List of unsolved problems in information theory explained

This article lists notable unsolved problems in information theory. These are separated into source coding and channel coding. There are also related unsolved problems^[1] in philosophy.

Channel coding

• Capacity of a network: The capacity of a general wireless network is not known. There are some specific cases for which the capacity is known, such as the AWGN channel and fading channel.^[2]
• Capacity of the broadcast channel: The capacity of the broadcast channel, or the case in which a single transmitter is sending information to many receivers, is unknown in general, though it is known for several specific cases.^{[3] [4]}
• Capacity of the interference channel (Two User): The capacity of the interference channel, in the case where there are two transmitter and receiver pairs that interfere among each other, is unknown in general. Capacity is known in special cases: strong interference regime, injective-deterministic. Capacity is known in approximate sense or within a range for: injective-semi-deterministic, additive white Gaussian noise with per block power constraint.
• Capacity of the two-way channel: The capacity of the two-way channel (a channel in which information is sent in both directions simultaneously) is unknown.^{[5] [6]}
• Capacity of Aloha: The ALOHAnet used a very simple access scheme for which the capacity is still unknown, though it is known in a few special cases.^[7]
• Capacity of the queue channel: Under a FIFO policy, whether the feedback capacity of the queue channel is strictly greater than the capacity without feedback is unknown for general service time distributions though it is known that the two quantities are equal when the service time distribution is memoryless.^[8]
• Quantum capacity

The capacity of a quantum channel is in general not known.^[9]

Source coding

• Lossy distributed source coding: The best way to compress correlated information sources using encoders that do not communicate with each other, preserving each source to within its distortion metric, is not known.

Further reading

• Book: Cover . Thomas . Gopinath . B. . Open Problems in Communication and Computation . 1987 . Springer-Verlag . 11 February 2021.
• Book: Selected Unsolved Problems in Coding Theory. 2010. Springer. New York. David Joyner. Jon-Lark Kim .
• Book: Longo, Giuseppe. Information theory: new trends and open problems. 1975. Springer . 9783211813782.
• Tse. David. It's Easier to Approximate. Information Theory Society Newsletter. 1996. 26 June 2013.

Notes and References

1. Web site: Adriaans. Pieter. Open Problems in the Study of Information and Computation. 21 June 2013.
2. Book: Cover. Thomas. Elements of Information Theory. registration. Wiley-Interscience. 978-0471062592. 1991-08-26.
3. Cover. Thomas. Comments on Broadcast Channels. IEEE Trans Inf Theory. Oct 1998. 44. 6. 2524. 10.1109/18.720547. 8985406 .
4. Web site: Sridharan. Arvind. Broadcast Channels. Notre Dame. 6 July 2014. 29 August 2017. https://web.archive.org/web/20170829005436/https://www3.nd.edu/~jnl/ee698g/materials/summaries/arvind.pdf. dead.
5. Shannon. Claude. Two-way communication channels. Proc Fourth Berkeley Sump on Mathematical Statistics and Probability. 1961. 1. 611.
6. meeuwissen. Erik. The Origin of Two-Way Channels. Proc ISIT. 16 Aug 1998. I. 185.
7. Médard. Muriel. Muriel Médard. Capacity of Time-Slotted ALOHA Packetized Multiple-Access Systems Over the AWGN Channel. IEEE Transactions on Wireless Communications. March 2004. 3. 2. 486–499. 11 July 2014. 10.1109/TWC.2003.821175. 791018 . https://web.archive.org/web/20111218191542/http://colemant.ece.illinois.edu/pubs/capacityALOHAtwireless.pdf. 18 December 2011. dead.
8. Anantharam. Venkat. Verdu. Sergio. Bits through queues. IEEE Trans Inf Theory. 1996. 42. 1. 4-18. 10.1109/18.481773 .
9. Book: Shor, Peter. Quantum Information Theory: Results and Open Problems . Alon N. . Bourgain J. . Connes A. . Gromov M. . Milman V.. Visions in Mathematics, GAFA 2000 Special Volume: Part II. 2000. Modern Birkhäuser Classics . Birkhäuser Basel. 816–838. http://www-math.mit.edu/~shor/papers/GAFA.pdf. 10.1007/978-3-0346-0425-3_9. 978-3-0346-0425-3.