Group code

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In coding theory, group codes are a type of code. Group codes consist of ${\displaystyle n}$ linear block codes which are subgroups of ${\displaystyle G^{n}}$, where ${\displaystyle G}$ is a finite Abelian group.

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

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

${\displaystyle G={\begin{pmatrix}{\begin{pmatrix}00\\11\end{pmatrix}}{\begin{pmatrix}01\\01\end{pmatrix}}{\begin{pmatrix}11\\01\end{pmatrix}}\\{\begin{pmatrix}00\\11\end{pmatrix}}{\begin{pmatrix}11\\11\end{pmatrix}}{\begin{pmatrix}00\\00\end{pmatrix}}\end{pmatrix}}}$

the elements of this matrix are ${\displaystyle 2\times 2}$ matrices which are endomorphisms. In this scenario, each codeword can be represented as ${\displaystyle g_{1}^{m_{1}}g_{2}^{m_{2}}...g_{r}^{m_{r}}}$ where ${\displaystyle g_{1},...g_{r}}$ are the generators of ${\displaystyle G}$.

• Group coded recording (GCR)

• Watkinson, John (1990). "3.4. Group codes". Coding for Digital Recording. Stoneham, MA, USA: Focal Press. pp. 51–61. ISBN 978-0-240-51293-8.
• Biglieri, Ezio; Elia, Michele (1993-01-17). "Construction of Linear Block Codes Over Groups". Proceedings. IEEE International Symposium on Information Theory (ISIT). p. 360. doi:10.1109/ISIT.1993.748676. ISBN 978-0-7803-0878-7. S2CID 123694385.
• Forney, George David; Trott, Mitch D. (1993). "The dynamics of group codes: State spaces, trellis diagrams and canonical encoders". IEEE Transactions on Information Theory. 39 (5): 1491–1593. doi:10.1109/18.259635.
• Vazirani, Vijay Virkumar; Saran, Huzur; Rajan, B. Sundar (1996). "An efficient algorithm for constructing minimal trellises for codes over finite Abelian groups". IEEE Transactions on Information Theory. 42 (6): 1839–1854. CiteSeerX 10.1.1.13.7058. doi:10.1109/18.556679.
• Zain, Adnan Abdulla; Rajan, B. Sundar (1996). "Dual codes of Systematic Group Codes over Abelian Groups". Applicable Algebra in Engineering, Communication and Computing (AAECC). 8 (1): 71–83.