- Ternary Golay code
There are two closely related

error-correcting code s known as**ternary Golay codes**. The code generally known simply as the**ternary Golay code**is a perfect (11, 6, 5) ternarylinear code ; the**extended ternary Golay code**is a (12, 6, 6) linear code obtained by adding a zero-sumcheck 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:$egin\{bmatrix\}11122010000\backslash 11210201000\backslash 12101200100\backslash 12012100010\backslash 10221100001end\{bmatrix\}$Any two different codewords differ in at least 5 positions.Every ternary word of length 11 has aHamming distance of at most 2 from exactly one codeword.The code can also be constructed as thequadratic residue code of length 11 over thefinite field **F**_{3}.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 aMathieu 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**F**_{3}.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 in1949 .Independently and slightly earlier the same code had also been developed by the Finnish football pool enthusiast Juhani Virtakallio who published it in1947 .**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

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