Ternary Golay code

There are two closely related error-correcting codes known as ternary Golay codes. The code generally known simply as the ternary Golay code is a perfect (11, 6, 5) linear code; the extended ternary Golay code is a (12, 6, 6) linear code obtained by adding a zero-sum check digit to the (11, 6, 5) code.

The complete weight enumerator of the extended ternary Golay code is

$x^{12}+y^{12}+z^{12}+22(x^{6}y^{6}+y^{6}z^{6}+z^{6}x^{6})+220(x^{6}y^{3}z^{3}+y^{6}z^{3}x^{3}+z^{6}x^{3}y^{3})$.

The perfect ternary Golay code can be constructed as the quadratic residue code of length 11 over the finite field F₃.

The automorphism group of the extended ternary Golay code is 2.M₁₂, where M₁₂ is a Mathieu group.

References

J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups, Springer-Verlag, New York, Berlin, Heidelberg, 1988.