Constructing the Extended Binary Golay Code

Coding theory is the subject which is concerned with how information can be sent over a noisy channel. A code then is a collection of codewords which are strings of a fixed number of letters from an alphabet. Some of these strings are codewords others are not. When a codeword is sent over a channel there is a probability less than /2 that each letter in the codeword will be changed, thus introducing errors into the received codeword. To counter this codes build redundancy into the information they propagate which hopefully makes them resistant to these errors. So, if a received codeword has only a small number of errors, then when it is decoded hopefully we obtain the correct original sent codeword. We formalize these notions in the next subsection. Probably the most famous binary (an alphabet of two letters) codes are the binary Golay codes. They exhibit some astounding qualities and are highly symmetrical objects. The codes have a huge number of constructions and this paper is concerned with examining some of those constructions. Section 1.1 introduces the basic notions that will be required to develop the binary Golay codes. Section 2 gives the first construction of the extended binary Golay code and proves some of the codes basic properties. Section 3 gives several more constructions of the extended binary Golay code.