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Good morning!

I have a question about arithmetic encoding on sheet 01-77. Shouldn't be any encoding injective? If I get it right, the encoding isn't injektive. For the given example on sheet 01-77 0.66 is a valid encoding for BAAC but also for BAA or BAACA. How can I know witch word is meant by a given codeword?

Thanks for all answers!

Best regards

Nick Jochum
in # Mandatory Modules Bachelor by (1.1k points)

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Yes you are right. When we decode 0.66, we first identify that in the start interval [0.0,1.0], it belongs to B's sub-interval [0.6,0.8] (see first line in figure). Then in the interval [0.6,0.8], we identify that encoding 0.66 belongs to A's sub-interval [0.6,0.72]  (second line in figure). Then in the interval [0.6,0.72], we narrow down 0.66 to A's sub-interval [0.6,0.672] (third line in figure). Continuing similarly, in the interval [0.6,0.672], encoding 0.66 belongs to C's sub-interval [0.6576,0.672] (fourth line in figure). As you can see, we may continue decoding like this forever since the number 0.66 will always belong to one of the sub-intervals. 

It is therefore necessary to inform the decoder when to stop. This can be done in a few ways. One way would be to use a special symbol as end-of-data. For instance, if symbol C is used in the above example to indicate end-of-data, the decoded word would be BAAC. Another way would be to specify total number of symbols in the word. For instance, if total number of symbols is specified as 5 in the above example, then the decoded word would be BAACA. In this case, there would be a fifth line in the above figure indicating that in the interval [0.6576,0.672], the encoding 0.66 belongs to A's sub-interval [0.6576,0.66624].

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