Golay code

The term Golay code stands for two closely related codes that have a prominent position in coding theory . They are (apart from trivial code and repeat codes ) up to isomorphism the only two perfect codes that can correct a mistake as more. They are named after the Swiss electrical engineer Marcel JE Golay . In both cases it is a quadratic remainder code and therefore in particular a cyclic code and a linear code .

The binary Golay code

Generator matrix for the extended binary Golay code

The binary Golay code is defined as the binary quadratic remainder code of length 23. As a linear code it has the parameters . This means that the code is a 12-dimensional sub-vector space of the 23-dimensional vector space with the minimum Hamming distance 7. It follows . The code is therefore 3-error correcting. ${\displaystyle G_{23}}$ ${\displaystyle (n,k,d)=(23,12,7)}$ ${\displaystyle \mathbb {F} _{2}^{23}}$ ${\displaystyle t=\left\lfloor {\frac {d-1}{2}}\right\rfloor =3}$

The parameters satisfy the equation

${\displaystyle q^{k}\sum _{i=0}^{t}{n \choose i}(q-1)^{i}=q^{n}}$

That is why the Golay binary code is perfect . ${\displaystyle G_{23}}$

The extended binary Golay code

If you add a parity bit to the binary Golay code , you get the extended binary Golay code with the parameters . This code is double straight , i.e. H. all code words have a Hamming weight that is divisible by 4 . ${\displaystyle G_{23}}$ ${\displaystyle G_{24}}$ ${\displaystyle (n,k,d)=(24,12,8)}$

The automorphism group of the extended binary Golay code is the Mathieu group , a sporadic group . ${\displaystyle M_{24}}$

The Golay Ternary Code

The ternary Golay code is defined as the ternary quadratic residue code of length 11. As a linear code it has the parameters . This means that the code is a 6-dimensional sub-vector space of the 11-dimensional vector space with the minimum distance 5. It follows . The code is therefore 2-error-correcting. Here, too, the parameters satisfy the above equation, so the Golay ternary code is also perfect. ${\displaystyle G_{11}}$ ${\displaystyle (n,k,d)=(11,6,5)}$ ${\displaystyle \mathbb {F} _{3}^{11}}$ ${\displaystyle t=\left\lfloor {\frac {d-1}{2}}\right\rfloor =2}$ ${\displaystyle G_{11}}$