List of unsolved problems in information theory

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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.

Contents

Channel coding

Source coding

Related Research Articles

Information theory is the mathematical study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field, in applied mathematics, is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering.

<span class="mw-page-title-main">Quantum information</span> Information held in the state of a quantum system

Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term.

ALOHAnet, also known as the ALOHA System, or simply ALOHA, was a pioneering computer networking system developed at the University of Hawaii.

Channel capacity, in electrical engineering, computer science, and information theory, is the theoretical maximum rate at which information can be reliably transmitted over a communication channel.

A cryptosystem is considered to have information-theoretic security if the system is secure against adversaries with unlimited computing resources and time. In contrast, a system which depends on the computational cost of cryptanalysis to be secure is called computationally, or conditionally, secure.

In telecommunications, an interference is that which modifies a signal in a disruptive manner, as it travels along a communication channel between its source and receiver. The term is often used to refer to the addition of unwanted signals to a useful signal. Common examples include:

Network calculus is "a set of mathematical results which give insights into man-made systems such as concurrent programs, digital circuits and communication networks." Network calculus gives a theoretical framework for analysing performance guarantees in computer networks. As traffic flows through a network it is subject to constraints imposed by the system components, for example:

In information theory, the noisy-channel coding theorem, establishes that for any given degree of noise contamination of a communication channel, it is possible to communicate discrete data nearly error-free up to a computable maximum rate through the channel. This result was presented by Claude Shannon in 1948 and was based in part on earlier work and ideas of Harry Nyquist and Ralph Hartley.

Weighted fair queueing (WFQ) is a network scheduling algorithm. WFQ is both a packet-based implementation of the generalized processor sharing (GPS) policy, and a natural extension of fair queuing (FQ). Whereas FQ shares the link's capacity in equal subparts, WFQ allows schedulers to specify, for each flow, which fraction of the capacity will be given.

Multiuser detection deals with demodulation of the mutually interfering digital streams of information that occur in areas such as wireless communications, high-speed data transmission, DSL, satellite communication, digital television, and magnetic recording. It is also being currently investigated for demodulation in low-power inter-chip and intra-chip communication. Multiuser detection encompasses both receiver technologies devoted to joint detection of all the interfering signals or to single-user receivers which are interested in recovering only one user but are robustified against multiuser interference and not just background noise.

The Blackwell channel is a deterministic broadcast channel model used in coding theory and information theory. It was first proposed by mathematician David Blackwell. In this model, a transmitter transmits one of three symbols to two receivers. For two of the symbols, both receivers receive exactly what was sent; the third symbol, however, is received differently at each of the receivers. This is one of the simplest examples of a non-trivial capacity result for a non-stochastic channel.

In telecommunications, dirty paper coding (DPC) or Costa precoding is a technique for efficient transmission of digital data through a channel subjected to some interference known to the transmitter. The technique consists of precoding the data in order to cancel the interference. Dirty-paper coding achieves the channel capacity without a power penalty and without requiring the receiver to know the interfering signal.

<span class="mw-page-title-main">MIMO</span> Use of multiple antennas in radio

In radio, multiple-input and multiple-output (MIMO) is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wireless communication standards including IEEE 802.11n, IEEE 802.11ac, HSPA+ (3G), WiMAX, and Long Term Evolution (LTE). More recently, MIMO has been applied to power-line communication for three-wire installations as part of the ITU G.hn standard and of the HomePlug AV2 specification.

A deletion channel is a communications channel model used in coding theory and information theory. In this model, a transmitter sends a bit, and the receiver either receives the bit or does not receive anything without being notified that the bit was dropped. Determining the capacity of the deletion channel is an open problem.

In graph theory, the Shannon capacity of a graph is a graph invariant defined from the number of independent sets of strong graph products. It is named after American mathematician Claude Shannon. It measures the Shannon capacity of a communications channel defined from the graph, and is upper bounded by the Lovász number, which can be computed in polynomial time. However, the computational complexity of the Shannon capacity itself remains unknown.

In mathematics and telecommunications, stochastic geometry models of wireless networks refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.

<span class="mw-page-title-main">Katalin Marton</span> Hungarian mathematician (1941–2019)

Katalin Marton was a Hungarian mathematician, born in Budapest.

In information theory, the interference channel is the basic model used to analyze the effect of interference in communication channels. The model consists of two pairs of users communicating through a shared channel. The problem of interference between two mobile users in close proximity or crosstalk between two parallel landlines are two examples where this model is applicable.

Mark McMahon Wilde is an American quantum information scientist. He is an Associate Professor in the School of Electrical and Computer Engineering at Cornell University, and he is also a Fields Member in the School of Applied and Engineering Physics and the Department of Computer Science at Cornell.

John Cronan Kieffer is an American mathematician best known for his work in information theory, ergodic theory, and stationary process theory.

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