# Analytical Properties of Shannon’s Capacity of Arbitrarily Varying Channels under List Decoding: Super-Additivity and Discontinuity Behavior

## Abstract

The common wisdom is that the capacity of parallel channels is usually additive. This was also conjectured by Shannon for the zero-error capacity function, which was later disproved by constructing explicit counterexamples demonstrating the zero-error capacity to be super-additive. Despite these explicit examples for the zero-error capacity, there is surprisingly little known for nontrivial channels. This paper addresses this question for the arbitrarily varying channel (AVC) under list decoding by developing a complete theory. The list capacity function is studied and shown to be discontinuous, and the corresponding discontinuity points are characterized for all possible list sizes. For parallel AVCs it is then shown that the list capacity is super-additive, implying that joint encoding and decoding for two parallel AVCs can yield a larger list capacity than independent processing of both channels. This discrepancy is shown to be arbitrarily large. Furthermore, the developed theory is applied to the arbitrarily varying wiretap channel to address the scenario of secure communication over AVCs.

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## References

- 1.Boche, H., Schaefer, R.F., and Poor, H.V., Characterization of Super-additivity and Discontinuity Behavior of the Capacity of Arbitrarily Varying Channels under List Decoding, in
*Proc. 2017 IEEE Int. Sympos. on Information Theory (ISIT’2017), Aachen, Germany*, June 25–30, 2017, pp. 2820–2824.CrossRefGoogle Scholar - 2.Shannon, C.E., A Mathematical Theory of Communication,
*Bell Syst. Tech. J.*, 1948, vol. 27, no. 3, pp. 379–423.MathSciNetCrossRefzbMATHGoogle Scholar - 3.Shannon, C.E., The Zero Error Capacity of a Noisy Channel,
*IRE Trans. Inf. Theory*, 1956, vol. 2, no. 3, pp. 8–19.MathSciNetCrossRefGoogle Scholar - 4.Lovász, L., On the Shannon Capacity of a Graph,
*IEEE Trans. Inf. Theory*, 1979, vol. 25, no. 1, pp. 1–7.MathSciNetCrossRefzbMATHGoogle Scholar - 5.Haemers, W., On Some Problems of Lovász Concerning the Shannon Capacity of a Graph,
*IEEE Trans. Inf. Theory*, 1979, vol. 25, no. 2, pp. 231–232.MathSciNetCrossRefzbMATHGoogle Scholar - 6.Alon, N., The Shannon Capacity of a Union,
*Combinatorica*, 1998, vol. 18, no. 3, pp. 301–310.MathSciNetCrossRefzbMATHGoogle Scholar - 7.Keevash, P. and Long, E., On the Normalized Shannon Capacity of a Union,
*Combin. Probab. Comput.*, 2016, vol. 25, no. 5, pp. 766–767.MathSciNetCrossRefzbMATHGoogle Scholar - 8.Ahlswede, R., Elimination of Correlation in Random Codes for Arbitrarily Varying Channels,
*Z. Wahrsch. Verw. Gebiete*, 1978, vol. 44, no. 2, pp. 159–175.MathSciNetCrossRefzbMATHGoogle Scholar - 9.Blackwell, D., Breiman, L., and Thomasian, A.J., The Capacities of Certain Channel Classes under Random Coding,
*Ann. Math. Statist.*, 1960, vol. 31, no. 3, pp. 558–567.MathSciNetCrossRefzbMATHGoogle Scholar - 10.Csiszár, I. and Narayan, P., The Capacity of the Arbitrarily Varying Channel Revisited: Positivity,
