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Shannon's determination of the capacity of the linear Gaussian channel has posed a magnificent challenge to succeeding generations of researchers. This paper surveys how this challenge has been met during the past half century. Orthogonal minimumbandwidth modulation techniques and channel capacity are discussed. Binary coding techniques for low-signal-to-noise ratio (SNR) channels and nonbinary coding techniques for high-SNR channels are reviewed. Recent developments, which now allow capacity to be approached on any linear Gaussian channel, are surveyed. These new capacity-approaching techniques include turbo coding and decoding, multilevel coding, and combined coding/precoding for intersymbol-interference channels.

Abstract-In this paper, we consider the problem of constructing measurements optimized to distinguish between a collection of possibly nonorthogonal quantum states. We consider a collection of pure states and seek a positive operator-valued measure (POVM) consisting of rank-one operators with measurement vectors closest in squared norm to the given states. We compare our results to previous measurements suggested by Peres and Wootters [11] and Hausladen et al.[10], where we refer to the latter as the square-root measurement (SRM). We obtain a new characterization of the SRM, and prove that it is optimal in a least-squares sense. In addition, we show that for a geometrically uniform state set the SRM minimizes the probability of a detection error. This generalizes a similar result of Ban et al. [7].Index Terms-Geometrically uniform quantum states, leastsquares measurement, quantum detection, singular value decomposition, square-root measurement (SRM).

Multidimensional constellations are desirable for representing fractional numbers of bits per two dimensions, useful for increasing SNR efficiency, and natural for use with multidimensional coded modulation. Desirable characteristics of such a constellation include good SNR efficiency, low implementation complexit), compatibility with coded modulation and with QAM modems, including small size and peak-to-average power ratio (PAR) of its constituent 2D constellation, phase symmetry, scalability, and capability of supporting an "opportunistic secondary channel." The gain in SNR efficiency of a multidimensional constellation (lattice code) consisting of the points from a lattice A within a region compared to a cubic constellation is shown to be approximately separable into the coding gain of the lattice A and the "shape gain" of the region , for large constellations. Similarly, the expansion of the associated constituent 2D constellation is shown to be approximately separable into a coding component CER, ( A ) and a shaping component CER,(l ). The N sphere is the region with the best shape gain, but also has large constellation expansion. Bounds for the best possible shape gain versus CER,( ) or PAR are given. Finally, generalized cross constellations are discussed; these constellations yield a modest shape gain with very low CER,( ) or PAR, are easily implemented, are well suited for use with coded QAM modems, and can be readily adapted to support a n opportunistic secondary channel.