In
coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are stud ...
, the hexacode is a length 6
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 a ...
of dimension 3 over the
Galois field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
of 4 elements defined by
:
It is a 3-dimensional subspace of the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
of dimension 6 over
.
Then
contains 45
codewords
In communication, a code word is an element of a standardized code or protocol. Each code word is assembled in accordance with the specific rules of the code and assigned a unique meaning. Code words are typically used for reasons of reliability, ...
of
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar q ...
4, 18 codewords of weight 6 and
the zero word. The full
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is th ...
of the hexacode is
. The hexacode can be used to describe the
Miracle Octad Generator
In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for manipulating the Mathieu groups, binary Golay code and Leech lattice.
Description
The Miracle Octad Generator is a 4x6 array of combinat ...
of R. T. Curtis.
References
*{{cite book , first = John H. , last = Conway , authorlink = John Horton Conway , author2=Sloane, Neil J. A. , authorlink2=Neil Sloane , year = 1998 , title = Sphere Packings, Lattices and Groups , url = https://archive.org/details/spherepackingsla0000conw_b8u0 , url-access = registration , edition = (3rd ed.) , publisher = Springer-Verlag , location = New York , isbn = 0-387-98585-9
Coding theory