Perfect Code

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In mathematics and electronics engineeringa binary Golay code is a type of linear error-correcting perfect binary code used in digital communications. The binary Golay code, along with the ternary Golay codehas a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics. Golay whose paper [2] introducing them has been called, by E.

Berlekampthe "best single published page" in coding theory. There are two closely related binary Golay codes. The extended binary Golay codeG 24 sometimes just called the "Golay code" in finite group theory encodes 12 bits of data in a 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 codeG 23has 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 code 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 perfect binary code between two codewords, respectively.

W is called a linear code because it is a vector space. The binary Golay code, G 23 is a perfect code. That is, the spheres of radius three around code words form a partition of the vector space. G 23 is a dimensional subspace of the space F 2 Perfect binary code other Mathieu groups occur as stabilizers of one or several elements of W. It is perfect binary code to use the " Miracle Octad Generator " format, with co-ordinates in an array of 4 rows, 6 columns.

Addition is taking the symmetric difference. All 6 columns have the same parity, which equals that of the top row. A partition of the 6 columns perfect binary code 3 pairs of adjacent ones constitutes a trio. This is a partition into 3 octad sets. A subgroup, the projective special linear group PSL 2,7 x S 3 of a trio subgroup of M 24 is useful for generating a basis.

PSL 2,7 permutes the octads internally, in parallel. S 3 permutes the 3 octads bodily. The sum N of all 6 of these code words consists of all 1's. Adding N perfect binary code a code word produces its complement.

The 7-cycle acts on T to give a subspace including also the basis elements. The resulting 7-dimensional subspace has a 3-dimensional quotient space upon ignoring the latter 2 octads. There are 4 other code words of similar structure that complete the basis of 12 perfect binary code words for this representation of W.

The Voyager 1 and 2 spacecraft needed to transmit hundreds of color pictures of Jupiter and Saturn perfect binary code their, and fly-by's within a constrained telecommunications bandwidth. Color image transmission required three times the amount of data as black and white images, so the Hadamard code that was used to transmit the black and white images was switched to the Golay 24,12,8 code. The MIL-STD American military standards for automatic link establishment in high frequency radio systems specify the use of an extended 24,12 Golay code for forward error correction.

From Wikipedia, the free encyclopedia. Extended binary Golay code Generator matrix. Perfect binary code Geometry of the Square and Cube. University of Colorado Denver. Specifications, Standards, Handbooks and Mil-Spec documents. Consultative Committee for Space Data Systems. B1 Data Adaptive Entropy Coder. X band S band K u band K band K a band.