Login
Login

Group code

{{Multiple issues|

{{confusing|date=May 2015}}

{{no footnotes|date=May 2015}}

}}

In coding theory, 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.

A systematic group code C is a code over G^n of order \left| G \right|^k defined by n-k homomorphisms which determine the parity check 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 endomorphisms of the group instead of symbols from the code's alphabet. For example, considering the generator matrix

:

G = \begin{pmatrix} \begin{pmatrix} 0 0 \\ 1 1 \end{pmatrix} \begin{pmatrix} 0 1 \\ 0 1 \end{pmatrix} \begin{pmatrix} 1 1 \\ 0 1 \end{pmatrix} \\

\begin{pmatrix} 0 0 \\ 1 1 \end{pmatrix} \begin{pmatrix} 11 \\ 1 1 \end{pmatrix} \begin{pmatrix} 0 0 \\ 0 0 \end{pmatrix}

\end{pmatrix}

the elements of this matrix are 2\times 2 matrices which are endomorphisms. In this scenario, each codeword can be represented as

g_1^{m_1} g_2^{m_2} ... g_r^{m_r}

where g_1,... g_r are the generators of G.

See also

References

{{Reflist}}

Further reading

  • {{cite book |title=Coding for Digital Recording |chapter=3.4. Group codes |author-first=John |author-last=Watkinson |publisher=Focal Press |location=Stoneham, MA, USA |date=1990 |isbn=978-0-240-51293-8 |pages=51–61}}
  • {{cite book |author-last1=Biglieri |author-first1=Ezio |author-last2=Elia |author-first2=Michele |doi=10.1109/ISIT.1993.748676 |chapter=Construction of Linear Block Codes Over Groups |title=Proceedings. IEEE International Symposium on Information Theory (ISIT) |page=360 |date=1993-01-17 |isbn=978-0-7803-0878-7|title-link=IEEE International Symposium on Information Theory |s2cid=123694385 }}
  • {{cite journal |author-first1=George David |author-last1=Forney |author-link1=George David Forney |author-first2=Mitch D. |author-last2=Trott |doi=10.1109/18.259635 |title=The dynamics of group codes: State spaces, trellis diagrams and canonical encoders |journal=IEEE Transactions on Information Theory |volume=39 |issue=5 |date=1993 |pages=1491–1593}}
  • {{cite journal |author-first1=Vijay Virkumar |author-last1=Vazirani |author-link1=Vijay Virkumar Vazirani |author-first2=Huzur |author-last2=Saran |author-first3=B. Sundar |author-last3=Rajan |doi=10.1109/18.556679 |title=An efficient algorithm for constructing minimal trellises for codes over finite Abelian groups |journal=IEEE Transactions on Information Theory |volume=42 |number=6 |date=1996 |pages=1839–1854|citeseerx=10.1.1.13.7058 }}
  • {{cite journal |author-first1=Adnan Abdulla |author-last1=Zain |author-first2=B. Sundar |author-last2=Rajan |title=Dual codes of Systematic Group Codes over Abelian Groups |journal=Applicable Algebra in Engineering, Communication and Computing |volume=8 |number=1 |pages=71–83 |date=1996}}

Category:Coding theory