Can a language have free phoneme order? (

In all natural human languages, the order of the sounds/signs (phonemes) in an utterance conveys information. But is it possible for a language to convey no information with phoneme order? And if so, what properties does such a language have?

This is a Summer of Math Exposition 4 (SoME4) submission.
SoME4 website: https://some.3b1b.co/
SoME4 announcement video:    • Summer of Math Exposition #4 | Teachers, I...  
Related CCC4 submission:    • Lĩhusã: the language with free phoneme ord...  

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Sections:
00:00 Intro
00:46 Abstraction
01:47 Are unordered languages possible?
03:56 How long are ordered utterances?
06:56 How long are unordered utterances?
12:02 Information theory
15:23 Outro

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Notes:
00:15: Thanks to NexMaster CZ for providing me with the Czech example sentence and helping me understand it! Literal English translation done by me.

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Assets used:
00:06: Drawing of a cat eating a fish modified from an illustration by Irasutoya/いらすとや (https://www.irasutoya.com/2016/06/blo...)
(Irasutoya provides free clip art, with remixing allowed, for both commercial and non-commercial use under weak conditions.)
00:15: Czech example sentence from the 19th century poem "Máj" by Karel Hynek Mácha, one of the most famous pieces of Czech literature

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