Welcome to the Group Kingdom.

Group-based code Encodes $$K$$ states (codewords) in $$n$$ coordinates labeled by elements of a group $$G$$. The number of codewords may be infinite for infinite groups, so various restricted versions have to be constructed in practice. Parents: Error-correcting code (ECC).
Encodes $$K$$ states (codewords) in $$n$$ coordinates over a group $$G$$ such that the codewords form a subgroup of $$G^n$$. Parents: Group-based code, Group-orbit code. Cousin of: Slepian group-orbit code.
Also known as a code in permutations. A family of codes that encode a finite set of size $$M$$ into a group $$S_n$$ of permutations of $$[n]=(1,2,...,n)$$. They can be derived from Lee-metric codes, Reed-Solomon codes [8], quadratic residue codes and most binary codes. Protection: Protects against errors in the Kendall tau distance on the space of permutations. The Kendall distance between permutations $$\sigma$$ and $$\pi$$ is defined as the minimum number of adjacent transpositions required to change $$\sigma$$ into $$\pi$$. Parents: Group-based code.

## References

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[2]
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H. Chadwick and L. Kurz, “Rank permutation group codes based on Kendall's correlation statistic”, IEEE Transactions on Information Theory 15, 306 (1969). DOI
[7]
Anxiao Jiang, M. Schwartz, and J. Bruck, “Error-correcting codes for rank modulation”, 2008 IEEE International Symposium on Information Theory (2008). DOI
[8]
A. Mazumdar, A. Barg, and G. Zemor, “Constructions of rank modulation codes”, 2011 IEEE International Symposium on Information Theory Proceedings (2011). DOI