Decoding the Golay codes
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TL;DR
This work introduces exceptionally simple decoding algorithms for the two extended Golay codes based on recent methods of Conway and Curtis of finding the unique blocks containing five points in either the (5,8,24) Steiner system or the ( 5,6,12) Steiners system.
Abstract
We introduce exceptionally simple decoding algorithms for the two extended Golay codes. The algorithms are based on recent methods of Conway and Curtis of finding the unique blocks containing five points in either the (5,8,24) Steiner system or the (5,6,12) Steiner system. These decoding methods are simple enough to enable decoding extended Golay codes by hand. Both of the methods involve relations between the extended Golay codes and other self-dual codes. Proofs are given explaining these relationships and why the decoding methods work. The decoding algorithms are explained and illustrated with many examples. [3 , chap. 12] has facts about the Mathieu group and some details about decoding the Golay codes.