Coding Theory
– Z_{4}-Codes
Codes over the alphabet Z_{4} have been studied very actively since 1994, when it was discovered [1] that certain very good but peculiar non-linear codes over the binary alphabet can be viewed as Gray images of linear codes over Z_{4}.
The Gray map takes a Z_{4}-symbol and maps it into a pair of bits as follows: 0->00, 1->01, 2->11, 3->10. One of the many remarkable properties of this map is that it is an isometry, i.e. the Hamming distance of the images of two symbols is equal to the Lee distance of the two symbols. Thus applying the Gray map to components of words in a good linear Z_{4}-code gives a good binary code (of a doubled length).
The Z_{4}-codes have also been used in the construction of large families of both binary and quadriphase sequences with excellent correlation properties. For a very wide range of parameters the best known families of binary sequences have been constructed as applications of the Z_{4}-theory.
Our research group has developed algebraic decoding algorithms for certain codes. The challenge is to develop ways of decoding with respect to the Lee metric. For some codes the algorithms developed by our group are the only known ones [5,6].
Another topic we have researched is the combinatorial designs constructed from codewords of some Z_{4}-code. For example, some infinite families of 3-designs can be obtained from Z_{4}-Goethals codes and their generalizations [7,10].
References:- A. Roger Hammons, Jr., P. Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, and Patrick Solé, The Z_{4}-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Transactions on Information Theory, 40(2):301-319, 1994.
- Sami Koponen and Jyrki Lahtonen, On the aperiodic and odd correlations of the binary Shanbhag-Kumar-Helleseth sequences, IEEE Transactions on Information Theory, 43(5):1593-1596, 1997. [doi]
- Xiang-Dong Hou, Sami Koponen, and Jyrki Lahtonen, The Reed-Muller code R(r,m) is not Z_{4}-linear for 3<=r<=m-1, IEEE Transactions on Information Theory, 44(2):798-799, 1998. [doi]
- Chunming Rong, Tor Helleseth, and Jyrki Lahtonen, On algebraic decoding of the Z_{4}-linear Calderbank-McGuire code, IEEE Transactions on Information Theory, 45(5):1423-1434, 1999. [doi]
- Kalle Ranto, On algebraic decoding of the Z_{4}-linear Goethals-like codes, IEEE Transactions on Information Theory, 46(6):2193-2197, 2000. [doi]
- Jyrki Lahtonen, Decoding the 6-error-correcting Z_{4}-linear Calderbank-McGuire code, Proceedings of the 2000 IEEE International Symposium on Information Theory, 446. [doi]
- Kalle Ranto, Infinite families of 3-designs from Z_{4}-Goethals codes with block size 8, SIAM Journal on Discrete Mathematics, 15(3):289-304, 2002. [doi]
- Kalle Ranto, Z_{4}-Goethals Codes, Decoding and Designs, TUCS Dissertation No. 42, 2002. [pdf-file]
- Tor Helleseth, Jyrki Lahtonen, and Kalle Ranto, A simple proof to the minimum distance of Z_{4}-linear Goethals-like codes, Journal of Complexity, 20(2-3):297-304, 2004. [doi]
- Jyrki Lahtonen, Kalle Ranto, and Roope Vehkalahti, 3-Designs from all Z_{4}-Goethals-like codes with block size 7 and 8, Finite Fields and Their Applications, 13(4):815-827, 2007. [doi]