Group code
Encyclopedia
In computer science
, group codes are a type of code
. Group codes consist of
linear block codes which are subgroups of 
, where 
is a finite Abelian group
.
A systematic group code
is a code over 
of order 
defined by 
homomorphisms which determine the parity check bits. The remaining 
bits are the information bits themselves.
which resemble generator matrices of linear block codes except that the elements of those matrices are endomorphism
s of the group instead of symbols from the code's alphabet. For example, consider the generator matrix
The elements of this matrix are
matrices which are endomorphisms. In this scenario, each codeword can be represented as
where
are the generator
s of
.
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, group codes are a type of code
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...
. Group codes consist of
linear block codes which are subgroups of , where is a finite Abelian groupAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
.
A systematic group code
is a code over of order defined by homomorphisms which determine the parity check bits. The remaining bits are the information bits themselves.Construction
Group codes can be constructed by special generator matricesGenerator matrix
In coding theory, a generator matrix is a basis for a linear code, generating all its possible codewords.If the matrix is G and the linear code is C,where w is a codeword of the linear code C, c is a row vector, and a bijection exists between w and c. A generator matrix for an q-code has...
which resemble generator matrices of linear block codes except that the elements of those matrices are endomorphism
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
s of the group instead of symbols from the code's alphabet. For example, consider the generator matrix
The elements of this matrix are
matrices which are endomorphisms. In this scenario, each codeword can be represented aswhere
are the generatorGenerating set of a group
In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
s of
.