Welcome to the Group Kingdom.

Group-based code Encodes $$K$$ states (codewords) in $$n$$ coordinates labeled by elements of a finite group $$G$$. Parents: Error-correcting code (ECC).
Stub. Parents: Group-based code. Cousin of: Rank-modulation code.
Also known as a code in permutations. A family of codes that encode a finite set of size $$M$$ into a set $$S_n$$ of permutations of $$[n]=(1,2,...,n)$$. They can be derived from Lee-metric codes, Reed-Solomon codes [5], 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

[1]
I. F. Blake, G. Cohen, and M. Deza, “Coding with permutations”, Information and Control 43, 1 (1979). DOI
[2]
P. J. Cameron, “Permutation codes”, European Journal of Combinatorics 31, 482 (2010). DOI
[3]
H. Chadwick and L. Kurz, “Rank permutation group codes based on Kendall's correlation statistic”, IEEE Transactions on Information Theory 15, 306 (1969). DOI
[4]
Anxiao Jiang, M. Schwartz, and J. Bruck, “Error-correcting codes for rank modulation”, 2008 IEEE International Symposium on Information Theory (2008). DOI
[5]
A. Mazumdar, A. Barg, and G. Zemor, “Constructions of rank modulation codes”, 2011 IEEE International Symposium on Information Theory Proceedings (2011). DOI