# Group code

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In [coding theory](/source/Coding_theory), **group codes** are a type of [code](/source/Coding_theory). Group codes consist of n {\displaystyle n} [linear block codes](/source/Linear_block_codes) which are subgroups of G n {\displaystyle G^{n}} , where G {\displaystyle G} is a finite [Abelian group](/source/Abelian_group).

A systematic group code C {\displaystyle C} is a code over G n {\displaystyle G^{n}} of order | G | k {\displaystyle \left|G\right|^{k}} defined by n − k {\displaystyle n-k} [homomorphisms](/source/Homomorphism) which determine the [parity check](/source/Parity_check) bits. The remaining k {\displaystyle k} bits are the information bits themselves.

## Construction

Group codes can be constructed by special [generator matrices](/source/Generator_matrix) which resemble generator matrices of linear block codes except that the elements of those matrices are [endomorphisms](/source/Endomorphism) of the group instead of symbols from the code's alphabet. For example, considering the generator matrix

- G = ( ( 00 11 ) ( 01 01 ) ( 11 01 ) ( 00 11 ) ( 11 11 ) ( 00 00 ) ) {\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 2 × 2 {\displaystyle 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 {\displaystyle g_{1}^{m_{1}}g_{2}^{m_{2}}...g_{r}^{m_{r}}} where g 1 , . . . g r {\displaystyle g_{1},...g_{r}} are the [generators](/source/Generating_set_of_a_group) of G {\displaystyle G} .

## See also

- [Group coded recording](/source/Group_coded_recording) (GCR)

## References

## Further reading

- Watkinson, John (1990). "3.4. Group codes". *Coding for Digital Recording*. Stoneham, MA, USA: [Focal Press](/source/Focal_Press). pp. 51–61. [ISBN](/source/ISBN_(identifier)) [978-0-240-51293-8](https://en.wikipedia.org/wiki/Special:BookSources/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)*](/source/IEEE_International_Symposium_on_Information_Theory). p. 360. [doi](/source/Doi_(identifier)):[10.1109/ISIT.1993.748676](https://doi.org/10.1109%2FISIT.1993.748676). [ISBN](/source/ISBN_(identifier)) [978-0-7803-0878-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7803-0878-7). [S2CID](/source/S2CID_(identifier)) [123694385](https://api.semanticscholar.org/CorpusID:123694385).

- [Forney, George David](/source/George_David_Forney); Trott, Mitch D. (1993). "The dynamics of group codes: State spaces, trellis diagrams and canonical encoders". *[IEEE Transactions on Information Theory](/source/IEEE_Transactions_on_Information_Theory)*. **39** (5): 1491–1593. [doi](/source/Doi_(identifier)):[10.1109/18.259635](https://doi.org/10.1109%2F18.259635).

- [Vazirani, Vijay Virkumar](/source/Vijay_Virkumar_Vazirani); Saran, Huzur; Rajan, B. Sundar (1996). "An efficient algorithm for constructing minimal trellises for codes over finite Abelian groups". *[IEEE Transactions on Information Theory](/source/IEEE_Transactions_on_Information_Theory)*. **42** (6): 1839–1854. [CiteSeerX](/source/CiteSeerX_(identifier)) [10.1.1.13.7058](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.7058). [doi](/source/Doi_(identifier)):[10.1109/18.556679](https://doi.org/10.1109%2F18.556679).

- Zain, Adnan Abdulla; Rajan, B. Sundar (1996). "Dual codes of Systematic Group Codes over Abelian Groups". *Applicable Algebra in Engineering, Communication and Computing*. **8** (1): 71–83.

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