In general, all lossless compression algorithms will produce output longer than the input for some values of the input.

Here is a sketchy proof:
there is a finite number of strings of length at most n. If none of them were compressed to a longer string, it would mean that their compressed values would be again the set of all strings of length at most n (i.e. the compression would be a permutation of said set). This implies that any string would be "compressed" to a string of the same length.
In other words, the only 1-1 injective (i.e. lossless) mappings that never cause length to increase are those that keep length the same.

Update: I changed 1-1 with injective, which makes the statement a bit stronger and more directly related to the sketch of proof. As a side note, I should point out that compression algorithms should tend to be 1-1 anyway, so as to avoid "wasting" short strings.
As to MidLifeXis's comment, of course by "compression algorithm" I meant something which strictly compresses at least one input, as pointed out by tilly.

Cheers

Antonio


The stupider the astronaut, the easier it is to win the trip to Vega - A. Tucket

In reply to Re^2: Efficient 7bit compression by abell
in thread Efficient 7bit compression by Limbic~Region

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