List of unsolved problems in information theory

This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.

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.

References

1. Adriaans, Pieter. "Open Problems in the Study of Information and Computation" . Retrieved 21 June 2013.
2. Cover, Thomas (1991-08-26). Elements of Information Theory . Wiley-Interscience. ISBN   978-0471062592.
3. Cover, Thomas (Oct 1998). "Comments on Broadcast Channels". IEEE Trans Inf Theory. 44 (6): 2524. doi:10.1109/18.720547. S2CID   8985406.
4. Sridharan, Arvind. "Broadcast Channels" (PDF). Notre Dame. Archived from the original (PDF) on 29 August 2017. Retrieved 6 July 2014.
5. Shannon, Claude (1961). "Two-way communication channels". Proc Fourth Berkeley Sump on Mathematical Statistics and Probability. 1: 611.
6. meeuwissen, Erik (16 Aug 1998). "The Origin of Two-Way Channels". Proc ISIT. I: 185.
7. Médard, Muriel (March 2004). "Capacity of Time-Slotted ALOHA Packetized Multiple-Access Systems Over the AWGN Channel" (PDF). IEEE Transactions on Wireless Communications. 3 (2): 486–499. doi:10.1109/TWC.2003.821175. S2CID   791018. Archived from the original (PDF) on 18 December 2011. Retrieved 11 July 2014.
8. Anantharam, Venkat; Verdu, Sergio (1996). "Bits through queues". IEEE Trans Inf Theory. 42 (1): 4–18. doi:10.1109/18.481773.
9. Shor, Peter (2000). "Quantum Information Theory: Results and Open Problems" (PDF). In Alon N.; Bourgain J.; Connes A.; Gromov M.; Milman V. (eds.). Visions in Mathematics, GAFA 2000 Special Volume: Part II. Modern Birkhäuser Classics. Birkhäuser Basel. pp. 816–838. doi:10.1007/978-3-0346-0425-3_9. ISBN   978-3-0346-0425-3.

Further reading

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