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	${\displaystyle [11,6,5]_{3}}$-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	${\displaystyle [12,6,6]_{3}}$-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 ${\displaystyle [11,6,5]_{3}}$-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.

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 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.

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