Hexacode
Encyclopedia
In coding theory
, the hexacode is length 6 linear code
of dimension 3 over the Galois field
of 4 elements defined by
It is a 3-dimensional subspace of the vector space of dimension 6 over
.
Then
contains 45 codewords of weight
4, 18 codewords of weight 6 and
the zero word. The full automorphism group of the hexacode is
. The hexacode can be used to describe the Miracle Octad Generator
of R. T. Curtis.
Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
, the hexacode is length 6 linear code
Linear code
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although Turbo codes can be seen as a hybrid of these two types. Linear codes allow for...
of dimension 3 over the Galois field
of 4 elements defined byIt is a 3-dimensional subspace of the vector space of dimension 6 over
.Then
contains 45 codewords of weightWeight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...
4, 18 codewords of weight 6 and
the zero word. The full automorphism group of the hexacode is
. The hexacode can be used to describe the Miracle Octad GeneratorMiracle Octad Generator
In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by for manipulating the Mathieu groups, binary Golay code and Leech lattice.-Description:...
of R. T. Curtis.