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). Parent of: Binary permutation-based code, Linear code over \(G\), Rank-modulation Gray code (RMGC). Cousins: Binary code, Galois-field \(q\)-ary code, Analog code, Ring code, Group-based quantum code.
Linear code over \(G\)[3][4][5] 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. Cousins: Linear binary code, Linear \(q\)-ary code, Lattice-based code, \(R\)-linear code, Linear code, Group GKP code. Cousin of: Slepian group-orbit code.
Rank-modulation Gray code (RMGC)[6][7] 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. Cousins: Gray code, 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]
F. R. Kschischang, P. G. de Buda, and S. Pasupathy, “Block coset codes for M-ary phase shift keying”, IEEE Journal on Selected Areas in Communications 7, 900 (1989). DOI
[4]
G. D. Forney, “Geometrically uniform codes”, IEEE Transactions on Information Theory 37, 1241 (1991). DOI
[5]
H.-A. Loeliger, “Signal sets matched to groups”, IEEE Transactions on Information Theory 37, 1675 (1991). DOI
[6]
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