Group code: Difference between revisions

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In [[Error correction and detection]], '''Group codes''' are <math>n</math> length [[Linear block codes]] which are subgroups of <math>G^n</math>, where <math>G</math> is a [[Finite Abelian group]].

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In [[coding theory]], '''group codes''' are a type of [[coding theory|code]]. Group codes consist of

A systematic group code <math>C</math> is a code over <math>G^n</math> of order <math>\left| G \right|^k</math> defined by <math>n-k</math> homomorphisms which determine the parity check bits. The remaining <math>k</math> bits are the information bits themselves.

In [[Error correction and detection]], '''Group codes''' are <math>n</math> length [[Linearlinear block codes]] which are subgroups of <math>G^n</math>, where <math>G</math> is a finite [[Finite Abelian group]].

A systematic group code <math>C</math> is a code over <math>G^n</math> of order <math>\left| G \right|^k</math> defined by <math>n-k</math> homomorphisms[[homomorphism]]s which determine the [[parity check]] bits. The remaining <math>k</math> bits are the information bits themselves.

== Construction ==

Group codes can be constructed by special [[generator matrix|generator matrices]] which resemble generator matrices of linear block codes except that the elements of those matrices are endomorphisms[[endomorphism]]s of the group instead of symbols from the code's alphabet. For example, considerconsidering the generator matrix below...

:<math>

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}

</math>

Thethe elements of this matrix are <math>2 \intotimes 2</math> matrices which are endomorphisms. In this scenario, each codeword can be represented as

<math>c = (c_1,...,c_n) = g_1^m_1 g_2^m2 ... g_r^ m_r</math>

where <math>g_1,^{m_1} g_2^{m_2} ... g_r^{m_r}</math> are the generators of <math>G</math>.

where <math>g_1,... g_r</math> are the [[Generating set of a group|generator]]s of <math>G</math>.

[[Category:Error detection and correction]]

== See also ==

* [[Group coded recording]] (GCR)

== 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<!-- Dave --> |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<!-- Alsaggaf --> |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<!-- Appl. Algebra Eng. Commun. Comput. --> |volume=8 |number=1 |pages=71–83 |date=1996}}

[[Category:Coding theory]]