\) from the Golay(23,12) code). Conversely, one can obtain a systematic Golay(24,12) code from a systematic Golay(23,12) code by adding a parity bit at the end. This means that \(c_{24} = \sum_{j=1}^{23} c_j\), since \(\sum_{j=1}^{24} c_j = 0\) for all words in a Golay(24,12) code.

The idea to decode a Golay(23,12) code with a Golay(24,12) decoder is first to restore the parity bit \(c_{24}\) and then apply the Golay(24,12) decoder. However, if there are errors in the received codeword, the restored parity bit can also be in error, increasing the number of errors in one.

The key remark is that both Golay(23,12) and Golay(24,12) are able to correct up to 3 errors. Therefore, we only care about restoring the parity bit correctly in the case when there are exactly 3 errors. If there are 2 or less errors, adding another error still gives a word decodable by the Golay(24,12) decoder.

Now note that if there are exactly 3 errors in \((c_1,\ldots,c_{23})\), then \(\sum_{j=1}^{23} c_j\) gives the opposite from the parity of the original codeword. Therefore, we should restore \(c_{24}\) as\[c_{24} = 1 + \sum_{j=1}^{23} c_j\]and then apply the Golay(24,12) decoder.

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