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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.

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?"

("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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closed as not constructive by Will Jagy, Qiaochu Yuan, Felipe Voloch, Andy Putman, Andrés Caicedo Feb 9 '12 at 7:21

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

Since this question was clearly written carefully, I suspect that the downvote and votes to close are because it could be seen as an attempt to disguise a blog entry as a question. Perhaps it might be better received if the "answer" were moved to an actual answer (with reference, in the question, to the fact that this is being done) and the question itself were elaborated a bit. Perhaps you should, in the question, either give several brief examples of the sort of thing you are looking for, or include a statement to the effect that you made this a very broad question because even in this... – Charles Staats Feb 7 '12 at 22:59
...generality, you could only think of one example (which is given below as an answer). You might also explain why a list of engineering problems would not suffice--e.g., stating that you are specifically interested in theoretical problems with practical applications. Come to think of it, if this is an accurate description of what you are after, "practical applications for theoretical open problems" might be a better title than what you have now. – Charles Staats Feb 7 '12 at 23:04
@Charles Staats Thank you for the comments and suggestions. Since it is CW you are welcome to make changes what you consider appropriate. To my taste - gist is clear words are not very important :) – Alexander Chervov Feb 8 '12 at 5:56
However "sooner" is better than "later" ... Does this opinion apply to $E=mc^2$ ? – Denis Serre Feb 8 '12 at 6:25
@Denis Well you are right, some extraordinary discoveries found by very very pure scientific reasons may have extraordinary applications much later. So probably my way on phrasing is not perfect. However I against the position that "Faraday invented electricity - so this justifies existence of pure science forever". So having visible open problems with practical impact is important. – Alexander Chervov Feb 8 '12 at 6:51