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[Note] The question has been asked at Matheoverflow. But there is no answers.

I believe that any non-trivial idea will sooner or later find application in real life. However "sooner" is better than "later":)

If we look at famous open problems - e.g. Millennium Prize problems - there is no clear practical outcome from their solution (imho). (Well, P vs NP of course will be of greatest importance if one will produce quick algrorithm to solve all NP problems - but it seems it is not plausible).

Question what are open problems which solution will have some "real life" outcome in visible future ?


My answer. It might be not perfect from various sides, but still...

Problem Construction of short and effective error-correcting codes.

Background Recent advances in error-corrrecting codes are LDPC and turbo-codes - invented in 90-ies (more recent polar codes) produced codes which are highly effective for quite a long length of the codewords (thousands bits or may be even dozens thousands). These codes are widely used in practive, e.g. 3G-smart-phones use turbo-codes. (See wikipedia articles on turbo-codes and LDPC)

However such codes are not effective for short sequences of bits (dozens bits). That is why a problem.

Practical importance In cell networks not only data is transmitted, but various auxilliary information - e.g. acknowledge receipts (data has been received or not), various channel quality indicators (measuring how bad are propogation conditions). Amount of such data is not big - dozens of bits - so we need short codes. Up to recent times this was not considered as important issue, because it might be just 1% of data transmission. But with the advance of smartphones and "always on-line" feature this appears to be critical problem. Since "always on-line" means sending very often and very short amounts of data, but for each data transmission one needs to send also auxilliary information - so the amount of auxilliary information becomes comparable with data transmission.

Literature This problem is mentioned in influential article in telecommunication community "Is the PHY layer dead?" http://repositori.upf.edu/handle/10230/13026

("PHY(=physical) layer" - is a part where most mathematical-consuming algorithms were concentrated (e.g. error-correcting codes, statstical algorithms for signal estimation etc.)). So the title has a flavor "math is not useful for wireless telecom anymore?" (However now RM (Resource managament) layer is consuming math (from game theory to optimization) so we may not worry :).

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Finding a (fast, practical) polynomial-time factoring algorithm would break RSA, turn public key cryptography on its head, and cause me to stop using online banking for the moment. –  Douglas B. Staple Apr 10 '13 at 16:16
    
Is this question on-topic? –  Douglas B. Staple Apr 10 '13 at 16:16
    
@DouglasB.Staple: Well it fits the chosen tag, and I'm also interested to know some answers. RSA would simply be replaced by something else in the meantime.. Theoretically RSA is broken by Shor's algorithm but I've yet to hear of a practical quantum computer. –  user21820 Aug 7 at 2:53

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