*Constraints, IEEE Trans. Inf. Theory*, 1988, vol. 34, no. 2, pp. 181–193.MathSciNetCrossRefzbMATHGoogle Scholar - 11.Blinovsky, V.M., Narayan, P., and Pinsker, M.S., Capacity of the Arbitrarily Varying Channel under List Decoding,
*Probl. Peredachi Inf.*, 1995, vol. 31, no. 2, pp. 3–19 [Probl. Inf. Transm. (Engl. Transl.), 1995, vol. 31, no. 2, pp. 99–113].Google Scholar - 12.Hughes, B.L., The Smallest List for the Arbitrarily Varying Channel,
*IEEE Trans. Inf. Theory*, 1997, vol. 43, no. 3, pp. 803–815.MathSciNetCrossRefzbMATHGoogle Scholar - 13.Csiszár, I. and Narayan, P., Arbitrarily Varying Channels with Constrained Inputs and States,
*IEEE Trans. Inf. Theory*, 1988, vol. 34, no. 1, pp. 27–34.MathSciNetCrossRefzbMATHGoogle Scholar - 14.Sarwate, A.D. and Gastpar, M., List-Decoding for the Arbitrarily Varying Channel under State Constraints,
*IEEE Trans. Inf. Theory*, vol. 58. 2012, no. 3, pp. 1372–1384.Google Scholar - 15.Ahlswede, R., A Note on the Existence of the Weak Capacity for Channels with Arbitrarily Varying Channel Probability Functions and Its Relation to Shannon’s Zero Error Capacity,
*Ann. Math. Statist.*, 1970, vol. 41, no. 3, pp. 1027–1033.MathSciNetCrossRefzbMATHGoogle Scholar - 16.Schaefer, R.F., Boche, H., and Poor, H.V., Super-Activation as a Unique Feature of Secure Communication in Malicious Environments,
*Information*, 2016, vol. 7, no. 2, Article 24 (21 pp.).Google Scholar - 17.MolavianJazi, E., Bloch, M., and Laneman, J.N., Arbitrary Jamming Can Preclude Secure Communication, in
*Proc. 47th Annual Allerton Conf. on Communication, Control, and Computing, Monticello, IL, USA, Sep. 30–Oct. 2*, 2009, pp. 1069–1075.Google Scholar - 18.Bjelaković, I., Boche, H., and Sommerfeld, J., Capacity Results for Arbitrarily Varying Wiretap Channels,
*Information Theory, Combinatorics, and Search Theory, Aydinian, H.K., Cicalese, F., and Deppe, C., Eds., Lect. Notes Comp. Sci*, vol. 7777, Berlin, Heidelberg: Springer, 2013, pp. 123–144.Google Scholar - 19.Boche, H. and Schaefer, R.F., Capacity Results and Super-Activation for Wiretap Channels with Active Wiretappers,
*IEEE Trans. Inf. Forensics Secur.*, 2013, vol. 8, no. 9, pp. 1482–1496.CrossRefGoogle Scholar - 20.Boche, H., Schaefer, R.F., and Poor, H.V., On the Continuity of the Secrecy Capacity of Compound and Arbitrarily Varying Wiretap Channels,
*IEEE Trans. Inf. Forensics Secur.*, 2015, vol. 10, no. 12, pp. 2531–2546.CrossRefGoogle Scholar - 21.Wiese, M., Nötzel, J., and Boche, H., A Channel under Simultaneous Jamming and Eavesdropping Attack—Correlated Random Coding Capacities under Strong Secrecy Criteria,
*IEEE Trans. Inf. Theory*, 2016, vol. 62, no. 7, pp. 3844–3862.MathSciNetCrossRefzbMATHGoogle Scholar - 22.Nötzel, J., Wiese, M., and Boche, H., The Arbitrarily Varying Wiretap Channel—Secret Randomness,
*Stability, and Super-Activation, IEEE Trans. Inf. Theory*, 2016, vol. 62, no. 6, pp. 3504–3531.CrossRefzbMATHGoogle Scholar - 23.Schaefer, R.F., Boche, H., and Poor, H.V.,
