Mathematical Definition
In mathematical terms, the extended binary Golay code consists of a 12-dimensional subspace W of the space V=F224 of 24-bit words such that any two distinct elements of W differ in at least eight coordinates. Equivalently, any non-zero element of W has at least eight non-zero coordinates.
- The possible sets of non-zero coordinates as w ranges over W are called code words. In the extended binary Golay code, all code words have the Hamming weights of 0, 8, 12, 16, or 24.
- Up to relabeling coordinates, W is unique.
The perfect binary Golay code is a perfect code. That is, the spheres of radius three around code words form a partition of the vector space.
The automorphism group of the binary Golay code is the Mathieu group . The automorphism group of the extended binary Golay code is the Mathieu group . The other Mathieu groups occur as stabilizers of one or several elements of W.
The Golay code words of weight eight are elements of the S(5,8,24) Steiner system.
Read more about this topic: Binary Golay Code
Famous quotes containing the words mathematical and/or definition:
“What he loved so much in the plant morphological structure of the tree was that given a fixed mathematical basis, the final evolution was so incalculable.”
—D.H. (David Herbert)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (18391894)