This paper derives upper and lower bounds on the capacity of the multiple-input single-output free-space optical intensity channel with signal-independent additive Gaussian noise subject to both an average-intensity and a peak-intensity constraint. In the limit where the signal-to-noise ratio (SNR) tends to infinity, the asymptotic capacity is specified, while in the limit where the SNR tends to zero, the exact slope of the capacity is given.

This paper studies the capacity of a general multiple-input multiple-output (MIMO) free-space optical intensity channel under a per-input-antenna peak-power constraint and a total average-power constraint over all input antennas. The focus is on the scenario with more transmit than receive antennas. In this scenario, different input vectors can yield identical distributions at the output, when they result in the same image vector under multiplication by the channel matrix. We first determine the most energy-efficient input vectors that attain each of these image vectors. Based on this, we derive an equivalent capacity expression in terms of the image vector, and establish new lower and upper bounds on the capacity of this channel. The bounds match when the signal-to-noise ratio (SNR) tends to infinity, establishing the high-SNR asymptotic capacity. We also characterize the low-SNR slope of the capacity of this channel.

This paper investigates fundamental properties of nonlinear binary codes by looking at the codebook matrix not row-wise (codewords), but column-wise. The family of weak flip codes is presented and shown to contain many beautiful properties. In particular the subfamily fair weak flip codes, which goes back to Berlekamp, Gallager, and Shannon and which was shown to achieve the error exponent with a fixed number of codewords M, can be seen as a generalization of linear codes to an arbitrary number of codewords. The fair weak flip codes are related to binary nonlinear Hadamard codes. Based on the column-wise approach to the codebook matrix, the r-wise Hamming distance is introduced as a generalization to the well-known and widely used (pairwise) Hamming distance. It is shown that the minimum r-wise Hamming distance satisfies a generalized r-wise Plotkin bound. The r-wise Hamming distance structure of the nonlinear fair weak flip codes is analyzed and shown to be superior to many codes. In particular, it is proven that the fair weak flip codes achieve the r-wise Plotkin bound with equality for all r. In the second part of the paper, these insights are applied to a binary erasure channel (BEC) with an arbitrary erasure probability 0 < δ < 1. An exact formula for the average error probability of an arbitrary (linear or nonlinear) code using maximum likelihood decoding is derived and shown to be expressible using only the r-wise Hamming distance structure of the code. For a number of codewords M satisfying M ≤ 4 and an arbitrary finite blocklength n, the globally optimal codes (in the sense of minimizing the average error probability) are found. For M = 5 or M = 6 and an arbitrary finite blocklength n, the optimal codes are conjectured. For larger M, observations regarding the optimal design are presented, e.g., that good codes have a large r-wise Hamming distance structure for all r. Numerical results validate our code design criteria and show the superiority of our best found nonlinear weak flip codes compared to the best linear codes.

This paper derives the asymptotic capacity for the multiple-input single-output free-space optical intensity channel in the regime of high signal-to-noise ratio (SNR). The asymptotic result is proven via upper and lower bounds on capacity at finite SNR.

Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral ${\chi }^2$-distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed.

The fading number of a general (not necessarily Gaussian) regular multiple-input multiple-output (MIMO) fading channel with arbitrary temporal and spatial memory is derived. The channel is assumed to be non-coherent, i.e., neither receiver nor transmitter have knowledge about the channel state, but they only know the probability law of the fading process. The fading number is the second term in the asymptotic expansion of channel capacity when the signal-to-noise ratio (SNR) tends to infinity. It is shown that the fading number can be achieved by an input that is the product of two independent processes: a stationary and circularly symmetric direction- (or unit-) vector process whose distribution needs to be chosen such that it maximizes the fading number, and a non-negative magnitude process that is independent and identically distributed (IID) and that escapes to infinity. Additionally, in the more general context of an arbitrary stationary channel model satisfying some weak conditions on the channel law, it is shown that the optimal input distribution is stationary apart from some edge effects.