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 ...

, group codes are a type of code. Group codes consist of

n

linear block codes which are subgroups of

G^n

, where

G

is a finite

Abelian group In mathematics, 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 the order in which they are written. That is, the group operation is com ...

. A systematic group code

C

is a code over

G^n

of order

\left,  G \^k

defined by

n-k

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

s which determine the

parity check A parity bit, or check bit, is a bit added to a string of binary code. Parity bits are a simple form of error detecting code. Parity bits are generally applied to the smallest units of a communication protocol, typically 8-bit octets (bytes) ...

bits. The remaining

k

bits are the information bits themselves.

Construction

Group codes can be constructed by special generator matrices which resemble generator matrices of linear block codes except that the elements of those matrices are

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a ...

s of the group instead of symbols from the code's alphabet. For example, considering the generator matrix :

G = \begin \begin  0 0 \\ 1 1 \end \begin  0 1 \\ 0 1 \end \begin  1 1 \\ 0 1 \end \\
\begin  0 0 \\ 1 1 \end \begin  11 \\ 1 1 \end \begin  0 0 \\ 0 0 \end
\end

the elements of this matrix are

2\times 2

matrices which are endomorphisms. In this scenario, each codeword can be represented as

g_1^ g_2^ ... g_r^

where

g_1,... g_r

are the generators of

G

Construction

See also

References

Further reading