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Mon, 03 Mar 2008
Uniquely-decodable codes revisited
Alan Morgan wrote to ask if there was a difference between uniquely-decodable (UD) codes for strings and for streams. That is, is there a code for which every finite string is UD, but for which some infinite sequence of symbols has multiple decodings. I pondered this a bit, and after a little guessing came up with an example: { "a", "ab", "bb" } is UD, because it is a suffix code. But the stream "abbbbbbbbbb..." can be decoded in two ways. After I found the example, I realized that I shouldn't have needed to guess, because I already knew that you sometimes have to see the last symbol of a string before you can know how to decode it, and in such a code, if there is no such symbol, the decoding must be ambiguous. The code above is UD, but to decode "abbbbbbbbbbbbbbb" you have to count the "b"s to figure out whether the first code word is "a" or "ab". Let's say that a code is UD+ if it has the property that no two infinite sequences of code words have the same concatenation. Can we characterize the UD+ codes? Clearly, UD+ implies UD, and the example above shows that the converse is not true. A simple argument shows that all prefix codes are UD+. So the question now is, are there UD+ codes that are not prefix codes? I don't know. [ Addendum 20080303: Gareth McCaughan points out that { "a", "ab" } is UD+ but not prefix. ]
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