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 computer data storage, data sto ...
, 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 Code word (communication), codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although t ...
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, subtr ...
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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
of dimension 6 over
.
Then
contains 45
codewords Code word may refer to:
* Code word (communication), an element of a standardized code or protocol
* Code word (figure of speech), designed to convey a predetermined meaning to a receptive audience, while remaining inconspicuous to others
** Proce ...
of
weight
In science and engineering, the weight of an object is a quantity associated with the gravitational force exerted on the object by other objects in its environment, although there is some variation and debate as to the exact definition.
Some sta ...
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 the g ...
of the hexacode is
.
The hexacode can be used to describe the
Miracle Octad Generator
of R. T. Curtis.
References
*{{cite book , first1 = John H. , last1 = Conway , authorlink = John Horton Conway , last2=Sloane, first2=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 , publisher = Springer-Verlag , location = New York , isbn = 0-387-98585-9
Coding theory