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) 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.

Properties

Ternary Golay code

The ternary Golay code consists of $3^6=729$ codewords. Its parity check matrix is:Any two different codewords differ in at least 5 positions.Every ternary word of length 11 has a Hamming distance of at most 2 from exactly one codeword.The code can also be constructed as the quadratic residue code of length 11 over the finite field F₃.

Used in a football pool with 11 games, the ternary Golay code corresponds to 729 bets and guarantees exactly one bet with at most 2 wrong outcomes.

Extended ternary Golay code

The complete weight enumerator of the extended ternary Golay code is:$x^\{12\}+y^\{12\}+z^\{12\}+22left(x^6y^6+y^6z^6+z^6x^6\; ight)+220left(x^6y^3z^3+y^6z^3x^3+z^6x^3y^3\; ight)$.

The automorphism group of the extended ternary Golay code is 2."M"₁₂, where "M"₁₂ is a Mathieu group.

The extended ternary Golay code can be constructed as the span of the rows of a Hadamard matrix of order 12 over the field F₃.

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).

History

The ternary Golay code has been constructed by Marcel J. E. Golay. He published it in a very short correspondence in 1949.Independently and slightly earlier the same code had also been developed by the Finnish football pool enthusiast Juhani Virtakallio who published it in 1947.

See also

* Binary Golay code

References

* M.J.E. Golay, Notes on digital coding, "Proceedings of the I.R.E." 37 (1949) 657
* I.F. Blake (ed.), "Algebraic Coding Theory: History and Development", Dowden, Hutchinson & Ross, Stroudsburg 1973
* J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, New York, Berlin, Heidelberg, 1988.
* Robert L. Griess, "Twelve Sporadic Groups", Springer, 1998.
* G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, "Covering Codes", Elsevier (1997) ISBN 0-444-82511-8
* Th. M. Thompson, "From Error Correcting Codes through Sphere Packings to Simple Groups", The Mathematical Association of America 1983, ISBN 0-88385-037-0