Ternary Golay code

Perfect ternary Golay code
Named after	Marcel J. E. Golay
Classification
Type	Linear block code
Block length	11
Message length	6
Rate	6/11 ~ 0.545
Distance	5
Alphabet size	3
Notation	$[11,6,5]_{3}$-code

Extended ternary Golay code
Named after	Marcel J. E. Golay
Classification
Type	Linear block code
Block length	12
Message length	6
Rate	6/12 = 0.5
Distance	6
Alphabet size	3
Notation	$[12,6,6]_{3}$-code

In coding theory, the ternary Golay codes are two closely related error-correcting codes. The code generally known simply as the ternary Golay code is an $[11,6,5]_{3}$-code, that is, it is a linear code over a ternary alphabet; the relative distance of the code is as large as it possibly can be for a ternary code, and hence, the ternary Golay code is a perfect 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. In finite group theory, the extended ternary Golay code is sometimes referred to as the ternary Golay code.

Properties

Ternary Golay code

The ternary Golay code consists of 3⁶ = 729 codewords. Its parity check matrix is

$\left[{\begin{array}{ccccccccccc}1&1&1&2&2&0&1&0&0&0&0\\1&1&2&1&0&2&0&1&0&0&0\\1&2&1&0&1&2&0&0&1&0&0\\1&2&0&1&2&1&0&0&0&1&0\\1&0&2&2&1&1&0&0&0&0&1\end{array}}\right].$

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.

The set of codewords with Hamming weight 5 is a 3-(11,5,4) design.

Extended ternary Golay code

The complete weight enumerator of the extended ternary Golay code is

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

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

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 was discovered by Golay (1949). It was independently discovered two years earlier by the Finnish football pool enthusiast Juhani Virtakallio, who published it in 1947 in issues 27, 28 and 33 of the football magazine Veikkaaja. (Barg 1993, p.25)

See also

• Binary Golay code

References

• Barg, Alexander (1993), "At the dawn of the theory of codes", The Mathematical Intelligencer, 15 (1): 20–26, doi:10.1007/BF03025254, ISSN 0343-6993, MR 1199273
• 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