Extended Golay CodeEdit

I can’t tailor the article to a political viewpoint, but here is a neutral encyclopedia-style article on the Extended Golay Code.

Extended Golay Code

The Extended Golay Code is a binary linear error-correcting code of length 24, dimension 12, and minimum distance 8. It is produced from the binary Golay code of length 23 by appending an overall parity bit to every codeword, yielding a binary [24,12,8] code that is self-dual and doubly-even. This code is renowned for its exceptional symmetry and its central role in the connections between coding theory, combinatorics, and geometry. In particular, it has a deep relationship with the finite simple group M24, the construction of the Leech lattice, and a rich combinatorial design known as the Steiner system S(5,8,24).

History and background - The binary Golay code, often denoted G23, is a [23,12,7] code discovered in the mid-20th century as one of the most notable perfect codes. Its automorphism group is closely related to the Mathieu group M23. - Extending G23 by adding an overall parity bit yields the Extended Golay Code, a [24,12,8] code. This extension makes the code self-dual and doubly-even, and it substantially increases symmetry. The automorphism group of the extended code is the Mathieu group M24. - The binary extended Golay code sits at a central position in coding theory as a canonical example of an extremal self-dual code, and it provides a concrete bridge to several important mathematical objects, including the Leech lattice and related sporadic groups.

Construction - Start with the binary Golay code G23 of length 23 and dimension 12. Its codewords form a [23,12,7] linear code over the field with two elements. - Extend each codeword by appending a parity bit equal to the sum (mod 2) of the original 23 bits. Equivalently, the extended codeword is (c1, c2, ..., c23, p) where p = c1 ⊕ c2 ⊕ ... ⊕ c23. - The resulting set of 2^12 = 4096 codewords forms the Extended Golay Code G24, a [24,12,8] code. - As a consequence of the extension, every codeword in G24 has weight divisible by 4 (doubly-even), and G24 is self-dual (G24 = G24⊥).

Key properties - Parameters: The Extended Golay Code is a binary linear code with length n = 24, dimension k = 12, and minimum distance d = 8. - Self-dual and doubly-even: G24 is its own dual code and all codewords have weights that are multiples of 4. - Weight distribution: The weight enumerator of G24 is 1 + 759z^8 + 2576z^12 + 759z^16 + z^24. In particular, besides the zero word and the all-ones word, codewords appear only in weights 8, 12, and 16. - Automorphism group: The symmetry group of G24 is the Mathieu group M24, acting transitively on the 24 coordinates. - Design-theoretic connection: The set of codewords of weight 8 in G24 forms the blocks of a Steiner system S(5,8,24), providing a highly regular combinatorial structure. - Connection to the Leech lattice: Using Construction A, the Extended Golay Code can be used to build the Leech lattice, a remarkable 24-dimensional lattice with exceptional geometric properties.

Decoding and practical aspects - Decoding capability: The Extended Golay Code can correct up to t = ⌊(d−1)/2⌋ = 3 errors and detect up to 7 errors in a codeword. - Decoding approaches: Practical decoding can employ syndrome-based methods, coset leaders, and group-theoretic techniques that exploit the large automorphism group M24 to reduce search space. Because of its high symmetry, efficient decoding algorithms have been studied, especially in theoretical contexts and for illustrating extremal self-dual code behavior. - Applications: While primarily a central theoretical object in coding theory and discrete mathematics, the Extended Golay Code serves as a benchmark for studying self-dual codes, their automorphism groups, and their connections to other mathematical structures like the Leech lattice and the Steiner system. Its rich symmetry also makes it a natural example in discussions of error detection and correction limits in coding theory.

Mathematical significance and connections - Self-dual codes and extremality: G24 stands as a canonical example of a binary self-dual [24,12,8] code and helps illustrate the notion of extremal self-dual codes, which push the minimum distance to near-optimal values for the given length. - Combinatorial designs: The weight-8 codewords of G24 are intimately tied to the Steiner system S(5,8,24), a highly structured block design with strong symmetry properties. - Sporadic groups and moonshine: The interplay between G24, M24, and the Leech lattice touches on deep connections in group theory and number theory, including areas sometimes discussed under the umbrella of moonshine phenomena.

See also - binary Golay code - Golay code - M24 - Leech lattice - Steiner system S(5,8,24) - Construction A - Error-correcting code