*Arbitrarily Varying Channels—A Model for Robust Communication in the Presence of Unknown Interference*, Communications in Interference Limited Networks, Utschick, W., Ed., Cham, Switzerland: Springer, 2016, pp. 259–283.CrossRefGoogle Scholar - 24.Boche, H. and Nötzel, J., Positivity,
*Discontinuity, Finite Resources, and Nonzero Error for Arbitrarily Varying Quantum Channels, J. Math. Phys.*, 2014, vol. 55, no. 12, p. 122201 (20 pp.).zbMATHGoogle Scholar - 25.Arendt, C., Nötzel, J., and Boche, H., Super-Activation of the Composite Independent Arbitrarily Varying Channel under State Constraints, in
*Proc. IEEE Global Communications Conf. (GLOBECOM’2017), Singapore, Dec. 4–8*, 2017, pp. 1–6.Google Scholar - 26.Mansour, A.S., Boche, H., Schaefer, R.F., The Secrecy Capacity of the Arbitrarily Varying Wiretap Channel under List Decoding,
*submitted for publication in Adv. Math. Commun.*, 2017.Google Scholar - 27.Nielsen, M.A. and Chuang, I.L.,
*Quantum Computation and Quantum Information*, Cambridge: Cambridge Univ. Press, 2010.CrossRefzbMATHGoogle Scholar - 28.Boche, H., Schaefer, R.F., and Poor, H.V., Identification over Channels with Feedback: Discontinuity Behavior and Super-Activation, in
*Proc. 2018 IEEE Int. Sympos. on Information Theory (ISIT’2018), Vail, CO, USA. June 17–22*, 2018, pp. 256–260.Google Scholar - 29.Boche, H., Schaefer, R.F., and Poor, H.V.,
*Identification Capacity of Channels with Feedback: Discontinuity Behavior, Super-Activation, and Turing Computability*, submitted for publication in IEEE Trans. Inf. Theory, 2018.Google Scholar - 30.Csiszár, I. and Körner, J.,
*Information Theory: Coding Theorems for Discrete Memoryless Systems*, Cambridge: Cambridge Univ. Press, 2011, 2nd ed.CrossRefzbMATHGoogle Scholar - 31.Wolfowitz, J.,
*Coding Theorems of Information Theory*, Berlin: Springer, 1978, 3rd ed.CrossRefzbMATHGoogle Scholar - 32.Ahlswede, R.,
*Transmitting and Gaining Data: Rudolf Ahlswede’s Lectures on Information Theory 2*, Ahlswede, A., Althöfer, I., Deppe, C., and Tamm, U., Eds., New York: Springer, 2015.CrossRefzbMATHGoogle Scholar - 33.Boche, H., Schaefer, R.F., and Poor, H.V., Undecidability of Strong Converses for Finite Compound Channels (in preparation).Google Scholar
- 34.Ahlswede, R. and Dueck, G., Identification via Channels,
*IEEE Trans. Inf. Theory*, 1989, vol. 35, no. 1, pp. 15–29.MathSciNetCrossRefzbMATHGoogle Scholar - 35.Boche, H. and Deppe, C., Secure Identification for Wiretap Channels; Robustness,
*Super-Additivity and Continuity, IEEE Trans. Inf. Forensics Secur.*, 2018, vol. 13, no. 7, pp. 1641–1655.CrossRefGoogle Scholar - 36.Boche, H. and Deppe, C., Secure Identification under Jamming Attacks, in
*Proc. 9th IEEE Int.Workshop on Information Forensics and Security (WIFS’2017), Rennes, France, Dec. 4–7*, 2017, pp. 1–6.Google Scholar - 37.Boche, H., Deppe, C., and Winter, A.,
*Secure and Robust Identification via Classical-QuantumChannels*, arXiv:1801.09967 [quant-ph], 2018.Google Scholar - 38.Boche, H., Deppe, C., and Winter, A., Secure and Robust Identification via Classical-Quantum Channels, in
*Proc. 2018 IEEE Int. Sympos. on Information Theory (ISIT’2018), Vail, CO, USA, June 17–22*, 2018, pp. 2674–2678.CrossRefGoogle Scholar