[math-fun] Non-stuttering codes
I have no idea at all whether the following problem is known in the literature: I'm looking for codes where the code words are "non-stuttering", i.e. have no two consecutive identical letters. The specific parameters I care about are a 4-letter alphabet with code words at minimum distance 4, probably with words in the vicinity of 9-10 letters. (Yes, for the application I have in mind, the 4-letter alphabet is {A,C,G,T}.) I can think of all sorts of very bad constructions for codes with this obnoxious property, but not anything that seems clever. (The map from all strings on {1,2,3} to non-stuttering strings on {0,1,2,3}, taking successive sums, is frustratingly useless, since it is very badly behaved with respect to Hamming distance.) Can anyone offer insight or references? --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
Certain codes (Hamming codes, e.g.) break the code words into a data portion and a parity portion. You could try having the data portion consist of all possible non-stuttering sequences that don't start with 0, which allows you to treat it as a number in 1 .. 3^m for the purposes of the parity portion. For example, if you had 2 data dimensions and one parity dimension, you would start from the code prefixes { 10, 12, 13, 20, 21, 23, 30, 31, 32 }. To get 23's parity dimension, you could take its position (6 of 9), reduce it mod 3 (0), and then append it using the x -> last digit +x+1 mod 4 mapping to get the final codword of 230. -Thomas C
participants (2)
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Michael Kleber -
Thomas Colthurst