A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two function spaces, and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of ideal systems which transmit at this maximum rate are discussed The equivalent number of binary digits per second for certain information sources is calculated.

This note discusses the problem of maximizing the rate of flow from one terminal to another, through a network which consists of a number of branches, each of which has a limited capacity. The main result is a theorem: The maximum possible flow from left to right through a network is equal to the minimum value among all simple cut-sets. This theorem is applied to solve a more general problem, in which a number of input nodes and a number of output nodes are used.

The zero error capacityC_oof a noisy channel is defined as the least upper bound of rates at which it is possible to transmit information with zero probability of error. Various properties ofC_oare studied; upper and lower bounds and methods of evaluation ofC_oare given. Inequalities are obtained for theC_orelating to the "sum" and "product" of two given channels. The analogous problem of zero error capacityC_oFfor a channel with a feedback link is considered. It is shown that while the ordinary capacity of a memoryless channel with feedback is equal to that of the same channel without feedback, the zero error capacity may be greater. A solution is given to the problem of evaluatingC_oF.

In any branch of applied mathematics, the vague and ambiguous concepts of a physical problem are given a more refined and idealized meaning. In information theory, one of the basic notions is that of the amount of information associated with a given situation. Information here, although related to the everyday of the word, should not be confused with it. In everyday usage, information usually implies something about the semantic content of a message. For the purposes of communication theory, the meaning of a message is generally irrelevant; what is significant is the difficulty in transmitting the message from one point to another.

A typical communication system consists of the following five elements: (1) An information source. This can be considered to be represented mathematically by a suitable stochastic process which chooses one message from a set of possible messages. The rate R of producing information is measured by the entropy per symbol of the process. (2) An encoding or transmitting element. Mathematically this amounts to a transformation applied to the message to produce the signal, i.e., the encoded message. (3) A channel on which the signal is transmitted from transmitter to receiver. During transmission the signal may be perturbed by noise. (4) A receiving and decoding (or demodulating) device which recovers the original message from the received signal. (5) The destination of the information, e.g., the human ear (for telephony) or the eye (for television). The characteristics of the destination may determine the significant elements of the information to be transmitted. For example, with sound transmission, precise recovery of the phases of components is not required because of the insensitivity of the ear to this type of distortion.

The word "information" has been given many different meanings by various writers in the general field of information theory. It is likely that at least a number of these will prove sufficiently useful in certain applications to deserve further study and permanent recognition. It is hardly to be expected that a single concept of information would satisfactorily account for the numerous possible applications of this general field. The present note outlines a new approach to information theory which is aimed specifically at the analysis of certain communication problems in which there exist a number of information sources simultaneously in operation. A typical example is that of a simple communication channel with a feedback path from the receiving point to the transmitting point. The problem is to make use of the feedback information for improving forward transmission, and to determine the forward channel capacity when the best possible use is made of this feedback information. Another more general problem is that of a communication system consisting of a large number of transmitting and receiving points with some type of interconnection network between the various points. The problem here is to formulate the best systems design whereby, in some sense, the best overall use of the available facilities is made. While the analysis sketched here has not yet proceeded to the point or a complete solution of these problems, partial answers have been found and it is believed that a complete solution may be possible.