The contributions of S.O. Rice to Shannon's formulations in communication theory are discussed. In 'Communication in the presence of noise-Probability of error for two encoding schemes' (see Bell Syst. Tech. J., vol.29, p.60-93, Jan. 1950) Rice considered two explicit construction schemes for choosing the transmitted signals of duration T and exhibited an exact formula for the probability of error in the decoded messages. Rice evaluated these unwieldy expressions and presented tables and curves showing the trade-off between rate, delay, and error probability. He obtained asymptotic expressions for the error probability for large delay, and showed that with his scheme for choosing the signals, Shannon's results could be obtained. He recognized from his numerical calculations that the error probability seemed to approach zero much more rapidly than this and that, in his asymptotics, he had evidently made approximations that might one day be improved.>

The digital transmission of signals by transmitters and receivers that are time-invariant finite-state machines were investigated in general form. An average distortion criterion is used. Some general structure theorems are proved and some examples given that show improvement in performance as the memories of the transmitter and receiver are increased. Many conjectures and unsolved problems are mentioned, and areas for further study are indicated.

A short personal account of some of the developments in Shannon-type information theory from 1950 to 1960.

Correlated information sequences\cdots ,X_{-1},X_0,X_1, \cdotsand\cdots,Y_{-1},Y_0,Y_1, \cdotsare generated by repeated independent drawings of a pair of discrete random variablesX, Yfrom a given bivariate distributionP_{XY} (x,y). We determine the minimum number of bits per characterR_XandR_Yneeded to encode these sequences so that they can be faithfully reproduced under a variety of assumptions regarding the encoders and decoders. The results, some of which are not at all obvious, are presented as an admissible rate region\mathcal{R}in theR_X - R_Yplane. They generalize a similar and well-known result for a single information sequence, namelyR_X \geq H (X)for faithful reproduction.

This correspondence provides counterexamples to two frequently proposed conjectures concerning group codes for the Gaussian channel. It is shown 1) that a code for which the list of distances from codewordito all other words is independent ofiis not necessarily a group code, and 2) a group code ofMwords need not possess a transitive symmetry group of orderM.

A class of equal-energy codes for use on the Gaussian channel is defined and investigated. Members of the class are eared group codes because of the manner in which they can be generated from a group of orthogonal matrices. Group codes possess an important symmetry property. Roughly speaking, all words in such a code are on an equal footing: each has the same error probability (under the assumptions of the usual model) and each has the same disposition of neighbors. A number of theorems about such codes are proved. A decomposition theorem shows every group code to be equivalent to a direct sum of certain basic group codes generated by real-irreducible representations of a finite group associated with the code. Some theorems on distances between words in group codes are demonstrated. The difficult problem of finding group codes with large nearest neighbor distance is discussed in detail and formulated in several ways. It is noted that linear (or group) codes for the binary channel can be regarded as very speciM cases of the group codes discussed. A definition of a group code for the Gaussian channel follows. An equal-energy codeCwith parametersMandnfor this channel is a collection ofMdistinct unitn-vectors,X_{1}, X_{2}, \cdots , X_{M}say, that span a Euclideann-space. Ann \times northogonal matrix0is said to be a symmetry ofCif theMvectorsY_{i} = 0X_{i}, i = 1, 2, \cdots , Mare again the collectionC. The set of all symmetries ofC, say0_{1}, 0_{2}, \cdots , 0_{g}, forms a group\cal{G}(C)under matrix multiplication. If\cal{G}(C)containsMelements0_{\alpha_{1}}, 0_{\alpha_{2}}, \cdots , 0_{\alpha M}such thatX_{i} = 0_{\alpha i}X_{1}, i = 1, 2, \cdots , M, thenCis called a group code.

This paper presents curves which are believed to represent the best thresholds obtainable with practical modulators that expand bandwidth and operate instantaneously on the baseband signal. The model from which the curves are obtained is described in detail.

It is pointed out that a frequently used mathematical model of the detection problem yields detection with arbitrarily small probability of error in many cases of engineering interest. Some comments are made as to why the model is inadequate from an engineering standpoint.

A generalization of Hamming's single error correcting codes is given along with a simple maximum likelihood detection scheme. For small redundancy these alphabets are unexcelled. The Reed-Muller alphabets are described as parity check alphabets and a new detection scheme is presented for them.

This paper is concerned with certain applications of the estimation theory of Fisher and Cramer[1] to the problem of estimating signal parameters in the presence of noise. Specifically, the situation to be treated is as follows. A received signal