[Solved]: Can we break the Shannon capacity?

Problem Detail: 

I have a friend working in wireless communications research. He told me that we can transmit more than one symbol in a given slot using one frequency (of course we can decode them at the receiver).

The technique as he said uses a new modulation scheme. Therefore if one transmitting node transmits to one receiving node over a wireless channel and using one antenna at each node, the technique can transmit two symbols at one slot over one frequency.

I am not asking about this technique and I do not know whether it is correct or not but I want to know if one can do this or not? Is this even possible? Can the Shannon limit be broken? Can we prove the impossibility of such technique mathematically?

Other thing I want to know, if this technique is correct what are the consequences? For example what would such technique imply for the famous open problem of the interference channel?

Any suggestions please? Any reference is appreciated.

Asked By : Learning

Answered By : Yuval Filmus

Shannon proved that the capacity of the channel is a limit on its capacity. This is actually the easy part of Shannon's theorem — it is much harder to show that capacity can always be achieved. A good source for this is Shannon's paper from 1948, A mathematical theory of communication, page 24, last paragraph before Section 14. As Shannon writes, the Shannon capacity is the limit more-or-less by definition.

While Shannon already found a formula for the capacity of a channel, namely the Shannon capacity, two directions are still open (though they have partial solutions):

1. Zero-error encoding, also known as block codes. The best capacity in this regime is a major open problem in coding theory.

2. Codes with efficient coding and decoding. Shannon's theorem is non-constructive (it uses a random code), and so the codes it implies cannot be used in practice. It is more difficult to construct codes having efficient encoding and decoding. There have been lots of developments in the recent years in constructing efficiently decodable capacity-achieving codes.

Another problem is the rate of convergence in Shannon's theorem - I'm not sure what the status of that is.

In your specific case, it seems that the improvement is not be breaking the Shannon barrier (which is impossible) but by effectively changing the channels, just as telephone lines can be used to carry high-speed network traffic.

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Question Source : http://cs.stackexchange.com/questions/23009