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

Linear code

In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although Turbo codes can be seen as a hybrid of these two types. Linear codes allow for...

; the extended ternary Golay code is a [12, 6, 6] linear code obtained by adding a zero-sum check digit

Check digit

A check digit is a form of redundancy check used for error detection, the decimal equivalent of a binary checksum. It consists of a single digit computed from the other digits in the message....

 to the [11, 6, 5] code.

Ternary Golay code

The ternary Golay code consists of 3⁶ = 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

Hamming distance

In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different...

 of at most 2 from exactly one codeword.
The code can also be constructed as the quadratic residue code

Quadratic residue code

A quadratic residue code is a type of cyclic code.There is a quadratic residue code of length pover the finite field GF whenever pand l are primes, p is odd andl is a quadratic residue modulo p....

 of length 11 over the finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

 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

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

Mathieu group

In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...

.

The extended ternary Golay code can be constructed as the span of the rows of a Hadamard matrix

Hadamard matrix

In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal...

 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

Steiner system

250px|right|thumbnail|The [[Fano plane]] is an S Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line....

 S(5, 6, 12).

History

The ternary Golay code has been constructed by Marcel J. E. Golay

Marcel J. E. Golay

Marcel J.E. Golay was a Swiss-born mathematician, physicist, and information theorist, who applied mathematics to real-world military and industrial problems. He was born in Neuchâtel, Switzerland.-Career:...

. He published it in a very short correspondence in 1949.
Independently and slightly earlier the same code had also been developed by the Finnish

Finland

Finland , officially the Republic of Finland, is a Nordic country situated in the Fennoscandian region of Northern Europe. It is bordered by Sweden in the west, Norway in the north and Russia in the east, while Estonia lies to its south across the Gulf of Finland.Around 5.4 million people reside...

football pool enthusiast Juhani Virtakallio who published it in 1947.