In Costa's dirty-paper channel, Gaussian random binning is able to eliminate the effect of interference which is known at the transmitter, and thus achieve capacity. We examine a generalization of the dirty-paper problem to a multiple access channel setup, where structured (lattice-based) binning seems to be necessary to achieve capacity. In the dirty-MAC, two additive interference signals are present, one known to each transmitter but none to the receiver.The achievable rates using Costa's Gaussian binning vanish if both interference signals are strong. In contrast, it is shown that lattice-strategies ("lattice precoding") can achieve positive rates, independent of the interference power.Furthermore, in some cases -which depend on the noise variance and power constraints -high-dimensional lattice strategies are in fact optimal. In particular, they are optimal in the limit of high SNR -where the capacity region of the dirty MAC approaches that of a clean MAC whose power is governed by the minimum of the users' powers rather than their sum. The rate gap at high SNR between lattice-strategies and optimum (rather than Gaussian) random binning is conjectured to be 1 2 log 2 (πe/6) ≈ 0.254 bit. Thus, the doubly-dirty MAC is another instance of a network setting, like the Körner-Marton problem, where (linear) structured coding is potentially better than random binning. Finally, it is shown that lattice strategies are at most 0.167 bit from the capacity region for all SNR. The results are also compared and contrasted to the single dirt multiple access channel case (considered by other researchers), where lattice strategies and Gaussian random binning have similar performance.

In Costa's dirty-paper channel, Gaussian random binning is able to eliminate the effect of interference which is known at the transmitter, and thus achieve capacity. We examine a generalization of the dirty-paper problem to a multiple access channel setup, where structured (lattice-based) binning seems to be necessary to achieve capacity. In the dirty-MAC, two additive interference signals are present, one known to each transmitter but none to the receiver.The achievable rates using Costa's Gaussian binning vanish if both interference signals are strong. In contrast, it is shown that lattice-strategies ("lattice precoding") can achieve positive rates, independent of the interference power.Furthermore, in some cases -which depend on the noise variance and power constraints -high-dimensional lattice strategies are in fact optimal. In particular, they are optimal in the limit of high SNR -where the capacity region of the dirty MAC approaches that of a clean MAC whose power is governed by the minimum of the users' powers rather than their sum. The rate gap at high SNR between lattice-strategies and optimum (rather than Gaussian) random binning is conjectured to be 1 2 log 2 (πe/6) ≈ 0.254 bit. Thus, the doubly-dirty MAC is another instance of a network setting, like the Körner-Marton problem, where (linear) structured coding is potentially better than random binning. Finally, it is shown that lattice strategies are at most 0.167 bit from the capacity region for all SNR. The results are also compared and contrasted to the single dirt multiple access channel case (considered by other researchers), where lattice strategies and Gaussian random binning have similar performance.

For general memoryless systems, the typical information theoretic solution -when exists -has a "single-letter"form. This reflects the fact that optimum performance can be approached by a random code (or a random binning scheme), generated using independent and identically distributed copies of some single-letter distribution. Is that the form of the solution of any (information theoretic) problem? In fact, some counter examples are known. The most famous is the "two help one" problem: Korner and Marton showed that if we want to decode the modulo-two sum of two binary sources from their independent encodings, then linear coding is better than random coding. In this paper we provide another counter example, the "doubly-dirty" multiple access channel (MAC). Like the KornerMarton problem, this is a multi-terminal scenario where side information is distributed among several terminals; each transmitter knows part of the channel interference but the receiver is not aware of any part of it. We give an explicit solution for the capacity region of a binary version of the doubly-dirty MAC, demonstrate how the capacity region can be approached using a linear coding scheme, and prove that the "best known single-letter region" is strictly contained in it. We also state a conjecture regarding a similar rate loss of single letter characterization in the Gaussian case. Index TermsMulti-user information theory, random binning, linear lattice binning, dirty paper coding, lattice strategies, Korner-Marton problem.

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