Group code explained
In coding theory, 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.
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 endomorphisms of the group instead of symbols from the code's alphabet. For example, considering the generator matrix
G=\begin{pmatrix}\begin{pmatrix}00\ 11\end{pmatrix}\begin{pmatrix}01\ 01\end{pmatrix}\begin{pmatrix}11\ 01\end{pmatrix}\\
\begin{pmatrix}00\ 11\end{pmatrix}\begin{pmatrix}11\ 11\end{pmatrix}\begin{pmatrix}00\ 00\end{pmatrix}
\end{pmatrix}
the elements of this matrix are
matrices which are endomorphisms. In this scenario, each codeword can be represented as
where
are the
generators of
.
See also
Further reading
- Book: Coding for Digital Recording . 3.4. Group codes . John . Watkinson . . Stoneham, MA, USA . 1990 . 978-0-240-51293-8 . 51–61.
- Book: Biglieri . Ezio . Elia . Michele . 10.1109/ISIT.1993.748676 . Construction of Linear Block Codes Over Groups . Proceedings. IEEE International Symposium on Information Theory (ISIT) . 360 . 1993-01-17 . 978-0-7803-0878-7. IEEE International Symposium on Information Theory . 123694385 .
- George David . Forney . George David Forney . Mitch D. . Trott . 10.1109/18.259635 . The dynamics of group codes: State spaces, trellis diagrams and canonical encoders . . 39 . 5 . 1993 . 1491–1593.
- Vijay Virkumar . Vazirani . Vijay Virkumar Vazirani . Huzur . Saran . B. Sundar . Rajan . 10.1109/18.556679 . An efficient algorithm for constructing minimal trellises for codes over finite Abelian groups . . 42 . 6 . 1996 . 1839–1854. 10.1.1.13.7058 .
- Adnan Abdulla . Zain . B. Sundar . Rajan . Dual codes of Systematic Group Codes over Abelian Groups . Applicable Algebra in Engineering, Communication and Computing (AAECC) . 8 . 1 . 71–83 . 1996.
