This paper considers a particular case of the problem of binary codes with the constraint that the codes restricted cerl3in subsets of columns must be contained in particular codes of the shorter lengths. Here we consider codes of even length 2 k, and of minimum distance ~ d. However, the code obtained by restricting the first k positions must have even weight and at the same time the code obtained by restricting the last k positions also must have even weight. If k = 2 n, so that the length is 4 n, n odd, and d = 2 n, we prove that the code has at most 8 n - 4 codewords. Further, 8n - 4 is attainable if and only if a Hadamard matrix of size 4 n exists. For n = 3, this yields 20 binary words of length 12 and distance ~ 6. where the number of 1 s in the first six and in the last six positions is even for every codeword in the code. This permits a file·transfer protocol control ftmction assignment for personal computers be chosen for 20 control functions using what amounts to pairs of upper-case alphabetic ASCII characters. In this case, the Hamming distance between the binary fomlS of every tWO different control fwtctions is at least six.

Techniques from coding theory are applied to study rigorously the capacity of the Hopfield associative memory. Such a memory stores n -tuple of \pm 1 's. The components change depending on a hard-limited version of linear functions of all other components. With symmetric connections between components, a stable state is ultimately reached. By building up the connection matrix as a sum-of-outer products of m fundamental memories, one hopes to be able to recover a certain one of the m memories by using an initial n -tuple probe vector less than a Hamming distance n/2 away from the fundamental memory. If m fundamental memories are chosen at random, the maximum asympotic value of m in order that most of the m original memories are exactly recoverable is n/(2 \log n) . With the added restriction that every one of the m fundamental memories be recoverable exactly, m can be no more than n/(4 \log n) asymptotically as n approaches infinity. Extensions are also considered, in particular to capacity under quantization of the outer-product connection matrix. This quantized memory capacity problem is closely related to the capacity of the quantized Gaussian channel.

Centimeter-band transmissions, three-dimensional data compression, and laser-based systems are discussed for future use in unmanned missions to deep space. Project Galileo, a planned two-in-one spacecraft that is to explore Jupiter and four of its moons as an orbiting weather satellite, while a canonical probe detaches from the orbiter and plunges into the Jovian atmosphere, is highlighted. Details of the Galileo communications link and interplanetary environment are described. The upgrading of NASA's Deep Space Network (DSN) is also reported. This network is being upgraded to nine antennas: six 34-meter antennas and three 64-meter ones; and by the time the Galileo craft reaches Jupiter, the DSN 64-meter antennas will have been enlarged to 70 meters. Proposals for more advanced communications links to be used for the late 20th and early 21st century space missions are outlined.

This paper reviews the progress made in deep space communication from its beginnings until now, describes the development and applications of NASA's Deep Space Network, and indicates directions for the future. Limiting factors in deep space communication are examined using the upcoming Voyager encounter with Uranus, centered on the downlink telemetry from spacecraft to earth, as an example. A link calculation for Voyager at Uranus over Australia is exhibited. Seven basic deep space communication functions are discussed, and technical aspects of spacecraft communication equipment, ground antennas, and ground electronics and processing are considered.

Binary cyclic redundancy codes for feedback communication over noisy digital links are considered. The standard 16-bit ADCCPt polynomial is designed for digital links that already have a low input bit error probability. For file transfer between personal computers over telephone circuits, the quality of the resulting digital circuit may be much lower. This leads to the consideration of 3-byte (24-bit) and 4-byte (32-bit) polynomials. Generator polynomials of a certain class are found that have minimum weight and yet achieve the bound on minimum distance for arbitrary codes. Particular polynomials for 24-bit and 32-bit redundancies are exhibited, of weight and distance 6 in the 24-bit case and weight 10 and distance 8 in the 32-bit case.

Suppose we have L digital links in tandem. Between successive links we have a repeater. The links are each white Gaussian channels with no bandwidth constraint. The repeater may either be a {\em binary repeater}, which sends on each bit separately having made a binary decision on each, or a {\em Shannon repeater}, which perfectly reproduces at one end the bits that were transmitted at the other end of the link. What is the loss in capacity if we use only binary repeaters and code for the entire L links at the transmit end? For large L , the capacity drops by a factor asymptotic to In L , and the normalized optimum time T per symbol T_{\opt}^{L} is also asymptotic to \ln L . Arbitrarily short L gives 0 capacity for L > 1 . More precise asymptotic results are obtained and compared with computed numerical values. These results show when it pays to code each link separately in digital data transmission.

A strategy for data acquisition from a very distant spacecraft is presented, when the system performance can be severely degraded by the earth's weather due to the high microwave frequency being used. Two cases are considered, one in which there is a certain minimum data rate to be maintained and one in which there is not. The goal is to maximize expected data return, where we assume that there ar...

