Rectified 5-cell: Difference between revisions

Revision as of 13:38, 23 March 2013

In computer science, 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 ${| G |}^{k}$ 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, consider the generator matrix

G = (\begin{matrix} (\begin{matrix} 00 \\ 11 \end{matrix}) (\begin{matrix} 01 \\ 01 \end{matrix}) (\begin{matrix} 11 \\ 01 \end{matrix}) \\ (\begin{matrix} 00 \\ 11 \end{matrix}) (\begin{matrix} 11 \\ 11 \end{matrix}) (\begin{matrix} 00 \\ 00 \end{matrix}) \end{matrix})

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 $G$ .

References

Template:Cite doi
G. D. Forney, M. Trott, Template:Doi-inline, IEEE Trans. Inform. theory, Vol 39 (1993), pages 1491-1593.
V. V. Vazirani, Huzur Saran and B. S. Rajan, Template:Doi-inline, IEEE Trans. Inform. Theory 42, No.6, (1996), 1839-1854.
A. A. Zain, B. Sundar Rajan, "Dual codes of Systematic Group Codes over Abelian Groups", Appl. Algebra Eng. Commun. Comput. 8(1): 71-83 (1996).

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+In [[computer science]], '''group codes''' are a type of [[coding theory|code]]. Group codes consist of
+<math>n</math> [[linear block codes]] which are subgroups of <math>G^n</math>, where <math>G</math> is a 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 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 [[endomorphism]]s of the group instead of symbols from the code's alphabet. For example, consider the generator matrix
+:<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>
+The elements of this matrix are <math>2\times 2</math> matrices which are endomorphisms. In this scenario, each codeword can be represented as
+<math>g_1^{m_1} g_2^{m_2} ... g_r^{m_r}</math>
+where <math>g_1,... g_r</math> are the [[Generating set of a group|generator]]s of <math>G</math>.
+== References ==
+* {{cite doi|10.1109/ISIT.1993.748676}}
+* G. D. Forney, M. Trott, {{doi-inline|10.1109/18.259635|The dynamics of group codes : State spaces, trellis diagrams and canonical encoders}}, ''IEEE Trans. Inform. theory'', Vol '''39''' (1993), pages 1491-1593.
+* V. V. Vazirani, Huzur Saran and B. S. Rajan,  {{doi-inline|10.1109/18.556679|An efficient algorithm for constructing minimal trellises for codes over finite Abelian groups}}, ''IEEE Trans. Inform. Theory'' '''42''', No.6, (1996), 1839-1854.
+* A. A. Zain, B. Sundar Rajan, "Dual codes of Systematic Group Codes over Abelian Groups", ''Appl. Algebra Eng. Commun. Comput.'' '''8'''(1): 71-83 (1996).
+[[Category:Coding theory]]

Rectified 5-cell: Difference between revisions

Revision as of 13:38, 23 March 2013

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