In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics.[1] These codes are named in honor of Marcel J. E. Golay whose 1949 paper[2] introducing them has been called, by E. R. Berlekamp, the "best single published page" in coding theory.[3]
There are two closely related binary Golay codes. The extended binary Golay code, G24 (sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 7-bit errors can be detected.
The other, the perfect binary Golay code, G23, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit). In standard coding notation, the codes have parameters [24, 12, 8] and [23, 12, 7], corresponding to the length of the codewords, the dimension of the code, and the minimum Hamming distance between two codewords, respectively.
^Thompson 1983
^Golay, Marcel J. E. (1949). "Notes on Digital Coding" (PDF). Proc. IRE. 37: 657. Archived from the original (PDF) on April 10, 2023.
^Berlekamp, E. R. (1974), Key Papers in the Development of Coding Theory, I.E.E.E. Press, p. 4
tool introduced by Rob T. Curtis for studying the Mathieu groups, binaryGolaycode and Leech lattice. The Miracle Octad Generator is a 4x6 array of combinations...
GPS etc. Berger code Constant-weight code Convolutional code Expander codes Group codesGolaycodes, of which the BinaryGolaycode is of practical interest...
the 24-bit binaryGolaycode are generated, and the 759 codewords with a Hamming weight of 8 correspond to the S(5,8,24) system. The Golaycode can be constructed...
In coding theory, the ternary Golaycodes are two closely related error-correcting codes. The code generally known simply as the ternary Golaycode is...
The group M24 is the permutation automorphism group of the extended binaryGolaycode W, i.e., the group of permutations on the 24 coordinates that map...
codes, Hadamard codes, Expander codes, Golaycodes, Reed–Muller codes and Polar codes. These examples also belong to the class of linear codes, and hence they...
inventing the field of information theory. The binaryGolaycode was developed in 1949. It is an error-correcting code capable of correcting up to three errors...
{\displaystyle (23,12)} binaryGolaycode over G F ( 2 ) {\displaystyle GF(2)} and the ( 11 , 6 ) {\displaystyle (11,6)} ternary Golaycode over G F ( 3 ) {\displaystyle...
octads from code words of Hamming weight 8 from the extended binaryGolaycode, one of five different Hamming weights the extended binaryGolaycode uses: 0...
Conway and Neil Sloane. The binary lexicographic codes are linear codes, and include the Hamming codes and the binaryGolaycodes. A lexicode of length n...
coefficients sum to zero. Binary complementary sequences were first introduced by Marcel J. E. Golay in 1949. In 1961–1962 Golay gave several methods for...
Hamming(7,4) code which adds three parity bits to four bits of data. In mathematical terms, Hamming codes are a class of binary linear code. For each integer...
24 objects. (In coding theory the term "binaryGolaycode" often refers to a shorter related length 23 code, and the length 24 code used here is called...
doubly even codes are the extended binary Hamming code of block length 8 and the extended binaryGolaycode of block length 24. These two codes are, in addition...
The codewords of the extended binaryGolaycode have a length of 24 bits and have weights 0, 8, 12, 16, or 24. This code can correct up to three errors...
Barker sequences or Barker codes, after the inventor Ronald Hugh Barker. The process is described in "Group Synchronisation of Binary Digital Systems" published...
linear codes. There are other, subtler relationships between Euclidean sphere packing and error-correcting codes. For example, the binaryGolaycode is closely...
Global Information
![{\displaystyle [24,12,8]_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94468240ab43ea76dd2a763b3ae1442fdc6bdded)
![{\displaystyle [23,12,7]_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17f5f280c4e4796b177454dd9e84c3b7ade8325)