907 research outputs found

    Upper Bounds on the Capacity of Binary Channels with Causal Adversaries

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    In this work we consider the communication of information in the presence of a causal adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codeword (x1,...,xn)(x_1,...,x_n) bit-by-bit over a communication channel. The sender and the receiver do not share common randomness. The adversarial jammer can view the transmitted bits xix_i one at a time, and can change up to a pp-fraction of them. However, the decisions of the jammer must be made in a causal manner. Namely, for each bit xix_i the jammer's decision on whether to corrupt it or not must depend only on xjx_j for jij \leq i. This is in contrast to the "classical" adversarial jamming situations in which the jammer has no knowledge of (x1,...,xn)(x_1,...,x_n), or knows (x1,...,xn)(x_1,...,x_n) completely. In this work, we present upper bounds (that hold under both the average and maximal probability of error criteria) on the capacity which hold for both deterministic and stochastic encoding schemes.Comment: To appear in the IEEE Transactions on Information Theory; shortened version appeared at ISIT 201

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

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    In the qq-ary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x=(x1,,xn){0,1,,q1}n\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1,\ldots,q-1\}^n symbol by symbol via a channel limited to at most pnpn errors and/or pnp^{*} n erasures. The channel is "online" in the sense that at the iith step of communication the channel decides whether to corrupt the iith symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols (x1,,xi)(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 x\mathbf{x}. In this work we study the capacity of qq-ary online channels for a combined corruption model, in which the channel may impose at most pnpn {\em errors} and at most pnp^{*} 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,pq,p, and pp^{*}.Comment: This is a new version of the binary case, which can be found at arXiv:1412.637

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

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    We consider the problem of communicating a message mm in the presence of a malicious jamming adversary (Calvin), who can erase an arbitrary set of up to pnpn bits, out of nn transmitted bits (x1,,xn)(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 xix_i depends on his observations (x1,,xi)(x_1,\ldots,x_i) was recently characterized to be 12p1-2p. 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 xix_i depends only on (x1,,xi1)(x_1,\ldots,x_{i-1}) (and is independent of the "current bit" xix_i) then the capacity increases to 1p1-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 mm) then no rate asymptotically larger than 12p1-2p is possible with vanishing probability of error, hence stochastic encoding (using private randomness at the encoder) is essential to achieve the capacity of 1p1-p against a one-bit-delayed Calvin.Comment: 21 pages, 4 figures, extended draft of submission to ISIT 201

    A characterization of the capacity of online (causal) binary channels

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    In the binary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x=(x1,,xn){0,1}n\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1\}^n bit by bit via a channel limited to at most pnpn corruptions. The channel is "online" in the sense that at the iith step of communication the channel decides whether to corrupt the iith bit or not based on its view so far, i.e., its decision depends only on the transmitted bits (x1,,xi)(x_1,\ldots,x_i). This is in contrast to the classical adversarial channel in which the error is chosen by a channel that has a full knowledge on the sent codeword x\mathbf{x}. In this work we study the capacity of binary online channels for two corruption models: the {\em bit-flip} model in which the channel may flip at most pnpn of the bits of the transmitted codeword, and the {\em erasure} model in which the channel may erase at most pnpn bits of the transmitted codeword. Specifically, for both error models we give a full characterization of the capacity as a function of pp. The online channel (in both the bit-flip and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we present and analyze a coding scheme that improves on the previously suggested lower bounds and matches the previously suggested upper bounds thus implying a tight characterization

    Network error correction with unequal link capacities

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    This paper studies the capacity of single-source single-sink noiseless networks under adversarial or arbitrary errors on no more than z edges. Unlike prior papers, which assume equal capacities on all links, arbitrary link capacities are considered. Results include new upper bounds, network error correction coding strategies, and examples of network families where our bounds are tight. An example is provided of a network where the capacity is 50% greater than the best rate that can be achieved with linear coding. While coding at the source and sink suffices in networks with equal link capacities, in networks with unequal link capacities, it is shown that intermediate nodes may have to do coding, nonlinear error detection, or error correction in order to achieve the network error correction capacity

    Generalized List Decoding

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    This paper concerns itself with the question of list decoding for general adversarial channels, e.g., bit-flip (XOR\textsf{XOR}) channels, erasure channels, AND\textsf{AND} (ZZ-) channels, OR\textsf{OR} channels, real adder channels, noisy typewriter channels, etc. We precisely characterize when exponential-sized (or positive rate) (L1)(L-1)-list decodable codes (where the list size LL is a universal constant) exist for such channels. Our criterion asserts that: "For any given general adversarial channel, it is possible to construct positive rate (L1)(L-1)-list decodable codes if and only if the set of completely positive tensors of order-LL with admissible marginals is not entirely contained in the order-LL confusability set associated to the channel." The sufficiency is shown via random code construction (combined with expurgation or time-sharing). The necessity is shown by 1. extracting equicoupled subcodes (generalization of equidistant code) from any large code sequence using hypergraph Ramsey's theorem, and 2. significantly extending the classic Plotkin bound in coding theory to list decoding for general channels using duality between the completely positive tensor cone and the copositive tensor cone. In the proof, we also obtain a new fact regarding asymmetry of joint distributions, which be may of independent interest. Other results include 1. List decoding capacity with asymptotically large LL for general adversarial channels; 2. A tight list size bound for most constant composition codes (generalization of constant weight codes); 3. Rederivation and demystification of Blinovsky's [Bli86] characterization of the list decoding Plotkin points (threshold at which large codes are impossible); 4. Evaluation of general bounds ([WBBJ]) for unique decoding in the error correction code setting
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