Information Theory and Reliable Communication

Sphere Packings, Lattices and Groups

We say that a network coding scheme is strongly 1-secure if a source node s can multicast n field elements {m1, · · · ,mn} to a set of sink nodes {t1, · · · , tq} in such a way that any single edge leaks no information on any S ⊂ {m1, · · · ,mn} with |S| = n − 1, where n = mintimax-flow(s, ti) is the maximum transmission capacity. We also say that a strongly h-secure network coding scheme is strongly (h + 1)secure if any h + 1 edges leak no information on any S ⊂ {m1, · · · ,mn} with |S| = n − (h + 1). In this paper, we show the first explicit algorithm which can construct strongly k-secure network coding schemes. In particular, it runs in polynomial time for fixed k.

Information Theoretic Security

In certain communication systems where information is to be transmitted from one point to another, additional side information is available at the transmitting point. This side information relates to the state of the transmission channel and can be used to aid in the coding and transmission of information. In this paper a type of channel with side information is studied and its capacity determined.

Channels with Side Information at the Transmitter

Mathematical Methods and Theory in Games, Programming, and Economics. 2 volumes. By Samuel Karlin. Pp. 433, 386. 75s. each. 1960. (Pergamon Press)

Consider a single-user or multiple-access channel with a large output alphabet. A method to approximate the channel by an upgraded version having a smaller output alphabet is presented and analyzed. The original channel is not necessarily symmetric and does not necessarily have a binary input alphabet. Also, the input distribution is not necessarily uniform. The approximation method is instrumental when constructing capacity achieving polar codes for an asymmetric channel with a non-binary input alphabet. Other settings in which the method is instrumental are the wiretap setting as well as the lossy source coding setting.

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Channel Upgradation for Non-Binary Input Alphabets and MACs

We study the arbitrarily varying channel (AVC) with input and state constraints, when the encoder has state information in a causal or noncausal manner. For the causal state information setting, we develop lower and upper bounds on the random code capacity. A lower bound on the deterministic code capacity is established in the case of a message-averaged input constraint. In the setting where a state constraint is imposed on the jammer, while the user is under no constraints, the random code bounds coincide, and the random code capacity is determined. Furthermore, for this scenario, a generalized non-symmetrizability condition is stated, under which the deterministic code capacity coincides with the random code capacity. For the noncausal state information setting, we determine the random code capacity of the AVC under input and state constraints. In addition, a condition on the channel is stated, under which the deterministic code capacity coincides with the random code capacity.

The Arbitrarily Varying Channel Under Constraints With Side Information at the Encoder

Communication over a random-parameter quantum channel when the decoder reconstructs the parameter sequence is considered in different scenarios. Regularized formulas are derived for the capacity-distortion regions with strictly-causal, causal, or non-causal channel side information (CSI) available at the encoder, and also without CSI. Single-letter characterizations are established in special cases. In particular, a single-letter formula is given for entanglement-breaking channels when CSI is not available. As a consequence, we obtain regularized formulas for the capacity of random-parameter quantum channels with CSI, generalizing previous results on classical-quantum channels.

Communication over Quantum Channels with Parameter Estimation

Deterministic identification (DI) is addressed for Gaussian channels with fast and slow fading, where channel side information is available at the decoder. In particular, it is established that the number of messages scales as $2^{n\log(n)R}$, where $n$ is the block length and $R$ is the coding rate. Lower and upper bounds on the DI capacity are developed in this scale for fast and slow fading. Consequently, the DI capacity is infinite in the exponential scale and zero in the double-exponential scale, regardless of the channel noise.

Deterministic Identification Over Fading Channels

In this paper, we study the arbitrarily varying broadcast channel (AVBC) when the state information is available at the transmitter in a causal manner. We establish the inner and outer bounds on both the random code capacity region and the deterministic code capacity region with degraded message sets. The capacity region is then determined for a class of channels satisfying a condition on the mutual information between the strategy variables and the channel outputs. As an example, we consider the arbitrarily varying binary symmetric broadcast channel. We show the cases where the condition holds and, hence, the capacity region is determined and other cases where there is a gap between the bounds. This gap shows that the minimax theorem does not hold for rate regions.

Uzi Pereg

Papers

Channel Upgradation for Non-Binary Input Alphabets and MACs

The Arbitrarily Varying Channel Under Constraints With Side Information at the Encoder

Communication over Quantum Channels with Parameter Estimation

Deterministic Identification Over Fading Channels

The Arbitrarily Varying Broadcast Channel With Causal Side Information at the Encoder