The Capacity of Online (Causal) qq-ary Error-Erasure Channels

In the $q$-ary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1,\ldots,q-1\}^n$ symbol by symbol via a channel limited to at most $pn$ errors and/or $p^{*} n$ erasures. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has a full knowledge on the sent codeword $\mathbf{x}$. In this work we study the capacity of $q$-ary online channels for a combined corruption model, in which the channel may impose at most $pn$ {\em errors} and at most $p^{*} n$ {\em erasures} on the transmitted codeword. The online channel (in both the error and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we give a full characterization of the capacity as a function of $q,p$, and $p^{*}$.Comment: This is a new version of the binary case, which can be found at arXiv:1412.637

Causal order as a resource for quantum communication

In theories of communication, it is usually presumed that the involved parties perform actions in a fixed causal order. However, practical and fundamental reasons can induce uncertainties in the causal order. Here we show that a maximal uncertainty in the causal order forbids asymptotic quantum communication, while still enabling the noisy transfer of classical information. Therefore causal order, like shared entanglement, is an additional resource for communication. The result is formulated within an asymptotic setting for processes with no fixed causal order, which sets a basis for a quantum information theory in general quantum causal structures.Comment: 5 pages, 1 figur

The benefit of a 1-bit jump-start, and the necessity of stochastic encoding, in jamming channels

We consider the problem of communicating a message $m$ in the presence of a malicious jamming adversary (Calvin), who can erase an arbitrary set of up to $pn$ bits, out of $n$ transmitted bits $(x_1,\ldots,x_n)$. The capacity of such a channel when Calvin is exactly causal, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends on his observations $(x_1,\ldots,x_i)$ was recently characterized to be $1-2p$. In this work we show two (perhaps) surprising phenomena. Firstly, we demonstrate via a novel code construction that if Calvin is delayed by even a single bit, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends only on $(x_1,\ldots,x_{i-1})$ (and is independent of the "current bit" $x_i$) then the capacity increases to $1-p$ when the encoder is allowed to be stochastic. Secondly, we show via a novel jamming strategy for Calvin that, in the single-bit-delay setting, if the encoding is deterministic (i.e. the transmitted codeword is a deterministic function of the message $m$) then no rate asymptotically larger than $1-2p$ is possible with vanishing probability of error, hence stochastic encoding (using private randomness at the encoder) is essential to achieve the capacity of $1-p$ against a one-bit-delayed Calvin.Comment: 21 pages, 4 figures, extended draft of submission to ISIT 201

Error Correcting Codes for Distributed Control

The problem of stabilizing an unstable plant over a noisy communication link is an increasingly important one that arises in applications of networked control systems. Although the work of Schulman and Sahai over the past two decades, and their development of the notions of "tree codes"\phantom{} and "anytime capacity", provides the theoretical framework for studying such problems, there has been scant practical progress in this area because explicit constructions of tree codes with efficient encoding and decoding did not exist. To stabilize an unstable plant driven by bounded noise over a noisy channel one needs real-time encoding and real-time decoding and a reliability which increases exponentially with decoding delay, which is what tree codes guarantee. We prove that linear tree codes occur with high probability and, for erasure channels, give an explicit construction with an expected decoding complexity that is constant per time instant. We give novel sufficient conditions on the rate and reliability required of the tree codes to stabilize vector plants and argue that they are asymptotically tight. This work takes an important step towards controlling plants over noisy channels, and we demonstrate the efficacy of the method through several examples.Comment: 39 page

Energy Optimal Transmission Scheduling in Wireless Sensor Networks

One of the main issues in the design of sensor networks is energy efficient communication of time-critical data. Energy wastage can be caused by failed packet transmission attempts at each node due to channel dynamics and interference. Therefore transmission control techniques that are unaware of the channel dynamics can lead to suboptimal channel use patterns. In this paper we propose a transmission controller that utilizes different "grades" of channel side information to schedule packet transmissions in an optimal way, while meeting a deadline constraint for all packets waiting in the transmission queue. The wireless channel is modeled as a finite-state Markov channel. We are specifically interested in the case where the transmitter has low-grade channel side information that can be obtained based solely on the ACK/NAK sequence for the previous transmissions. Our scheduler is readily implementable and it is based on the dynamic programming solution to the finite-horizon transmission control problem. We also calculate the information theoretic capacity of the finite state Markov channel with feedback containing different grades of channel side information including that, obtained through the ACK/NAK sequence. We illustrate that our scheduler achieves a given throughput at a power level that is fairly close to the fundamental limit achievable over the channel.Comment: Accepted for publication in the IEEE Transactions on Wireless Communication