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Issue No.08 - August (2006 vol.55)

pp: 929-946

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2006.124

ABSTRACT

A symbol permutation invariant balanced (SPI-balanced) code over the alphabet {\hbox{\rlap{Z}\kern 2.0pt{\hbox{Z}}}}_m=\{0,1,\ldots,m-1\} is a block code over {\hbox{\rlap{Z}\kern 2.0pt{\hbox{Z}}}}_m such that each alphabet symbol occurs as many times as any other symbol in every codeword. For this reason, every permutation among the symbols of the alphabet changes an SPI-balanced code into an SPI-balanced code. This means that SPI--balanced words are "the most balanced” among all possible m{\hbox{-}}\rm ary balanced word types and this property makes them very attractive from the application perspective. In particular, they can be used to achieve m{\hbox{-}}\rm ary DC-free communication, to detect/correct asymmetric/unidirectional errors on the m{\hbox{-}}\rm ary asymmetric/unidirectional channel, to achieve delay-insensitive communication, to maintain data integrity in digital optical disks, and so on. This paper gives some efficient methods to convert (encode) m{\hbox{-}}\rm ary information sequences into m{\hbox{-}}\rm ary SPI-balanced codes whose redundancy is equal to roughly double the minimum possible redundancy r_{min}. It is proven that r_{min} \simeq [(m-1)/2]\log_{m}n-(1/2)[1-(1/\log_{2\pi}m)]m-(1/\log_{2\pi}m) for any code which converts k information digits into an SPI-balanced code of length n=k+r. For example, the first method given in the paper encodes k information digits into an SPI-balanced code of length n=k+r, with r=(m-1)\log_{m}k+O(m\log_{m}\log_{m}k). A second method is a recursive method, which uses the first as base code and encodes k digits into an SPI-balanced code of length n=k+r, with r\simeq(m-1)\log_{m}n-\log_{m}[(m-1)!].

INDEX TERMS

Coding and information theory, error control codes, balanced codes, constant weight codes, digital communication, m{\hbox{-}}\rm ary communication, line codes, DC-free communication, delay-insensitive communication.

CITATION

Raffaele Mascella, "Efficient m-Ary Balanced Codes which Are Invariant under Symbol Permutation",

*IEEE Transactions on Computers*, vol.55, no. 8, pp. 929-946, August 2006, doi:10.1109/TC.2006.124