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.
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