Ternary Golay code

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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) ternary 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

x¹² + y¹² + z¹² + 22(x⁶y⁶ + y⁶z⁶ + z⁶x⁶) + 220(x⁶y³z³ + y⁶z³x³ + z⁶x³y³).

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.

Consider all codewords of the extended code which have just six nonzero digits. The sets of positions at which these nonzero digits occur form the Steiner system S(5, 6, 12).

[edit] References

• J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups, Springer-Verlag, New York, Berlin, Heidelberg, 1988.
• Griess, Robert L.: "Twelve Sporadic Groups", Springer-Verlag, 1998.