Consider the following unequal error protection scenario. One special message, dubbed the “red alert” message, is required to have an extremely small probability of missed detection. The remainder of the messages must keep their average probability of error and probability of false alarm below a certain threshold. The goal then is to design a codebook that maximizes the error exponent of the red alert message while ensuring that the average probability of error and probability of false alarm go to zero as the blocklength goes to infinity. This red alert exponent has previously been characterized for discrete memoryless channels. This paper completely characterizes the optimal red alert exponent for additive white Gaussian noise channels with block power constraints.
In @cite_14 , Borade, Nakibo g lu, and Zheng study bit''-wise and message''-wise unequal error protection (UEP) problems and error exponents. The red alert problem is a message-wise UEP problem in which one message is special and the remaining messages are standard. While @cite_14 focuses on general DMCs near capacity, Lemma 1 of that paper develops a general sharp bound on the red alert exponent for DMCs at any rate below capacity (both with and without feedback). Specializing to the exponent achieved at capacity, let @math denote the input alphabet, @math the channel transition matrix, and @math the capacity-achieving output distribution of the DMC. Then, the optimal red alert exponent at capacity is where @math is the KL divergence. We also mention recent work by Nakibo g lu @cite_1 @cite_5 that considers the generalization where a strictly positive error exponent is required of the standard messages.
@cite_5 @cite_14 @cite_1
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An information-theoretic framework for unequal error protection is developed in terms of the exponential error bounds. The fundamental difference between the bit-wise and message-wise unequal error protection ( UEP) is demonstrated, for fixed-length block codes on discrete memoryless channels (DMCs) without feedback. Effect of feedback is investigated via variable-length block codes. It is shown that, feedback results in a significant improvement in both bit-wise and message-wise UEPs (except the single message case for missed detection). The distinction between false-alarm and missed-detection formalizations for message-wise UEP is also considered. All results presented are at rates close to capacity. Suppose that Alice wishes to send messages to Bob through a communication channel C1, but her transmissions also reach an eavesdropper Eve through another channel C2. This is the wiretap channel model introduced by Wyner in 1975. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of the mutual information between the message and Eve's observations. Wyner showed that the situation is characterized by a single constant Cs, called the secrecy capacity, which has the following meaning: for all e >; 0, there exist coding schemes of rate R ≥ Cs-e that asymptotically achieve the reliability and security objectives. However, his proof of this result is based upon a random-coding argument. To date, despite consider able research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. In this paper, we use polar codes to construct a coding scheme that achieves the secrecy capacity for a wide range of wiretap channels. Our construction works for any instantiation of the wiretap channel model, as long as both C1 and C2 are symmetric and binary-input, and C2 is degraded with respect to C1. Moreover, we show how to modify our construction in order to provide strong security, in the sense defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will also be satisfied unless the main channel C1 is noiseless, although we believe it can be always satisfied in practice. We compute the channel capacity of non-binary fingerprinting under the Marking Assumption, in the limit of large coalition size c. The solution for the binary case was found by Huang and Moulin. They showed that asymptotically, the capacity is 1 (c22 ln 2), the interleaving attack is optimal and the arcsine distribution is the optimal bias distribution. In this paper we prove that the asymptotic capacity for general alphabet size q is (q - 1) (c22 lnq). Our proof technique does not reveal the optimal attack or bias distribution. The fact that the capacity is an increasing function of q shows that there is a real gain in going to non-binary alphabets. We study a fingerprinting game in which the number of colluders and the collusion channel are unknown. The encoder embeds fingerprints into a host sequence and provides the decoder with the capability to trace back pirated copies to the colluders. Fingerprinting capacity has recently been derived as the limit value of a sequence of maximin games with mutual information as their payoff functions. However, these games generally do not admit saddle-point solutions and are very hard to solve numerically. Here under the so-called Boneh-Shaw marking assumption, we reformulate the capacity as the value of a single two-person zero-sum game, and show that it is achieved by a saddle-point solution. If the maximal coalition size is k and the fingerprinting alphabet is binary, we show that capacity decays quadratically with k. Furthermore, we prove rigorously that the asymptotic capacity is 1 (k221n2) and we confirm our earlier conjecture that Tardos' choice of the arcsine distribution asymptotically maximizes the mutual information payoff function while the interleaving attack minimizes it. Along with the asymptotics, numerical solutions to the game for small k are also presented.