Previous work has shown that the capacity of the single-mode photon-counting channel in the region in which quantum effects are most pronounced is 1/(kT \ln 2) bit/J. Here T is the quantum noise temperature, and k is Boltzmann's constant. However, to achieve this, the number of photons per second must be very low, so this does not determine the capacity of the average power-limited photon channel in bits per second. Here the previous work is extended and the capacity is determined in bits per second. This includes a determination of the optimum counting time. The capacity in bits per second is for low T asymptotic to P_{s}/(kT \ln 2) , where P_{s} is the average signal power. This is high enough to suggest that photon counting be seriously considered for applications such as a deep space to near Earth link. Some related problems are also considered including a peak power limitation.

The capacity region for the discrete memoryless multiple-access channel without time synchronization at the transmitters and receivers is shown to be the same as the known capacity region for the ordinary multiple-access channel. The proof utilizes time sharing of two optimal codes for the ordinary multiple-access channel and uses maximum likelihood decoding over shifts of the hypothesized transmitter words.

This paper proves that there exists a fixed random coding Strategy for block coding a memoryless information source to achieve the absolute epsilon entropy of the source. That is, the strategy can be chosen independent of the block length. The principal new tool is an easy result on the semicontinuity of the relative entropy functional of one probability distribution with respect to another. The theorem generalizes a result from rate-distortion theory to the "zero-infinity" case.

This paper shows that the epsilon entropy in the sup norm of a wide variety of processes with continuous paths on the unit interval is finite. In fact, the class coincides with the class of processes for which proofs of continuity have been given from a covariance condition. This suggests the conjecture that the epsilon entropy of any process continuous on the unit interval is finite in the sup norm of continuous functions. The epsilon entropy considered in this paper is defined as the minimum Shannon entropy of any partition by sets of diameter at most epsilon of the space of continuous functions on the unit interval, where the probability is the one inherited from the given process. The proof proceeds by constructing partitions and estimating their entropy using probability bounds.

This article studies efficient data transmission, or "data compression", from the standpoint of the theory of epsilon entropy. The notion of the entropy of a "data source" is defined. This quantity gives a precise measure of the amount of channel capacity necessary to describe a data source to within a given fidelity, epsilon, with probability one, when each separate "experiment" must be transmitted without storage from experiment to experiment. We also define the absolute epsilon entropy of a source, which is the amount of capacity needed when storage of experiments is allowed before transmission. The absolute epsilon entropy is shown to be equal to Shannon's rate distortion function evaluated for zero distortion, when suitable identifications are made. The main result is that the absolute epsilon entropy and the epsilon entropy have ratio close to one if either is large. Thus, very little can be saved by storing the results of independent experiments before transmission.

This paper uses a family of bivariate extreme-value distributions to estimate the probability of a large exceedance of a random variable given that a certain other random variable not independent of the first has exceeded a certain value. A simple method of reasonably good efficiency is given for estimating a bivariate extreme-value distribution from independent bivariate samples. The method is used to analyze the performance of a spacecraft command receiver which has an indication of data quality so that commands likely to be in error can be rejected.

The signal-to-noise ratio in a given equivalent bandwidth must be maintained above a minimum value during a deep-space mission in order to provide predictable performance, and in order to maintain effective telemetry communications with spacecraft at interplanetary distances it is necessary for earth stations to use maximum antenna gain. One solution is to extend telecommand range by reducing system noise temperature in the spacecraft receiver and reducing the bandwidth of the receiver's phase-lock loop. Low-noise preamplifiers have not yet been applied in planetary spacecraft in order to maintain simplicity and reliability. An answer to the problem seems to be high-power transmitters (earth-based) with outputs in the 1000-GW range.

This paper shows that the epsilon entropy of any mean-continuous Gaussian process on $L_2\lbrack 0, 1 \rbrack$ is finite for all positive $\epsilon$. The epsilon entropy of such a process is defined as the infimum of the entropies of all partitions of $L_2\lbrack 0, 1 \rbrack$ by measurable sets of diameter at most $\epsilon$, where the probability measure on $L_2$ is the one induced by the process. Fairly tight upper and lower bounds are found as $\epsilon \rightarrow 0$ for the epsilon entropy in terms of the eigenvalues of the process.

This paper studies the product epsilon entropy of mean-continuous Gaussian processes. That is, a given mean-continuous Gaussian process on the unit interval is expanded into its Karhunen expansion. Along the $k$th eigenfunction axis, a partition by intervals of length $\epsilon_k$ is made, and the entropy of the resulting discrete distribution is noted. The infimum of the sum over $k$ of these entropies subject to the constraint that $\mathbf{\sum} \mathbf{\epsilon}_k^2 \leqq \mathbf{\epsilon}^2$ is the product epsilon entropy of the process. It is shown that the best partition to take along each eigenfunction axis is the one in which 0 is the midpoint of an interval in the partition. Furthermore, the product epsilon entropy is finite if and only if $\mathbf{\sum} \lambda_k \log \lambda_k^{-1}$ is finite, where $\lambda_k$ is the $k$th eigenvalue of the process. When the above series is finite, the values of $\mathbf{\epsilon}_k$ which achieve the product entropy are found. Asymptotic expressions for the product epsilon entropy are derived in some special cases. The problem arises in the theory of data compression, which studies the efficient representation of random data with prescribed accuracy