Binary Golay code

The term Golay code represents two closely related codes which occupy a prominent position in coding theory. They are (apart from trivial codes and repetition codes), up to isomorphism the only two perfect codes that can correct more than one error. They are named after the Swiss electrical engineer Marcel JE Golay. In both cases, it is a quadratic residue code, and thus in particular a cyclic code and a linear code.

The binary Golay code

Binary Golay code is defined as the quadratic residues binary code of length 23, as it has the linear code parameters. This means that the code is a 12 -dimensional subspace of the 23- dimensional vector space having the minimum Hamming distance of 7. There follows. The code is so 3 - error- correcting.

The parameters satisfy the equation

Therefore, the binary Golay code is perfect.

The extended binary Golay code

If you hang the binary Golay code, a parity bit, one obtains the extended binary Golay code with the parameters. This code is just twice, that is, all codewords have a divisible by 4 Hamming weight.

The automorphism group of the extended binary Golay code is the Mathieu group, a sporadic group.

The ternary Golay code

The ternary Golay code is defined as the quadratic residues ternary code of length 11, as it has the linear code parameters. This means that the code is a 6 -dimensional subspace of the 11- dimensional vector space with the minimum distance 5. There follows. The code is thus 2 - error- correcting. Again, the parameters satisfy the above equation, including the ternary Golay code is perfect.

• Coding Theory