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Group-based code
Encodes \(K\) states (codewords) in \(n\) coordinates labeled by elements of a finite group \(G\).
Parents:
Error-correcting code (ECC).
Parent of:
Binary permutation-based code, Rank-modulation code.
Cousins:
Binary code, Lattice-based code, Group-based quantum code.

Binary permutation-based code[1][2]
Stub.
Parents:
Group-based code.
Cousin of:
Rank-modulation code.

Rank-modulation code[3][4]
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.
Cousins:
Reed-Solomon (RS) code, Binary permutation-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